Stochastic embedding of dynamical systems

Stochastic embedding of dynamical systemsJacky Cresson and Sébastien Darses Citation: J. Math. Phys. 48, 072703 (2007); doi: 10.1063/1.2736519 View online: http://dx.doi.org/10.1063/1.2736519 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v48/i7 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Stochastic embedding of dynamical systemsJacky Cressona�

Laboratoire de Mathématiques Appliqués de Pau, CNRS UMR 5142,Université de Pau et des l’Adour, Batiment IPRA,Avenue de l’Université, BP 1155, 64013 Pau Cedex, Franceand IHÉS, Le Bois Marie, 35 Route de Chartres, F-91440 Bures sur Yvette, France

Sébastien Darsesb�

Laboratoire de Probabilitiés et Modèles Aléatoires Université Pierre et Marie CurieParis VI, Boîte Courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France

�Received 7 January 2007; accepted 11 April 2007; published online 27 July 2007�

Most physical systems are modeled by an ordinary or a partial differential equation,like the n-body problem in celestial mechanics. In some cases, for example, whenstudying the long term behavior of the solar system or for complex systems, thereexist elements which can influence the dynamics of the system which are not wellmodeled or even known. One way to take these problems into account consists oflooking at the dynamics of the system on a larger class of objects that are eventu-ally stochastic. In this paper, we develop a theory for the stochastic embedding ofordinary differential equations. We apply this method to Lagrangian systems. Inthis particular case, we extend many results of classical mechanics, namely, theleast action principle, the Euler-Lagrange equations, and Noether’s theorem. Wealso obtain a Hamiltonian formulation for our stochastic Lagrangian systems. Manyapplications are discussed at the end of the paper. © 2007 American Institute ofPhysics. �DOI: 10.1063/1.2736519�

INTRODUCTION

Ordinary as well as partial differential equations play a fundamental role in most parts ofmathematical physics. The story begins with Newton’s formulation of the law of attraction and thecorresponding equations which describe the motion of mechanical systems. Regardless the beautyand usefulness of these theories in the study of many important natural phenomena, one must keepin mind that they are based on experimental facts and as a consequence are only an approximationof the real world. The basic example we have in mind is the motion of the planets in the solarsystem which is usually modeled by the famous n-body problem, i.e., n points of mass mi whichare only submitted to their mutual gravitational attraction. If one looks at the behavior of the solarsystem for finite time, then this model is a very good one. However, this is not true when onelooks at the long term behavior, which is, for instance, relevant when dealing with the so-calledchaotic behavior of the solar system over billions years, or when trying to predict ice ages over avery large range of time. Indeed, the n-body problem is a conservative system �in fact, a Lagrang-ian system� and many nonconservative effects, such as tidal forces between planets, will be ofincreasing importance along the computation. These nonconservative effects push the model out-side the category of Lagrangian systems. You can go further by considering effects due to thechanging in the oblateness of the sun. In this case, we do not even know how to model suchperturbations, and one is not sure of staying in the category of differential equations.1

a�Electronic mail: [email protected]�Electronic mail: [email protected] that in the context of the solar system we have two different problems: first, if one uses only Newton’s gravitationallaw, one must take into account the entire universe to model the behavior of the planets. This by itself is a problem which

JOURNAL OF MATHEMATICAL PHYSICS 48, 072703 �2007�

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As a first step, this paper proposes tackling this problem by introducing a natural stochasticembedding procedure for ordinary differential equations. This consists of looking for the behaviorof stochastic processes submitted to constraints induced by the underlying differential equation.2

We point out that this strategy is different from the standard approach based on stochastic differ-ential equations or stochastic dynamical systems, where one gives a meaning to ordinary differ-ential equations perturbed by a small random term. In our work, no perturbations of the underlyingequation are carried out.

A point of view that bears some resemblance to ours is contained in Arnold’s materializationof resonances �see Ref. 4, pp. 303–304�, whose main underlying idea can be briefly explained asfollows: the divergence of the Taylor expansion of the arctan function at 0 for �x��1 can beproved by computing the coefficients of this series. However, this does not explain the reason forthis divergence behavior. One can obtain a better understanding by extending the function to thecomplex plane and by looking at its singularities at ±i. The same idea can be applied in the contextof dynamical systems. In this case, we look for the obstruction to linearization of a real systems inthe complex plane. Arnold has conjectured that this is due to the accumulation of periodic orbitsin the complex plane along the real axis. In our case, one can try to understand some properties ofthe trajectories of dynamical systems by using a suitable extension of its domain of definition. Inour work, we give a precise sense to the concept of differential and partial differential equations inthe class of stochastic processes. This procedure can be viewed as a first step toward the general“stochastic programme” as described by Mumford in Ref. 45.

Our embedding procedure is based on a simple idea: in order to write down differential orpartial differential equations, one uses derivatives. An ordinary differential equation is nothing elsebut a differential operator of order one.3 In order to embed ordinary differential equations, onemust first extend the notion of derivative so that it makes sense in the context of stochasticprocesses. By extension, we mean that our stochastic derivative reduces to the classical derivativefor deterministic differentiable processes. Having this extension, one easily defines in a uniqueway the stochastic analog of a differential operator and, as a consequence, a natural embedding ofan ordinary differential equation on stochastic processes.

Of course, one can think that such a simple procedure will not produce anything new for thestudy of classical differential equations. This is not the case. The main problem that we study inthis paper is the embedding of natural Lagrangian systems which are of particular interest forclassical mechanics. In this context, we obtain some numerous surprising results, from the exis-tence of a coherent least action principle with respect to the stochastic embedding procedure to aderivation of a stochastic Noether theorem and passing by a new derivation of the Schrödingerequation. All these points will be described with details in the following.

Two companion papers16,7 give an application of this method to derive new results on theformation of planets in a protoplanetary nebulae, in particular, a proof of the existence of aso-called Titus-Bode law for the spacing of planets around a given star.

The paper is organized as follows. In the first part, we develop our notion of a stochasticderivative and study in detail all its properties. Section I A gives a review of the stochasticcalculus developed by Nelson.47 In particular, we discuss the classical definition of the backwardand forward Nelson derivatives, denoted by D and D*, with respect to dynamical problems. Wealso define a class of stochastic process called good diffusion processes for which one can com-pute explicitly the Nelson derivatives. In Sec. I B we define what we call an abstract extension ofthe classical derivative. Using the Nelson derivatives, we define an extension of the ordinary

can be studied by using the classical perturbation theory of ordinary differential equations. This is different if we want tospeak of the “real” solar system for which we must consider effects that we ignore. In that case, even the validation of thelaw of gravitation as a real law of nature is not clear. I refer to Ref. 14 for more details on this point.2This strategy is part of a general programme called the embedding procedure in Ref. 13 and which can be used to embedordinary differential equations not only on stochastic processes but on general functional spaces. The strategy of embed-ding theories is in germ in Refs. 11 and 12 in the context of the nondifferentiable embedding of ordinary differentialequations.3In this case, we can also speak of vector fields.

072703-2 J. Cresson and S. Darses J. Math. Phys. 48, 072703 �2007�


derivative on stochastic processes, which we call the stochastic derivative. As pointed out previ-ously, one imposes that the stochastic derivative reduces to the classical derivative on differen-tiable deterministic processes. This constraint ensures that the stochastic analog of a partial dif-ferential equation �PDE� contains the classical PDE. Of course such a gluing constraint is notsufficient to define a rigid notion of stochastic derivative. We study several natural constraintswhich allow us to obtain a unique extension of the classical derivative on stochastic processes as

D� =D + D*

2+ i�

D − D*

2, � = ± 1. �1�

By extending this operator to complex valued stochastic processes, we are able to define the iterateof D, i.e., D2=D �D and so on. The main surprise is that the real part of D2 corresponds to thechoice of Nelson for acceleration in his dynamical theory of Brownian motion. However, thisresult depends on the way we extend the stochastic derivative to complex valued stochasticprocesses. We discuss several alternatives which cover well known variations on the Nelsonacceleration.

In Sec. I C we study the product rule satisfied by the stochastic derivative which is a funda-mental ingredient of our stochastic calculus of variation. We also introduce an important class ofstochastic processes, called Nelson differentiable, which have the property to have a real valuedstochastic derivative. These processes play a fundamental role in the stochastic calculus of varia-tion as they define the natural space of variations for stochastic processes.

The second part of this article deals specifically with the definition of a stochastic embeddingprocedure for ordinary differential equations.

Section II A is associated with a differential operator of a given form acting on sufficientlyregular functions a unique operator acting on stochastic processes and defined simply by replacingthe classical derivative by the stochastic derivative. This is this procedure that we call the sto-chastic embedding procedure. Note that the form of this procedure acts on differential operators ofa given form. Although the procedure is canonical for a given form of operator, it is not canonicalfor a given operator.

The previous embedding is formal and does not take constraints which are of dynamicalnature, like the reversibility of the underlying differential equation. As reversibility plays a centralrole in physics, especially in celestial mechanics which is one domain of application of our theory,we discuss this point in detail. We introduce an embedding which respects the reversibility of theunderlying equation. Doing this, we see that we must restrict attention to the real part of ouroperator, which is the unique one to possess this property in our setting. We then recover underdynamical and algebraic arguments studies dealing with particular choice of stochastic derivativesin order to derive quantum mechanics from classical mechanics under the Nelson approach.

The third part is mainly concerned with the application of the stochastic embedding to La-grangian systems.

We consider autonomous4 Lagrangian systems L�x ,v�, �x ,v��U�Rd�Rd, where U is anopen set, which satisfy a number of conditions, one of it being that it must be holomorphic withrespect to the second variable which represent the derivative of a given function. Such kind ofLagrangian functions are called admissible. Using the stochastic embedding procedure, we canassociate with the classical Euler-Lagrange equation a stochastic one which has the form

�L

�x�X�t�,DX�t�� = D� �L

�v�X�t�,DX�t�� SEL�

where X is a real valued stochastic process.Lagrangian systems possess very special features, the main one is that the Euler-Langrange

equation comes from a variational principle. We then are lead to look for a stochastic variationalprinciple giving the stochastic Euler-Lagrange equation. We remark that the Lagrangian function

4This restriction is due to technical difficulties.

072703-3 Stochastic embedding of dynamical systems J. Math. Phys. 48, 072703 �2007�


L keep sense on stochastic processes and can be considered as a functional. As a consequence, wecan search for the existence of a least action principle which gives the stochastic Euler-Lagrange�SEL� equation. The existence of such a stochastic least action principle is far from being trivialwith respect to the embedding procedure. Indeed, it must follows from a stochastic calculus ofvariations which is not developed apart from this procedure. Our problem can then be formalize asthe following diagram:

L�x,dx/dt� →LAP

EL

↓S ↓S

L�X,DX� →SLAP ?

�SEL�

, �2�

where LAP is the least action principle, S is the stochastic embedding procedure, �EL� is theclassical Euler-Lagrange equation associated with L, and SLAP the, at this moment, unknownstochastic least action principle. The existence of such a principle is called the coherence problem.

Section III B develops a stochastic calculus of variations for functionals of the form

E��a

b

L�X�t�,DX�t��dt� , �3�

where E denotes the classical expectation. Introducing the correct notion of extremals and varia-tions, we obtain two different stochastic analog of the LAP depending on the regularity class wechoose for the admissible variations. The main point is that for variations in the class of Nelson-differentiable process, the extremals of our functional coincide with the SEL equation obtained viathe stochastic embedding procedure. This result is called the coherence lemma. In the reversiblecase, i.e., taking as a stochastic derivative only the real part of our operator, we obtain the sameresult but in this case one can consider general variations.

In Sec. III C, we provide a first study of what dynamical data remain from the classicaldynamical system under the stochastic embedding procedure. We have focused on symmetries ofthe underlying equation and as a consequence on first integrals. We prove a stochastic analog ofthe Noether theorem. This allows us to define a natural notion of first integral for stochasticdifferential equations. This part also puts in evidence the need for a geometrical setting governingLagrangian systems which is the analog of symplectic manifolds.

Section III D deals with the SEL equation for natural Lagrangian systems, i.e., associated withLagrangian functions of the form

L�x,v� = T�v� − U�x� , �4�

where U is a smooth function and T is a quadratic form. In classical mechanics U is the potentialenergy and T the kinetic energy. The main result of this chapter is that by restricting our attentionto good diffusion processes and up to a well chosen function �, called the wave function, the SELequation is equivalent to a nonlinear Schrödinger equation. Moreover, by specializing the class ofstochastic processes, we obtain the classical Schrödinger equation. In that case, we can give a veryinteresting characterization of stochastic processes which are solution of the SEL equation. Indeed,the square of the modulus of � is equal to the density of the associated stochastic process solution.

In Sec. III E, we define a natural notion of stochastic Hamiltonian system. This result can beseen as a first attempt to put in evidence the stochastic analog of a symplectic structure. We definea stochastic momentum process and prove that up to a suitable modification of the stochasticembedding procedure called the Hamiltonian stochastic embedding and reflecting the fact that the“speed” of a given stochastic process is complex, we obtain a coherent picture with the classicalformalism of Hamiltonian systems. This first result is called the Legendre coherence lemma as itdeals with the coherence between the Hamiltonian stochastic embedding procedure and the Leg-endre transform. Secondly, we develop a Hamilton LAP and we prove again a coherence lemma,i.e., that the following diagram commutes:



H�x�t�,p�t�� ——→SH

H�X�t�,P�t��Hamilton LAP↓ ↓stochastic Hamilton LAP,

�HE� ——→SH

�SHE�

where SH denotes the Hamiltonian stochastic embedding procedure.The last chapter discusses many possible developments of our theory from the point of view

of mathematics and applications.

I. THE STOCHASTIC DERIVATIVE

A. About Nelson stochastic calculus

1. About measurement and experiments

In this section, we explain what we think are the basis of all possible extensions of theclassical derivative. The setting of our discussion is the following.

We consider an experimental setup which produces a dynamics. We assume that each dynam-ics is observed during a time which is fixed, for example, �0,T�, where T�R*+. For each experi-ment i , i�N, we denote by Xi�t� the dynamical variable which is observed for t� �0,T�.

Assume that we want to describe the kinematic of such a dynamical variable. What is thestrategy?

The usual idea is to model the dynamical behavior of a variable by ordinary differentialequations or PDEs. In order to do this, we must first try to have access to the speed of the variable.In order to compute a significant quantity, we can follow at least two different strategies.

�a� We do not have access to the variable Xi�t�, t� �0,T�, but to a collection of measurements ofthis dynamical variable. Assume that we want to compute the speed at time t. We can onlycompute an approximation of it for a given resolution h greater than a given threshold h0.Assume that for each experiment we are able to compute the quantity

vi,h�t� =Xi�t + h� − Xi�t�

h. �1.1�

We can then try to look for the behavior of this quantity when h varies. If the underlyingdynamics is not too irregular, then we can expect a limit for vi,h�t� when h goes to zero thatwe denote by vi�t�.We then compute the mean value

v�t� =1

ni=1

n

vi�t� . �1.2�

If the underlying dynamics is not too irregular, then v�t� can be used to model the problem.In the contrary the basic idea is to introduce a random variable.

Remark that due to the intrinsic limitation for h, we never have access to vi�t� so that thisprocedure cannot be implemented.

�b� Another idea is to look directly for the quantity

vh,n�t� =1

ni=1

n

vi,h�t� . �1.3�

Contrary to the previous case, if there exists a well defined mean value vh�t� when n goes toinfinity, then we can find an approximation vh,n as close as we want to vh. Indeed it sufficesto do sufficiently many experiences. We then look for the limit of vh�t� when h goes to zero.

For regular dynamics these two procedures lead to the same result as all these quantities arewell defined and converge to the same quantity. This is not the case when we deal with highly



irregular dynamics. In that case the second procedure is easily implemented contrary to the firstone. The only problem is that we lose the geometrical meaning of the resulting limit quantity withrespect to individual trajectories as one directly takes a mean on all trajectories before taking thelimit in h.

This second alternative can be formalized using stochastic processes and leads to the Nelsonbackward and forward derivatives that we define in the next section.

We have taken the opportunity to discuss these notions because the previous remarks provethat one cannot justify the form of the Nelson derivatives using a geometrical argument like thenondifferentiability of trajectories for a Brownian motion. This is, however, the argument used byNelson �see, Ref. 46, p. 1080� in order to justify the fact that we need a substitute for the classicalderivative when studying Wiener processes. This misleadingly suggests that the forward andbackward derivatives capture this nondifferentiability in their definition, which is not the case.

2. The Nelson derivatives

Let �� ,A ,P� be a probability space, where A is the � algebra of all measurable events and Pis a probability measure defined on A. Let P= Pt� and F= Ft� be an increasing and a decreasingfamily of sub-�-algebras, respectively. We denote by E�·�B� the conditional expectation withrespect to any sub-�-algebra B�A. We denote by L2�� the Hilbert space of all square integrablerandom variables, endowed with the Hilbertian norm � · �L2��. We denote by I an open interval ofR.

Definition 1.1: A d-dimensional random process X�·� defined on I�� is an SO process if X�·�has continuous sample paths, X�·� is F and P adapted, for all t� I, X�t��L2��, and the mapping

t→X�t� from I to L2�� is continuous.Definition 1.2: A d-dimensional random process X�·� is an S1 process if it is an SO process

such that

DX�t� = limh→0+

E�X�t + h� − X�t�h

�Pt� , �1.4�

and

D*X�t� = limh→0+

E�X�t� − X�t − h�h

�Ft� �1.5�

exist in L2�� for t� I, and the mappings t�DX�t� and t�D*X�t� are both continuous from I toL2��.

Definition 1.3: We denote by C1�I� a completion of the totality of S1 processes in the norm

�X� = supt�I

��X�t��L2�� + �DX�t��L2�� + �D*X�t��L2�� . �1.6�

Remark 1.1: The main point in the previous definitions for a forward and a backwardderivative of a stochastic process is that the forward and backward filtrations are fixed by theproblem. As a consequence, we do not have an intrinsic quantity only related to the stochasticprocess. A possible alternative definition is the following.

Definition 1.4: Let X be a stochastic process and ��X� ��*�X�� the forward (backward)adapted filtration. We define, when they exist, the quantities

dX�t� = limh→0+

h−1E�X�t + h� − X�t��Xs,s � I � �− �,t�� , �1.7�

d*X�t� = limh→0+

h−1E�X�t� − X�t − h��Xs,s � I � �t, + �� . �1.8�

In this case, we obtain intrinsic quantities, only related to the stochastic process. However,



these new operators behave very badly from an algebraic point of view. Indeed, without stringentassumptions on stochastic processes, we do not have linearity of d or d*.

This difficulty is not apparent as long as one restrict attention to a single stochastic process.

3. Good diffusion processes

Let I= �0,1�. Let �� ,A ,P� be a probability space endowed with an increasing filtration �Pt�and a decreasing filtration �Ft�.

We introduce special classes of diffusion processes for which we can explicitly compute theNelson derivatives.

The first one allows to compute the first order derivatives D and D*.Definition 1.5: We denote by �1 the set of diffusion processes X satisfying the following

conditions.

�i� X solves a stochastic differential equation,

dX�t� = b�t,X�t��dt + ��t,X�t��dW�t�, X�0� = X0, �1.9�

where X0�L2��, W�·� is a P-Brownian motion, and b : I�Rd→Rd and � : I�Rd→Rd

� Rd are Borel measurable functions satisfying the hypothesis: there exists a constant Ksuch that for every x ,y�Rd we have

supt

��t,x� − ��t,y�� + �b�t,x� − b�t,y�� K�x − y� , �1.10�

supt

��t,x�� + �b�t,x�� K�1 + �x�� . �1.11�

�ii� For any t� I, X�t� has a density pt�x� at point x.�iii� Setting aij = ��*�ij, for any i� 1, . . . ,n�, for any t0� I,

�t0

1 �Rd

�� j�aij�t,x�pt�x��dxdt + � , �1.12�

and the functions

�t,x� →� j�aij�t,x�pt�x��

pt�x�

satisfy Eqs. (1.10) and (1.11).�iv� X�t�ªX�1− t� is a Brownian diffusion [which is ensured by (i)–(iii)] with respect to �F1−t�.

The second one allows to compute, under very strong conditions, the second order derivativesDD*, D*D, D2, and D*

2.Definition 1.6: We denote by �2 the subset of �1 whose diffusion processes X are such that

their drift b and the function

�t,x� →� j�aij�t,x�pt�x��

pt�x�

are bounded, belong to C1,2�I�Rd�, and have all its first and second order derivatives bounded.Remark 1.2:

�a� Hypothesis �i� ensures that Eq. �1.9� has a unique t—continuous strong solution X�·�.�b� Hypotheses �i�–�iii� allow to apply Theorem 2.3 of Ref. 42 �p. 217�.�c� We may wonder in which cases hypothesis �ii� holds. Theorem 2.3.2 of Ref. 51 �p. 111�

gives the existence of a density for all t�0 under the Hörmander hypothesis which isinvolved by the stronger condition that the matrix diffusion ��* is elliptic at any point x. A



simple example is given by a stochastic differential equation where b is a C��I�Rd� functionwith all its derivatives bounded, and where the diffusion matrix is a constant equal to cId. Inthis case, pt�x� belongs to C��I�Rd�; moreover, if X0 has a differentiable and everywherepositive density p0�x� with respect to the Lebesgue measure such that p0�x� andp0�x�−1�p0�x� are bounded, then b�t ,x�−c� log�pt�x�� is bounded as noticed in the proof ofProposition 4.1 in Ref. 57. So hypothesis �ii� seems not to be such a restrictive condition.

According to Theorem 2.3 of Ref. 42 and thanks to �iv�, we will see that �1�C1�I� and thatwe can compute DX and D*X for X��1 �see Theorem 1.1�.

4. The Nelson derivatives for good diffusion processes

A useful property of good diffusions processes is that their Nelson’s derivatives can be ex-plicitly computed. Precisely, we have the following.

Theorem 1.1: Let X��1 solving dX�t�=b�t ,X�t��dt+��t ,X�t��dW�t�. Then X is Markov dif-fusion with respect to the increasing filtration �Pt� and the decreasing filtration �Ft�. Moreover,DX and D*X exists with respect to these filtrations and for almost all t,

DX�t� = b�t,X�t�� , �1.13�

D*X�t� = b*�t,X�t�� , �1.14�

where x→pt�x� denotes the density of X�t� and

b*i �t,x� = bi�t,x� −

� j�aij�t,x�pt�x��pt�x�

,

with the convention that the term involving 1/ pt�x� is 0 if pt�x�=0.Proof: The proof uses essentially Theorem 2.3 of Millet et al.42 Set I= �0,1�.

�1� Let X��1. Then X is a Markov diffusion with respect to the increasing filtration �Pt�, so

E�X�t + h� − X�t�h

�Pt� = E�1

h�

t

t+h

b�s,X�s��ds�Pt�and

E E�X�t + h� − X�t�h

�Pt� − b�t,X�t�� 1

hE�

t

t+h

�b�s,X�s�� − b�t,X�t��ds

=1

h�

t

t+h

E�b�s,X�s�� − b�t,X�t��ds .

Using the fact that b is Lipschizt and that t→E�Xt� is locally integrable, we can concludethat for almost all t,

1

h�

t

t+h

E�b�s,X�s�� − b�t,X�t��ds ——→h→0

0 a.s.

Therefore for almost all t, DX�t� exists and DX�t�=b�t ,X�t��.�2� As X��1, we can apply Theorem 2.3 of Ref. 42. Hence X�t�=X�1− t� is a diffusion process

with respect to the increasing filtration �F1−t� and whose generator reads



Ltf = bi�i f +1

2aij�ij f ,

with aij�1− t ,x�=aij�t ,x� and

bi�1 − t,x� = − bi�t,x� +� j�aij�t,x�pt�x��

pt�x�.

We have

E�X�t� − X�t − h�h

�Ft� = E� X�1 − t� − X�1 − t + h�h

�F1−t� = − E�1

h�

1−t

1−t+h

b�s,X�s��ds�F1−t� .

�1.15�

Using the same calculations and arguments as above since hypothesis �iii� implies that

t → E� � j�aij�t,X�t��pt�X�t��pt�X�t��

is locally integrable, we obtain that for almost all t, D*X�t� exists and is equal to

− b�1 − t,X�1 − t�� .

�

In the case of fractional Brownian motion of order H�1/2, the Nelson derivatives do notexist. However, one can define new operators extending the Nelson derivatives in the fractionalcase H�1/2. We refer to the work of Darses and Nourdin17 for more details.

5. A remark about reversed processes

This part reviews basic results about reversed processes, with a special emphasis to diffusionprocesses. We use Nelson’s stochastic calculus.

Let X be a process in the class C1�I�. We denote by X the reversed process: X�t�=X�1− t�, with

his “past” Pt and his “future” Ft. As a consequence, we also have x�C1�I�.Using the operators d and d* defined in Definition 1.4, we have the following.

Lemma 1.1: d*x�t�=−dx�1− t�=−dx�t�.Proof: The definition of d* gives immediately

d*x�t� = lim�→0+

E� x�1 − t� − x�1 − t + ��

Ft� .

However, Ft=�x�s� , ts1�=�x�u� ,0u1− t�= P1−t. Thus

d*x�t� = lim�→0+

− E� x�1 − t + �� − x�1 − t��

P1−t� = − dx�1 − t� = − dx�t� .

�

The same computation is not at all possible when dealing with the operators D and D*.

B. Stochastic derivative

In this part, we construct a natural extension5 of the classical derivative on real stochasticprocesses as a unique solution to an algebraic problem. This stochastic derivative turns out to be

5A precise meaning of this word will be given in the sequel.



necessarily complex valued. Our construction relies on Nelson’s stochastic calculus.47 We thenstudy properties of our stochastic derivative and establish a number of technical results, includinga generalization of Nelson’s product rule47 as well as the stochastic derivative for functions ofdiffusion processes. We also compute the stochastic derivative in some classical examples. Themain point is that, after a natural extension to complex processes, the real part of the secondderivative of a real stochastic process coincides with Nelson’s mean acceleration. We define aspecial class of processes called Nelson differentiable, which will be of importance for the sto-chastic calculus of variations developed in Sec. III B. This part is self-contained and all basicresults about Nelson’s stochastic calculus are reminded.

1. The abstract extension problem

In this section, we discuss in a general abstract setting what kind of analog of the classicalderivative we are waiting for on stochastic processes.

Let K=R or C. In the sequel, we denote by SK the whole set of K-valued stochastic processes.Let K� be an extension of K. We denote by PK a given subset of SK�.

6 We simply denote SK by Swhen a result is valid for both K=R or K=C.

We first remark that real7 valued functions naturally embed in stochastic processes.Indeed, let f :R→R be a given function. We denote by Xf the deterministic stochastic process

defined for all ��, t�R, by

Xf�t�� = f�t� . �1.16�

We denote by :RR→S the map associated with f �RR the stochastic process Xf.We denote by Pdet �Sdet� the subset of P �S� consisting of deterministic processes and by Pdet

k

�Sdetk � the set �Ck�, k�0. In the sequel, we assume that P always contains �C0�.

As a consequence, we have a natural action of the classical derivative on the set of differen-tiable deterministic processes Pdet

k , k�1 that we denote again by d/dt, i.e., for all f �C1, and allt�R, ��,

� d

dtXf��t�� ª Xdf/dt�t�� .

Equivalently, the following diagram commutes:

C1→

Sdet

f → Xf

↓d/dt ↓d/dt

C0→

Sdet

f → Xf ªd

dtXf

. �1.17�

Definition 1.7: Let K=R or C and K� be an extension of K. An extension of d/dt on PK is anoperator �, i.e., a map � : PK→SK� such that

�i� � coincides with d/dt on Pdet1 and

�ii� � is R linear.

6We do not give more precisions on this set for the moment; the set P can be the whole set of real or complex valuedstochastic processes or a particular class such as diffusion processes, etc.7Our aim was first to study dynamical systems over Rn. However, as we will see, we will need to consider complex valuedobjects.



Condition �i�, which is a gluing condition on the classical derivative, is necessary as long asone wants to relate classical differential equations with their stochastic counterpart.

Condition �ii� is more delicate. Of course, one has linearity of � on the differential. A naturalidea is then to preserve fundamental algebraic properties of d /dt, R linearity being one of them.This condition is not so stringent if, for example, we consider K=C. However, following this pointof view, one can ask for more precise properties like the Leibniz rule,

d�X · Y�dt

=dX

dt· Y + X ·

dY

dt, ∀ X,Y � Pdet

1 . �1.18�

In what follows, we construct a stochastic differential calculus based on Nelson’s derivatives.

2. Stochastic differential calculus

In this part, we extend the classical differential calculus to stochastic processes using a pre-vious work of Nelson47 on the dynamical theory of Brownian motion. We define a stochasticderivative and review its properties.

(a) Reconstruction problem and extension. Let us begin with some heuristic remarks support-ing our definition and construction of a stochastic derivative.

Our aim is to construct a “natural” operator on C1�I� which reduces to the classical derivatived/dt over differentiable deterministic processes.8 The basic idea underlying the whole constructionis that, for example, in the case of the Brownian motion, the trajectories are nondifferentiable. Atleast, this is the reason why Nelson47 introduces the left and right derivatives DX and D*X for agiven process X. If we refer to geometry, forgetting for a moment processes for trajectories, thefundamental property of the classical derivative dx /dt�t0� of a trajectory x�t� at point t0 is toprovide a first order �geometric� approximation of the curve in a neighborhood of t0. One wants toconstruct an operator that we denote by D, such that the data of DX�t0� allows us to give anapproximation of X in a neighborhood of t0. The difference is that we must know two quantities,namely, DX and D*X, in order to obtain the information.9 For computational reasons, one wants anoperator with values in a field F. This field must be a natural extension of R �as we want to recoverthe classical derivative� and at least of dimension 2. The natural candidate to such a field is C. Onecan also recover C by saying that we must consider not only R but the doubling algebra whichcorresponds to C.

This informal discussion leads us to build a complex valued operator D :C1�I�→PC�I�, withthe following constraints:

�i� (Gluing property� For X� Pdet1 , DX�t�=dX /dt.

�ii� The operator D is R linear.�iii� �Reconstruction property� For X�C1�I�, let us denote by

DX = A�DX,D*X� + iB�DX,D*X� ,

where A and B are linear R valued mappings by �ii�. We assume that the mapping

�DX,D*X� � �A�DX,D*X�,B�DX,D*X��

is invertible.

Lemma 1.2: The operator D has the form

D�X = �aDX + �1 − a�D*X� + i�b�DX − D*X�, � = ± 1,

where a ,b�R and b�0.

8A rigorous meaning to this sentence will be given in the sequel.9This remark is only valid for general stochastic processes. Indeed, as we will see, for diffusion processes, there is a closeconnection between DX and D*X, which allows to simplify the definition of D.



Proof: We denote by A�X�=aDX+bD*X and B�X�=cDX+dD*X. If X�C1�I�, we have DX=D*X=dX /dt, and �i� implies

a + b = 1, c + d = 0.

We then obtain the desired form. By �iii�, we must have b�0 in order to have invertibility. �

In order to rigidify this operator, we impose a constraint coming from the analogy with theconstruction of the scale derivative for non differentiable functions in Ref. 11.

�iv� If D*=−D, then A�X�=0, B�X�=D.We then obtain the following result.Lemma 1.3: An operator D satisfying conditions �i�–�iv� is of the form

D� =D + D*

2+ i�

D − D*

2, � = ± 1. �1.19�

Proof: Using Lemma 1.2, �iii� implies the relations 2a−1=0 and 2b=1, so a=b=1/2. �

We then introduce the following notion of stochastic derivative:Definition 1.8: We denote by D� the operators defined by

D� =D + D*

2+ i�

D − D*

2, � = ± 1.

(b) Extension to complex processes. In order to embed second order differential equations, weneed to define the meaning of D2 and more generally of Dn, n�N. The basic problem is that,contrary to what happens for the ordinary differential operator d /dt, even if we consider realvalued processes X, the derivative DX is a complex one. As a consequence, one must extend D tocomplex processes.

For the moment, let us denote by DC the extension to be defined by D to complex processes.Let F be a field containing C to be defined, and DC :CC

1�I�→SF. There are essentially two possi-bilities to extend the stochastic derivative leading to the same definition: an algebraic and ananalytic one.

(1) Algebraic extension. Let us assume that

�i� the operator DC is R linear.

Let Z=X+ iY be a complex process, where X and Y are two real processes. By R linearity, wehave

DC�Z� = DCX + DC�iY� .

As DC reduce to D on real processes, we obtain

DC�Z� = DX + DC�iY� ,

which reduce the problem of the extension to find a suitable definition of D on purelyimaginary processes.

We now make an assumption about the image of DC.�ii� The operator DC is C valued.

This assumption is far from being trivial and has many consequences. One of them is thatwhatever the definition of DC�iY� is, we will obtain a complex quantity which mixes the compo-nents of DX in a nontrivial way.

Remark 1.3: One may wonder if another choice is possible as, for example, using quater-nions in order to avoid this mixing problem. However, a heuristic idea behind the complex natureof D is that it corresponds to a fundamental property of Nelson processes, the (in general)nondifferentiable character of trajectories. Then, the doubling of the underlying algebra is related



to a symmetry breaking.10 The computation of D2 is not related to such phenomenon.In the following, we give two different extensions of D to complex processes under hypoth-

eses �i� and �ii�. The basic problem is the following.Let Y be a real process. We denote

DY = S�Y� ± iA�Y� , �1.20�

where

S�Y� = �D + D*

2��Y� and A�Y� = �D − D*

2��Y� , �1.21�

and S and A stand for the symmetric and antisymmetric operators with respect to the exchange ofD with D*.

We denote

DC�iY� = R�Y� + iI�Y� , �1.22�

where R�Y� and I�Y� are two real processes.One can ask if we expect for special relations between R�Y�, I�Y� and S�Y�, A�Y�.(2) C-linearity. If no relations are expected for, the natural hypothesis is to assume C linearity

of DC, i.e.,

DC�iY� = iDY . �1.23�

As a consequence, we obtain the following definition for the operator DC.We denote by CC

1�I� the set of stochastic processes of the form Z=X+ iY, with X ,Y �C1�I�.Definition1.9: The operator DC :CC

1 →PC is defined by

DC,��X + iY� = D�X + i�D�Y, � = ± 1,

where X ,Y �C1.In the sequel, we simply denote by DC for DC,�.The following lemma gives a strong reason to choose such a definition of DC. We denote by

DCn = DC � ¯ � DC.

Lemma 1.3: We have

DC2 =

DD* + D*D

2+ i

D2 − D*2

2. �1.24�

Proof. One uses the C linearity of operator D. �

We note that the real part of D2 is the mean acceleration as defined by Nelson.47

Remark 1.4: In Ref. 47 (pp. 81–82), Nelson discusses natural candidates for the stochasticanalog of acceleration. More or less, the idea is to consider quadratic combinations of D and D*,respecting a gluing property with the classical derivative.

Let Qa,b,c,d�x ,y�=ax2+bxy+cyx+dy2 be a real noncommutative quadratic form such that a+b+c+d=1. A possible definition for a stochastic acceleration is Q�D ,D*�.

We remark that the condition a+b+c+d=1 implies that when D=D*, we have Q�D ,D*�=D=D*.

The simplest examples of this kind are D2, D*2, DD*, and D*D.

We can also impose a symmetry condition in order to take into account that we do not wantto give a special importance to the mean-forward or mean-backward derivative, by assuming thatQ�x ,y�=Q�y ,x�, so that Q is of the form

10This reduces to DX=D*X for deterministic differentiable processes, namely, the invariance under h→−h.



Qa�x,y� = a�x2 + y2� + �1 − 2a�xy + yx

2, a � R .

The simplest example in this case is obtained by taking a=0, i.e.,

Q0�D,D*� =DD* + D*D

2.

This last one corresponds to Nelson’s mean acceleration and coincides with the real part of ourstochastic derivative.

It must be pointed out that Nelson discusses only five possible candidates where at least athree parameter family can be defined by Qa,b,c,1−a−b−c�D ,D*�. His five candidates correspond tothe simplest cases we have described.

The choice of Q0�D ,D*� as a mean acceleration is justified by Nelson using a GaussianMarkov process X�t� in equilibrium, satisfying the stochastic differential equation

dX�t� = − �X�t�dt + dW�t� .

We will return to this problem below.(3) Analytic extension. We first remark that D and D* possess a natural extension to complex

processes. Indeed, let X=X1+ iX2, with Xi�C1�I�, then

D�X1 + iX2� = D�X1� + iD�X2� and D*�X1 + iX2� = D*�X1� + iD*�X2� .

As a consequence, the quantities S�Y� and A�Y� introduced in the previous section for real valuedprocesses make sense for complex processes, and the quantity A�X�+ iS�X� is well defined for thecomplex process X�CC

1�I�. As a consequence, we can naturally extend D�X� to complex processesby simply posing

D�X� =D + D*

2+ �i

D − D*

2,

with the natural extension of D and D*.(4) Symmetry. A possible way to extend D is to assume that the regular part of DC�iY� is equal

to the imaginary part of D�Y�, i.e., that the geometric meaning of the complex and real part of DYis exchanged. We then impose the following relation:

R�Y� = �A�Y� .

This leads to the following extension: the operator DC :CC1 →PC would be defined by

DC,��X + iY� = D�X − i�D�Y, � = ± 1,

where X ,Y �C1.(c) Stochastic derivative for functions of diffusion process. In the following, we need to

compute the stochastic derivative of f�t ,Xt�, where Xt is a diffusion process and f is a smoothfunction.

Definition 1.10: We denote by Cb1,2�I�Rd� the set of functions f : I�Rd→R, �t ,x�� f�t ,x�

such that �t f , �f , and �xixjf exist and are bounded.

In the sequel, we denote by �ij f for �xixjf .

Our main result is the following lemma:Lemma 1.4: Let X��1 and f �C1,2�I�Rd�. Then, we have

Df�t,X�t�� = ��t f + DX�t� · �f +1

2aij�ij f��t,X�t�� , �1.25�



D*f�t,X�t�� = ��t f + D*X�t� · �f −1

2aij�ij f��t,X�t�� . �1.26�

Proof: Let X��1 and f �C1,2�I�Rd�. Thus f belongs to the domain of the generators Lt and

Lt of the diffusions X�t� and X�t�. Moreover these regularity assumptions allow us to use the samearguments as in the proof of Theorem 1.1 in order to write

Df�t,X�t�� = �t f�t,X�t�� + Lt�f�t, · ��X�t��

= ��t f + bi�i f +1

2aij�ij f��t,X�t��

= ��t f + DX�t� · �f +1

2aij�ij f��t,X�t��

and

D*f�t,X�t�� = �t f�t,X�t�� − L1−t�f�t, · ��X�t�� = ��t f + D*X�t� · �f −1

2aij�ij f��t,X�t�� .

�

We deduce immediately the following corollary.Corollary 1.1: Let X��1 and f �C1,2�I�Rd�. Then, we have

D�f�t,X�t�� = ��t f + D�X�t� · �f +i�

2aij�ij f��t,X�t�� . �1.27�

andCorollary 1.2: Let X��1 with a constant diffusion coefficient � and f �C1,2�I�Rd�. Then,

we have

D�f�t,X�t�� = ��t f + D�X�t� · �f +i��2

2�f��t,X�t�� . �1.28�

(d) Examples. We compute the stochastic derivative in some famous examples such as theOrnstein-Uhlenbeck process and a Brownian motion in an external force.

For each example, the underlying space �1 is constructed using the natural filtration generatedby the stochastic process considered and its reversed.

(1) The Ornstein-Uhlenbeck process. A good model of the Brownian motion of a particle withfriction is provided by the Ornstein-Uhlenbeck equation,

X��t� = − �X��t� + ��t� ,

X�0� = X0, �1.29�

X��0� = V0,

where X�t� is the position of the particle at time, � is the friction coefficient, � is the diffusioncoefficient, X0 and V0 are given Gaussian variables, and � is “white noise.” The term −�X��t�represents a frictional damping term.

The stochastic differential equation satisfied by the velocity process V�t�ªY��t� is given by

dV�t� = − �V�t�dt + �dW�t� ,

�1.30�V�0� = V0.



We can explicitly compute DV and D2V.Lemma 1.5: Let V�·� be a solution of

dV�t� = − �V�t�dt + �dW�t� ,

�1.31�V�0� = V0,

where V0 has a normal distribution with mean zero and variance �2 /2�.Then V�C2��0, +�� and

DV�t� = − i�V�t� , �1.32�

D2V�t� = − �2V�t� . �1.33�

Proof: The solution is a Gaussian process explicitly given by

∀t � 0, V�t� = V0e−�t + ��0

t

e−��t−s�dW�s� . �1.34�

Therefore, we can compute the expectation and the variance of the normal variable V�t�,

E�V�t�� = E�V0�e−�t,

�1.35�

Var�V�t�� =�2

2�+ �Var�V0� −

�2

2��e−2�t.

We notice, as in Ref. 26 that if V0 has a normal distribution with mean zero and variance �2 /2�,then X is a stationary Gaussian process which distribution pt�x� at each time t reads

pt�x� =��

��e−�x2/�2

. �1.36�

As a consequence, we have for all t�0,

�2�x ln�pt�x�� = − 2�x . �1.37�

Moreover,

DV�t� = − �V�t� , �1.38�

and according to Theorem 1.1, we obtain

D*V�t� = − �V�t� − �2�x ln�pt�V�t�� = �V�t� . �1.39�

Therefore DV�t�=−i�V�t�, and using the C linearity of D, we obtain D2V�t�=−�2V�t�, whichconcludes the proof. �

(2) Brownian particle submitted to an external force. In some examples of random mechanics,one has to consider the stochastic differential system,

dX�t� = V�t�dt ,

X�0� = X0,

dV�t� = − �V�t�dt + K�X�t��dt + �dW�t� ,



V�0� = V0. �1.40�

The processes X and V may represent the position and the velocity of a particle of mass m beingunder the influence of an external force F=−�U, where U is a potential. Set K=F /m. The “free”case K=0 is the above example.

When K�x�=−�2x �a linear restoring force�, the system can also be seen as the randomharmonic oscillator. In this case, it can be shown that if �X0 ,V0� has an appropriate Gaussiandistribution, then �X�t� ,V�t�� is a stationary Gaussian process in the same way as before.

Let us come back to the general case.First, we remark that X is Nelson differentiable and we have DX�t�=D*X�t�=V�t�. Moreover,

Nelson claims in Ref. 47 �pp. 83–84� that when the particle is in equilibrium with a specialstationary density,

DV�t� = − �V�t� + K�X�t�� , �1.41�

D*V�t� = �V�t� + K�X�t�� . �1.42�

We can summarize these results with the computation of D,

DX�t� = V�t� , �1.43�

D2X�t� = K�X�t�� − i�V�t� . �1.44�

C. Properties of the stochastic derivatives

1. Product rules

In Sec. III B, we develop a stochastic calculus of variations. In many problems, we will needthe analog of the classical formula of integration by parts, based on the following identity, calledthe product or Leibniz rule,

d

dt�fg� =

df

dtg + f

dg

dt�P� ,

where f ,g are two given functions.Using the previous work of Nelson,47 we generalize this formula for our stochastic derivative.

We begin by recalling the fundamental result of Nelson on a product rule formula for backwardand forward derivatives.

Theorem 1.2: Let X ,Y �C1�I�, then we have

d

dtE�X�t� · Y�t�� = E�DX�t� · Y�t� + X�t� · D*Y�t�� . �1.45�

We refer to Ref. 47 �pp. 80–81� for a proof.Remark 1.5: It must be pointed out that this formula mixes the backward and forward

derivatives. As a consequence, even without our definition of the stochastic derivative, which takesinto account these two quantities, the previous product rule suggests the construction of anoperator which mixes these two terms in a “symmetrical” way.

We now take up the various consequences of this formula regarding our operator D. Astraightforward calculation gives the following.

Lemma 1.6: Let X ,Y �C1�I�, we then have

d

dtE�X�t� · Y�t�� = E�Re�DX�t�� · Y�t� + X�t� · Re�DY�t�� , �1.46�



E�Im�DX�t�� · Y�t�� = E�X�t� Im�DY�t�� . �1.47�

Lemma 1.7: Let X ,Y �CC1�I�. We write X=X1+ iX2 and Y =Y1+ iY2, where Xi ,Yi�C1�I�.

Therefore

E�D�X · Y + X · D�Y� =d

dtg�X�t�,Y�t�� + r�X�t�,Y�t�� , �1.48�

where

g�X,Y� = E�X · Y� �1.49�

and

r�X,Y� = − 2E�Y1 Im�D�X2�� − 2E�Y2 Im�D�X1�� + i�2E�Y1 Im�D�X1�� − 2E�Y2 Im�D�X2�� .

�1.50�

Proof: We have

YD�X = Y1 Re�D�X1� − Y1 Im�D�X2� − Y2 Im�D�X1� − Y2 Re�D�X2� + i�Y1 Im�D�X1�

+ Y1 Re�D�X2� + Y2 Re�D�X1� − Y2 Im�D�X2�� . �1.51�

In a symmetrical way, we obtain

XD�Y = X1 Re�D�Y1� − X1 Im�D�Y2� − X2 Im�D�Y1� − X2 Re�D�Y2� + i�X1 Im�D�Y1�

+ X1 Re�D�Y2� + X2 Re�D�Y1� − X2 Im�D�Y2�� . �1.52�

Forming the sum of these expressions and using Lemma 1.6, we obtain Eq. �1.48�. �

The next lemma will be of importance in Sec. III B for the derivation of the stochastic analogof the E-L equations.

Lemma 1.8: Let X ,Y �CC1�I�. We write X=X1+ iX2 and Y =Y1+ iY2, where Xi ,Yi�C1�I�.

Therefore, we have

E�D�X · Y + X · D−�Y� =d

dtg�X�t�,Y�t�� , �1.53�

where g�X ,Y�=E�X1 ·Y1−X2 ·Y2�+ iE�Y1 ·X2+Y2 ·X1�=E�X ·Y�.Proof: We have

YD�X = Y1R�D�X1� − Y1I�D�X2� − Y2I�D�X1� − Y2R�D�X2� + i�Y1I�D�X1� + Y1R�D�X2�

+ Y2R�D�X1� − Y2I�D�X2�� , �1.54�

and in a symmetrical way,

XD−�Y = �X1 + iX2��D�Y1 + iD�Y2� = X1R�D�Y1� + X1I�D�Y2� + X2I�D�Y1� − X2R�D�Y2�

+ i�− X1I�D�Y1� + X1R�D�Y2� + X2R�D�Y1� + X2I�D�Y2�� . �1.55�

We form the sum of these expressions and we use Lemma 1.6 to obtain Eq. �1.48�. �

A new algebraic structure. A convenient way to write Eq. �1.53� is to use the followingHermitian product.

For all X ,Y �PC, we denote by � the product

X � Y = X · Y , �1.56�

where · denotes the usual scalar product.Formula �1.53� is then equivalent to



DE�X � Y� = E�DX � Y + X � DY� , �1.57�

where we have implicitly used the fact that D reduces to d/dt when this quantity has a sense.This new form leads us to the introduction of the following algebraic structure, which is, as far

as we know, new. Let � be the canonical mapping,

�:PC � PC → PC

X � Y � X � Y .�1.58�

We define for D the quantity ��D�=D � 1+1 � D, which we will call the coproduct of D. Then,denoting by E the classical mapping which takes the expectation of a given stochastic process, weobtain the following diagram:

PC � PC →��D�

PC � PC

X � Y → DX � Y + X � DY

↓� ↓�

X � Y → DX � Y + X � DY

↓E ↓E

E�X � Y� →D

E�DX � Y + X � DY� .

�1.59�

This structure is similar to the classical algebraic structure of the Hopf algebra. The difference isthat we perturb the classical relations by a linear mapping, here given by E. It will be interestingto study this kind of structure in full generality.

2. Nelson-differentiable processes

(a) Definition. We define a special class of processes, called Nelson-differentiable processes,which will play an important role in the stochastic calculus of variations of Sec. III B.

Definition 1.11: A process X�C1�I� is called Nelson differentiable if DX=D*X.Notation 1.1: We denote by N1�I� the set of Nelson-differentiable processes.A better definition is perhaps to use D instead of D and D* saying that Nelson-differentiable

processes have a real stochastic derivative.The main idea behind this definition is that we want to define a class P of processes in C1�I�

such that if X�C1�I�, then for all Y �P, we have

Im�D�X + Y�� = Im�DX� .

This condition imposes that Im�DY�=0.This condition will appear more clearly in Sec. III B concerning the stochastic calculus of

variations.Remark 1.6: We must keep in mind that our definition of the stochastic derivative follows the

idea of the scale calculus developed in Ref. 11 to study nondifferentiable functions. In that context,the existence of an imaginary part for the scale derivative of a function is seen as a resurgence ofits nondifferentiability. In particular, when the underlying function is differentiable, then the scalederivative is real. That is why we have chosen to call processes such that D=D* Nelson differen-tiable.

The definition of Nelson-differentiable processes is only given for processes in C1�I�. It is notat all clear to know what is the correct extension to CC

1�I�. As we have no use of such kind ofnotion on CC

1�I�, we do not discuss this point here.Of course a difficult problem is to characterize these processes. The next section discusses

some examples.(b) Examples of Nelson-differentiable process.We give examples of Nelson-differentiable processes. I.3.2.2.1. Differentiable deterministic



process. It is probably the first and the simplest example. Let x�·� be a differentiable deterministicprocess defined on I��. The past P and the future F are trivial,

∀t � I, Pt = Ft = � ,�� .

As a consequence, we have

∀t � I, Dx�t� = D*x�t� = x��t� ,

where x� is the usual derivative of x.(2) A very special random example. Let X�C1�I�. In Ref. 47, Nelson shows that X is a

constant �i.e., X�t� is the same random variable for all t� if and only if ∀t� I, DX�t�=D*X�t�=0.So it provides us a random example of N1�I� process.

(3) The random harmonic oscillator. The random harmonic oscillator satisfies the stochasticdifferential equation

dX�t� = V�t�dt ,

dV�t� = − �V�t�dt − �2X�t�dt + �dW�t� , �1.60�

X�0� = X0, V�0� = V0.

As a consequence, we have X�t�=�0t V�s�ds with E��0

b�V�s��2ds�� b�0�, and X has a strongderivative in L2. We then obtain DX�t�=D*X�t�=V�t�. Finally, we have X�N1��0,b�� andDX�t�=V�t�.

(c) About the structure of Nelson-differentiable diffusion processes. Using Theorem 1.1, wecan find a sufficient and a necessary condition for a diffusion process to be a Nelson-differentiableprocess.

Lemma 1.9: Let X��1, then X�N1�I� if and only if for all i� 1, . . . ,d�, all t� I, and x�Rd such that pt�x��0,

j=1

d

� j�aij�t,x�pt�x�� = 0. �1.61�

A consequence of this lemma is that a diffusion belonging to �1 with a constant diffusioncoefficient cannot be a Nelson-differentiable process.

Remark 1.7: When the diffusion equation is time homogeneous and the solutions have adensity, we note that this density must be a stationary density.

The difficulty to solve Eq. �1.61� relies on the fact that the density pt�x� is related to thecoefficient aij�t ,x� via the Fokker-Planck equation �Kolmogorov forward equation�.

In order to give examples of Nelson-differentiable process, we give a criteria for Nelson-differentiable process constructed from a diffusion.

Proposition 1.1: Let X��1 and Y�t�= f�t ,X�t��, where f �Cb1,2�R ,Rd�. Then Y is a Nelson-

differentiable process if and only if f solves the following PDE:

�t f�t,x� + i,j� � j�aij�t,x�pt�x��

pt�x��i f�t,x� + aij�t,x��ij f�t,x�� = 0. �1.62�

We can simplify this condition for suitable diffusion processes:Corollary 1.3: Let X��1 with a constant diffusion coefficient and a stationary density p�·�.

Let Y�t�= f�X�t��, where f �C2�Rd�. Then Y is a Nelson-differentiable process if and only if fsolves the following PDE:

�x ln p�x� · �xf�x� + �f�x� = 0. �1.63�



Despite the simple form of this PDE, its resolution seems to be a very difficult problem sincewe work on an unbounded domain. This will be studied in a forthcoming paper.

(d) Product rule and Nelson-differentiable processes.Corollary 1.4: Let X ,Y �CC

1�I�. If X is Nelson differentiable, then

E�D�X�t� · Y�t� + X�t� · D�Y�t�� =d

dtE�X�t�,Y�t�� . �1.64�

Proof: This is a simple consequence of the fact that if X=X1+ iX2 is Nelson-differentiable,then Im�D�X1�=Im�D�X2�=0. �

II. STOCHASTIC EMBEDDING PROCEDURES

A. Stochastic embedding of differential operators

A natural question concerning ordinary and PDEs concerns their behavior under small randomperturbations. This problem is particularly important in natural phenomena where we know thatmodels are only an approximation of the real setting. For example, the study of the long termbehavior of the solar system is usually done by running numerical computations on the n-bodyproblem. However, many effects in the solar systems are not included in this model and can be ofimportance if one looks for a long term integration, as nonconservative effects �due to tidal forcesbetween planets� and the oblateness of the sun which is not yet modeled by a differential equation.

The main problem is then to find the correct analog of a given differential equation taking intoaccount the following facts.

�i� The classical equation is a good model at least in first approximation.�ii� One must extend this equation to stochastic processes.

Using the stochastic derivative introduced in the previous part, we give a natural embeddingof PDE or ordinary differential equation into stochastic PDE or ordinary differential equations. Itmust be pointed out that we do not perturb the classical equation by a random noise or anythingelse. In this respect we are far from the usual way of thinking underlying the fields of stochasticdifferential equations or stochastic dynamical systems.

Of course, having this natural embedding, we can naturally define what a stochastic pertur-bation of a differential equation is. This is simply a stochastic perturbation of the stochasticembedding of the given equation. The main point is that we stay in the same class of objectsdealing with perturbations, which is not the case in the stochastic theory of differential equations,where we jump from classical solutions to stochastic processes in one step using, for example,Ito’s stochastic calculus. 11

In this part we first give a general embedding procedure for PDEs. We discuss classicalexamples, in particular, first and second order differential equations. The case of Lagrangiansystems is studied in detail in Sec. III B. An important part of classical differential equationscoming from mechanics are reversible. This property is not conserved by the previous stochasticembedding procedure. We define a special embedding called reversible, which preserves this

property, meaning that if X is a solution of the stochastic embedded equation, then X, the reversedprocess, is again a solution.

1. Stochastic embedding of differential operators

In this part, we first give an abstract embedding procedure based on an extension of theclassical derivative defined in the previous part. We then specialize our embedding procedureusing the stochastic derivative.

11This remark is also valid for all the theories of this kind, using your favorite stochastic calculus, like Malliavin calculus,for example.



(a) Abstract embedding. Let A be a ring, we denote by A�x� the ring of polynomials withcoefficients in A.

Definition 2.1: A differential operator is an element of A�d/dt�.Let O�A�d/dt�, the differential operator O is of the form

O = a0�•,t� + a1�•,t�d

dt+ ¯ + an�•,t�

dn

dtn , ai � A, = 0, . . . ,n , �2.1�

for a given n�N, called the degree of O.The action of O on a given function x :R→Rd, t�x�t� is denoted O ·x and defined by

O · x = i=0

n

ai�x�t�,t�dx

dt. �2.2�

Definition 2.2: Abstract stochastization. Let O�A�d/dt� be a differential operator of the form

O = a0�•,t� + a1�•,t�d

dt+ ¯ + an�•,t�

dn

dtn , ai � A, i = 0, . . . ,n , �2.3�

where n�N is given.The stochastic embedding of O with respect to the extension � : P→S is an element O� of P��

defined by

O� = a0�•,t� + a1�•,t�� + ¯ + an�•,t��n, ai � A, i = 0, . . . ,n , �2.4�

where �n=� � ¯ ��.The action of O� on a given stochastic process X, denoted by O� ·X, is defined by

O� · X = i=0

n

ai�X,t��iX , �2.5�

where the notation ai�X , t� stands for the stochastic process defined for all �� by

ai�X,y�� = ai�X��,t�,t� . �2.6�

The main property of this embedding is the fact that

�O��Pdetn = O , �2.7�

so that the classical differential equation associated with O and given by

O · x = 0 �E�

is contained in the stochastic differential equation

O� · X = 0 . �SE�

(b) Nelson stochastic embedding. Using the stochastic derivative, we have a particular sto-chastic embedding procedure. Let A=C1�Rd�R�.

Definition 2.3: (Stochastization) Let O�A�d/dt� be a differential operator of the form

O = a0�•,t� + a1�•,t�d

dt+ ¯ + an�•,t�

dn

dtn , ai � A, i = 0, . . . ,n , �2.8�

where n�N is given.The stochastic embedding of O with respect to the stochastic extension D� is an element Ostoc

of A�D�� defined by



Ostoc = a0�•,t� + a1�•,t�D� + ¯ + an�•,t�D�n , ai � A, i = 0, . . . ,n . �2.9�

We denote by S the operator associated with an operator O of form �2.8� the operator Ostoc. Asa consequence, we will frequently use the notation S�O� for Ostoc.

In some occasions, in particular, for the E-L equation, we will need to consider differentialoperators in a nonstandard form. Precisely, we need to consider operators like

Ba =d

dt� a�•,t� . �2.10�

This notation means that Ba acts on a given function as

Ba · x =d

dt��a�x�t�,t�� . �2.11�

The basic idea is to define the stochastic embedding of Ba as follows:Definition 2.4: The stochastic embedding of the basic brick Ba is given by

Ba = D � a�•,t� . �2.12�

However, classical properties of the differential calculus allow us to write Ba equivalently as

Ba · x = a��x�dx

dt. �2.13�

The stochastic embedding of this new form of Ba is given by

Ba · X = a��X�DX . �2.14�

The main problem is that, in general, we do not have

Ba = Ba, �2.15�

as in the classical case.This reflects the fact that S acts on operators of a given form and not on operators as an

abstract element of a given algebra. In particular, this is not a mapping.Nevertheless, there exists a class of functions a such that Eq. �2.15� is valid.Lemma 2.1: Equation (2.15) is satisfied on the set � with constant diffusion if a is an

harmonic function.Proof: This follows easily from Corollary 1.2. �

In the sequel we study some basic properties of this embedding procedure on differentialequations.

2. First examples

(a) First order differential equations. Let us consider a first order differential equation

dx

dt= f�x,t� �1-ODE� ,

where x�R and f :R�R→R is a given function. The stochastic embedding of �1-ODE� leads to

DX = F�X,t� �1-SODE� ,

where F is real valued.The reality of F imposes important constraints on solutions of 1-�SODE�. Indeed, we must

have



DX = D*X ,

so that X belongs to the class of Nelson-differentiable processes.In our general philosophy, ordinary differential equations are only coarse approximations to

reality which must include stochastic behavior in its foundation. A stochastic perturbation of a firstorder differential equation is then highly nontrivial. Indeed, we must consider SODE’s of the form

DX = F�X,t� + �G�X,t� ,

where G�X , t� is now complex valued. As a consequence, we allow solutions to leave the Nelson-differentiable class.

(b) Second order differential equations. Let us consider a second order differential equation

d2x

dt2 + a�x�dx

dt+ b�x� = 0 �2-ODE� ,

where x�R, and a ,b :R→R are given functions. The stochastic embedding of 2-�ODE� leads to

D2X + a�X�DX + b�X� = 0.

In this case, contrary to what happens for first order differential equations, we have no realitycondition which constrains our stochastic process.

In order to study such kind of equations, one can try to reduce it to a first order equation, usingstandard ideas. We denote by Y =DX, then the second order equation is equivalent to the followingsystem of first order stochastic differential equations:

DX = Y ,

�2.16�DY = − a�X�Y − b�X� .

One must be careful to take Y �CC1�I� as Y is a priori a complex stochastic process. This remark

is of importance since if we apply the stochastic embedding procedure12 to the classical system offirst order differential equations,

dx

dt= y ,

�2.17�dy

dt= − a�x�y − b�x� ,

by saying that we apply separately the stochastic embedding on each differential equations, weobtain stochastic equation �2.16� but with Y �C1�I�, which imposes strong constraints on thesolutions of our equations.

This example proves that the stochastic embedding procedure is not so easy to define if onewants to deal with systems of differential equations. We will return to this problem concerning thestochastic embedding of Hamiltonian systems.

B. Reversible stochastic embedding

1. Reversible stochastic derivative

In our construction of the stochastic derivative, we have imposed some constraints as, forexample, the gluing to the classical derivative on differentiable deterministic processes. We have

12Note that we have not defined the stochastic embedding procedure on systems of differential equations.



moreover kept some properties of the classical derivative such as linearity. However, we have notconserved more important properties of the classical derivative which are used in the study ofclassical differential equations. For example, let us consider

d2x

dt2 = f�x�, �E�

which is the basic equation of Newton’s mechanics. An important property of this kind of equa-tions is its reversibility.31

Let t→x�t� be a solution of �E�. We denote by x�t�=x�−t�. Then, we have

d2x

dt2 =d

dt�−

dx

dt�− t�� =

d2x

dt2 �− t� = f�x�− t�� = f�x�t�� ,

proving that the reversed solution x�t� is again a solution of the same equation. In this case, we saythat the differential equation is reversible.

The reversibility argument used the following important property:

d

dt�x�− t�� = −

dx

dt�− t� . �R�

The natural way to introduce a notion of reversibility is then to look for the stochastic

differential equation satisfied by X�t�=X�−t��C1�I� the reversed processes. However, in general,

we do not have access to DX or D*X. As a consequence, a definition using this characterization isnot effective. In the following, we follow a different strategy.

A convenient way to characterize the reversibility of a given differential equation, describedby a differential operator

O = i

aidi

dti � R�d/dt� , �2.18�

is to prove that this operator is invariant under the substitution

r:R�d/dt� → R�d/dt� , �2.19�

which is R linear and defined by

r�d/dt� = − d/dt . �2.20�

We then introduce in our setting the following analogous substitution.Definition 2.5: The reversibility operator R :C�D ,D*�→C�D ,D*� is a C morphism defined by

R�D� = − D*, R�D*� = − D . �2.21�

We have the following immediate consequence of the definition.Lemma 2.2: The reversibility operator is an involution of C�D ,D*�.This operator acts nontrivially on our stochastic derivative. Precisely, we have the following:Lemma 2.3:

R�D� = − D . �2.22�

The complex nature of the stochastic derivative induces new phenomenon which are differentfrom the classical case. For example, we have

R�D2� = D2, �2.23�

contrary to what happens for r.



We now define our notion of a reversible stochastic equation.Definition 2.6: (Reversibility) Let O�C�D ,D*�, then the stochastic equation O ·X=0 is re-

versible if and only if R�O� ·X=0.A natural problem is the following.Reversibility problem. Find an operator such that the stochastic embedding of a reversible

equation is again a reversible equation in the sense of Definition 2.6.Let us consider the family of stochastic derivatives D�, �=0, ±1. Without assuming a par-

ticular form for the underlying equation, the preservation of the reversible character reduces toprove that the operator � which is chosen satisfies

R�� = − � . �2.24�

In the family of stochastic derivatives D�, �=0, ±1, only one case is possible.Lemma 2.4: A reversibility of a differential equation is always preserved under a stochastic

embedding if and only if this embedding is associated with the stochastic derivative D0.Proof: Essentially this follows from Eq. �2.22�. If we want to preserve reversibility, then the

operator D� must satisfy R�D��=−D�. This is only possible if D� is real, i.e., �=0. �

It must be pointed out that the operator

D0 =D + D*

2

has been obtained by different authors using the following argument.If we use only D �or D*�, then we give a special importance to the future �or past� of the

process, which has no physical justification. As a consequence, one must construct an operatorwhich combines these two quantities in a more or less symmetric way. The simplest combinationis a linear one aD+bD* with equal coefficients a=b. The gluing to the classical derivative leadsto a=b=1/2.

The problem with this construction is that this argument is used on diffusion processes, whereD and D* are not free. As a consequence, working with D is the same �even if the connection withD* is not trivial� than working with D*. We cannot really justify the use of D0. It must be pointedout that Nelson47 does not use D0 in his derivation of the Schrödinger equation, but simply D.

Here, this operator is obtained by specialization of D�, which form is imposed by our con-struction �linearity, gluing to the classical derivative, and reconstruction property�. The reconstruc-tion property imposes that ��0 unless we work with diffusion processes.

Imposing a new constraint on the reversibility on this operator leads us to �=0. The operatorD0 is of course defined on C1�I�, but in order to satisfy the whole constraints of our construction,we must restrict its domain to diffusion processes.

We can of course find reversible equations without using D0 but D�. We keep the notationsand conventions of Sec. II A. We first define the action of R on a given operator of the form

O = i=0

n

ai�•,t��− 1�iDi. �2.25�

Definition 2.7: The action of R on Eq. (2.15) is denoted R�O� and defined by

R�O� = i=0

n

ai�•,t�Di. �2.26�

Definition 2.6 of a reversible equation can then be extended to cover operators of form �2.25�.

2. Reversibility of the stochastic Newton equation

Using this definition, we can prove that the stochastic equation,



D�2 X = − �U�X�, �E�

is reversible.Indeed, we have the following.Lemma 2.5: Equation �E� is reversible.Proof: We have

R�D�2 + �U� · X = D�

2 X + �U�X� ,

=D�2 X + �U�X� . �2.27�

As U is real valued and X are real stochastic processes, we deduce from �E� that

D�2 X = − �U�X� = − �U�X� . �2.28�

We deduce that

R�D�2 + �U� · X = 0, �2.29�

which concludes the proof. �

3. Iterates

There exists a fundamental difference between D0 and D�, ��0. The operator D0 send realstochastic processes to real stochastic processes in the contrary of D�, ��0, which leads tocomplex stochastic processes. As a consequence, the n-ième iterates of D0 is simply defined by

D0n = D0 � ¯ � D0, �2.30�

without problem, where a special extension of D�, ��0 to complex stochastic processes must bediscussed.

4. Reversible stochastic embedding

Using D0, we can define a stochastic embedding which conserves the fundamental property ofreversibility of a given equation. We keep notations from Sec. II A.

Definition 2.8: (Reversible stochastization) Let O�A�d/dt� be a differential operator of theform

O = a0�•,t� + a1�•,t�d

dt+ ¯ + an�•,t�

dn

dtn , ai � A i = 0, . . . ,n ,

where n�N is given.The reversible stochastic embedding of O is an element Orev of C1�I��D0� defined by

Orev = a0�•,t� + a1�•,t�D0 + ¯ + an�•,t�D0n, ai � C1�I�, i = 0, . . . ,n . �2.31�

A differential equation �E� is defined by a differential operator O�A�d/dt�, i.e., an equationof the form

O · x = 0 �E� ,

where x is a function.Using stochastization, the reversible stochastic analog of �E� is defined by

Orev · X = 0 �RSE� ,

where X is a stochastic process.



5. Reversible versus general stochastic embedding

The reversible stochastic embedding leads to very different results than the general stochasticembedding. We can already see this difference on first order differential equations. Let us consider

dx

dt= f�x� ,

where x�R and f is a real valued function. The reversible stochastic embedding gives

D0X = f�X� .

Contrary to what happens for the stochastic embedding, this equation does not impose for thesolution to be a Nelson-differentiable processes.

For second order systems, a new phenomenon appears, which makes the reversible embeddinginteresting. Let us consider a second order equation of the form

d2x

dt2 = f�x� . �Ef�

This equation is equivalent to the system

dx

dt= y ,

dy

dt= f�x� .

�Sf�

The stochastic embedding of �E� is given by

D�2 X = f�X�, X � C1.

This equation is equivalent to the stochastic system

D�X = Y ,

D�Y = f�X� ,��

with X�C1 and Y �CC1 as long as ��0.

However, the elementary stochastic embedding of �S� is given by �*�, but for X�C1 and Y�C1.

As a consequence, we have coherence between the associated systems �*� and the embeddedsystems if and only if �=0.

III. STOCHASTIC EMBEDDING OF LAGRANGIAN AND HAMILTONIAN SYSTEMS

A. Stochastic Lagrangian systems

Most of classical mechanics can be formulated using Lagrangian formalism.5,1 Lagrangianmechanics contains important problems, such as the n-body problem. Using our framework, westudy Lagrangian dynamical systems under stochastic perturbations.13

Our approach is first to embed classical Lagrangian systems, in particular, the associated E-Lequation in order to obtain an idea of what kind of equation govern stochastic Lagrangian systems.We then develop a stochastic calculus of variations. We obtain an analog of the LAP14 which gives

13For the n-body problem, which is usually used to study the long term behavior of the solar system �Ref. 41�, this problemis of crucial importance. Indeed, the n-body problem is only an approximation of the real problem, and even if somenumerical simulations take into account relativistic effects �Ref. 32�, this is not sufficient �Re. 43�.14In our case, the word least action is misleading and a better terminology is stationary �see below�.



a second stochastic Euler-Lagrange equation, denoted by �SEL� in the sequel. We then prove thefollowing surprising result, called the coherence lemma: we have S�EL�= �SEL�.

The principal interest of Lagrangian systems is that the action of a group of symmetries leadsto first integrals of motion, i.e., functions which are constants on solutions of the equations ofmotion. The celebrated theorem of Noether gives a precise relation between symmetries and firstintegrals. We prove a stochastic analog of the Noether theorem.

Finally, we prove that the stochastic embedding of Newton’s Lagrangian systems lead to anonlinear Schrödinger’s equation for a given wave function whose modulus is equal to the prob-ability density of the underlying stochastic process.

1. Reminder about Lagrangian systems

We refer to Ref. 4 for more details, as well as Ref. 1.Lagrangian systems play a central role in dynamical systems and physics, in particular, for

mechanical systems. A Lagrangian system is defined by a Lagrangian function, commonly denotedby L, and depending on three variables x, v, and t which belongs in the sequel to R. As Lagrangiansystems come from mechanics, x stands for position, v for speed, and t for time. In what follows,we consider a special type of Lagrangian function called admissible in the following.

Definition 3.1: An admissible Lagrangian function is a function L such that

�i� the function L�x ,v , t� is defined on Rd�Cd�R, holomorphic in the second variable andreal for v�R; and

�ii� L is autonomous, i.e., L does not depend on time.

Condition �i� is fundamental. This condition is necessary in order to apply the stochastizationprocedure �see below�. The fact that we only consider autonomous Lagrangian function is due totechnical difficulties in order to take into account backward and forward filtrations in the compu-tation of the SEL equation �see below�.

Remark 3.1: In applications, admissible Lagrangian functions L are analytic extensions tothe complex domain of real analytic Lagrangian functions. For example, the classical NewtonianLagrangian L�x ,v�= �1/2�v2−U�x�, defined on an open15 subset of R�R, with an analytic po-tential is an admissible Lagrangian function.

A Lagrangian function L being given, the equation

d

dt� �L

�v� =

�L

�x, �EL�

is called the E-L equations.An important property of the E-L equation is that it derives from a variational principle,

namely, the LAP �see, Ref. 5 p. 59�. Precisely, a curve � : t�x�t� is an extremal16 of the functional

Ja,b�� = �a

b

L�x�t�, x�t�,t�dt

on the space of curves passing through the points x�a�=xa and x�b�=xb, if and only if it satisfiesthe E-L equation along the curve x�t�.

2. Stochastic E-L equations

We now apply our stochastic procedure S to an admissible Lagrangian.Lemma 3.1: Let L�x ,v� :Rd�Cd→C be an admissible Lagrangian function. The SEL equa-

tion obtained from (EL) by the stochastic procedure is given by

15This Lagrangian function is not always defined on R�R. An example is given by Newton’s potential U�x�=1/x, x�R*.16We refer to Ref. 5, chap. 3, Sec. 12 for an introduction to the calculus of variations.



D�� L

�v�X�t�,D�X�t�� =

�L

�x�X�t�,D�X�t�� . S�EL�

Proof: The E-L equation associated with L�x ,v� can be seen as the following differentialoperator:

OEL =d

dt�

�L

�v−

�L

�x,

acting on �x�t� , x�t��. The embedding of OEL gives

OEL = D� ��L

�v−

�L

�x.

As OEL acts on �x�t� , x�t��, the operator OEL acts on �X�t� ,D�X�t��. This concludes theproof. �

The free parameter �� −1,0 ,1� can be fixed depending on the nature of the extension used.It must be pointed out that there exist crucial differences between all these extensions due to

the fact that D� is complex valued for �= ±1 and real for �=0. Indeed, let us consider thefollowing admissible Lagrangian function:

L�x,v� =1

2v2 − U�x� ,

where U is a smooth real valued function. Then, equation S�EL� gives

D�V = �U�X� ,

where V=D�X. When �= ±1, this equation imposes strong constraints on X due to the real natureof U�X�, namely, that D�

2 X�N1�I�.On the contrary, when �=0, i.e., in the reversible case, these intrinsic conditions disappear.

3. The coherence problem

Up to now, the stochastic embedding procedure can be viewed as a formal manipulation ofdifferential equations. Moreover, as most classical manipulations on equations do not commutewith the stochastic embedding, this procedure is not canonical.17 In order to rigidify this construc-tion and to make precise the role of this stochastic embedding procedure, we study the followingproblem, called the coherence problem.

We know that the E-L equations are obtained via a LAP on a functional. The main problem isthe existence of a stochastic analog of this LAP, that we can call as a SLAP, compatible with thestochastic embedding procedure.

L�x�t�, x�t�� →S

L�X�t�,DX�t��

LAP↓ ↓SLAP?

�EL� →S �SEL� .

�3.1�

In the next chapter, we develop the necessary tools to answer to this problem, i.e., a stochasticcalculus of variations. Note that due to the fact that the stochastic Lagrangian as well as the SELequation are fixed, this problem is far from being trivial. The main result of the next chapter is theLagrangian coherence lemma which says precisely that the SEL equation obtained via the sto-chastic embedding procedure coincide with the characterization of extremals for the functional

17We return to this problem in our discussion of a stochastic symplectic geometry which can be used to bypass this kindof problem.



associated with the stochastic Lagrangian function using the stochastic calculus of variations. Asa consequence, we obtain a rigid picture involving the stochastic embedding procedure and a firstprinciple via the SLAP.

This picture will be then extended in another chapter when dealing with the Hamiltonian partof this theory.

B. Stochastic calculus of variations

The embedding procedure allows us to associate a SEL equation to a stochastic Lagrangianfunction. A basic question is then the existence of an analog of the LAP. In this section, wedevelop a stochastic calculus of variations for our Lagrangian function following a previous workof Yasue.66 Our main result, called the coherence lemma, states that the SEL equation can beobtained as an application of a SLAP. Moreover, this derivation is consistent with the stochasticembedding procedure.

1. Functional and L-adapted process

In the sequel we denote by I a given open interval �a ,b�, ab.We first define the stochastic analog of the classical functional.Definition 3.2: Let L be an admissible Lagrangian function. Set

� = �X � C1�I�,E��I

�L�X�t�,D�X�t��dt� �� .

The functional associated with L is defined by

FI:�� → C

X � E��a

b

L�X�t�,D�X�t��dt� . � �3.2�

In what follows, we need a special notion which we will call L adaptation, as Yasue in Ref. 66Definition 3.3: Let L be an admissible Lagrangian function. A process X�C1�I� is called

L-adapted if

�i� for all t� I, �xL�X�t� ,D�X�t�� is Pt and Ft measurable, and �xL�X�t� ,D�X�t��L2��; and�ii� �vL�X�t� ,D�X�t��C1�I�.

The set of all L-adapted processes will be denoted by L.

2. Space of variations

Calculus of variations is concerned with the behavior of functionals under variations of theunderlying functional space, i.e., objects of the form �+h, where � belongs to the functional spaceand h is a given functional space of variations. Special care must be taken in our case to definewhat is the class of variations we are considering. In general, this problem is not really pointed outas both variations and curves can be taken in the same functional space �see Ref. 5, p. 56, footnote26�. We introduce the following terminology.

Definition 3.4: Let � be a subspace of C1�I� and X�C1�I�. A � variation of X is a stochasticprocess of the form X+Z, where Z��. Moreover, set

�� = Z � �, ∀ X � �,Z + X � �� .

We note that �� is a subspace of �.In the sequel, we consider two subspaces of variations: N1�I� and C1�I�.



The choice of C1�I� is natural. However, doing this we can obtain stochastic processes withcompletely different behavior than X.18

What is the specific property of X�C1�I� that we want to keep?If we refer to the construction of the stochastic derivative, then a main point is the existence

of an imaginary part in D�X.19 This property is related to the nondifferentiability of the underly-ing stochastic process. We are then lead to search for variations Z which conserve this imaginarypart. As a consequence, we must consider Nelson-differentiable processes introduced in the pre-vious part,20 and denoted by N1�I�.

3. Differentiable functional and critical processes

We now define our notion of differentiable functional. Let � be a subspace of C1�I�.Definition 3.5: Let L be an admissible Lagrangian function and FI the associated functional.

The functional FI is called � differentiable at an X��L if for all Z��,

FI�X + Z� − FI�X� = dFI�X,Z� + R�X,Z� , �3.3�

where dFI�X ,Z� is a linear functional of Z�� and R�X ,Z�=o��Z��.The stochastic analog of a critical point is then defined by the following.Definition 3.6: A �-critical process for the functional FI is a stochastic process X��L

such that dFI�X ,Z�=0 for all Z�� such that Z�a�=Z�b�=0.(a) The �=C1�I� case. Our main result is the following.Lemma 3.2: Let L be an admissible Lagrangian with all second derivatives bounded. Then

the functional FI defined by Eq. �3.2� is C1�I� differentiable at any X��L, and C1�I��=C1�I�.For all Z�C1�I�, the differential is given by

dFI�X,Z� = E��a

b � �L

�x�X�u�,D�X�u�� − D−�� L

�v�X�u�,D�X�u��Z�u�du�

+ g�Z,�vL��b� − g�Z,�vL��a� , �3.4�

where

g�Z,�vL��s� = E�Z�u��vL�X�u�,D�X�u�� . �3.5�

Proof: Thanks to the Taylor expansion of L, we have for Z�C1�I�,

L�X + Z,D�X + Z�� − L�X,DX� = �xL�X,DX�Z + �vL�X,DX�DZ + �0

1

�1 − s��x2L�Ts�Z2

+ �xv2 L�Ts�ZDZ + �v

2L�Ts��DZ�2�ds , �3.6�

where Ts= �X+sZ ,DX+sDZ�.Since Z�C1�I� and X��L, the expectation

18Of course, this is not the case in the classical case: one consider x�C��I� and z�C��I� such that x+h�C��I� is verysimilar to x. For example, we do not choose z�C0�I� which leads to radically new behavior of x+z with respect to x.19Of course, as long as �= ±1. This is of importance since we will be able to choose a more general variations space in thiscase.20An analogous problem is considered in Ref. 12, where a nondifferentiable variational principle is defined.



E��a

b

��xL�X�s�,DX�s��Z�s� + �vL�X�s�,DX�s��DZ�s��ds�is well defined. The Cauchy-Schwarz inequality shows that supI E��ZDZ��, supI E��Z�2�, andsupI E��DZ�2� are O��Z�2� since Z�C1�J�. Moreover, �xv

2 L, �x2L, and �v

2L are bounded. Therefore,C1�I��=C1�I� and we can write

FJ�X + Z� − FJ�X� = E��a

b

��xL�X�s�,DX�s��Z�s� + �vL�X�s�,DX�s��DZ�s��ds� + o��Z�� .

�3.7�

Using product rule �1.24�, we deduce Eq. �3.4�. �

(b) The �=N1�I� case. Our main result is the following.Lemma 3.3: Let L be an admissible Lagrangian with all second derivatives bounded. The

functional FI defined by Eq. (3.2) is N1�I� differentiable at any X��L, and N1�I��=N1�I�. Forall Z�N1�I�, the differential is given by

dFI�X,Z� = E��a

b � �L

�x�X�u�,D�X�u�� − D�� L

�v�X�u�,D�X�u��Z�u�du�

+ g�Z,�vL��b� − g�Z,�vL��a� , �3.8�

where

g�Z,�vL��s� = E�Z�u��vL�X�u�,D�X�u�� . �3.9�

Proof: In the same way, Eq. �3.7� holds. As Z�N1�I�, we can use product rule �1.35�. So wededuce Eq. �3.8� from Eq. �3.7�. �

4. Least action principles

As for the computation of the differential of functionals, we must consider two cases: �=C1�I� and �=N1�I�.

(a) The �=C1�I� case. The main result of this section is the following analog of the LAP forLagrangian mechanics.

Theorem 3.1: (Global least action principle) Let L be an admissible Lagrangian with allsecond derivatives bounded. A necessary and sufficient condition for a process X��L�C3�I� to be a C1�I�-critical process of the associated functional FI is that it satisfies

�L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t�� = 0. �3.10�

We call this equation the Global Stochastic Euler-Lagrange equation �GSEL�.We have conserved the terminology of LAP even if we have no notion of extremals for our

complex valued functional.Proof: The proof can be reduced to the case I= �0,1�. Let X�C3�I� be a solution of

�L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t�� = 0, �3.11�

then X is a C1�I�-critical process for the functional FI.Conversely, let X��L�C3�I� be a C1�I�-critical process for the functional FI, i.e.,

dFI�X ,Z�=0, or equivalently

Re�dFI�X,Z�� = Im�dFI�X,Z�� = 0.

We define



Zn�1��u� = �n

�1��u� · Re� �L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t��

and

Zn�2��u� = �n

�2��u� · Im� �L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t�� ,

where ��n�i��n�N are sequences of C��0,1�→R+� deterministic bump functions on �0, 1�, i.e., for

all n�N, �n�0�=�n�1�=0, and �n=1 on ��n ,�n� with 0�n�n1, limn→� �n=0,limn→� �n=1.

Thus, for all n�N,

Re�dFI�X,Zn�1�� = E�

0

1

�n�u�Re2� �L

�x�X�u�,D�X�u�� − D−�� L

�v�X�u�,D�X�u��du = 0.

By the bounded dominated convergence theorem, we deduce that

E�0

1

Re2� �L

�x�X�u�,D�X�u�� − D−�� L

�v�X�u�,D�X�u��du = 0.

The same argument leads to

E�0

1

Im2� �L

�x�X�u�,D�X�u�� − D−�� L

�v�X�u�,D�X�u��du = 0.

Therefore for almost all t� �0,1� and almost all ��,

�L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t�� = 0.

�

(b) The �=N1�I� case An easy consequence of Lemma 3.3 isLemma 3.4: Let L be an admissible Lagrangian with all second derivatives bounded. A

solution of the equation

�L

�x�X�t�,D�X�t�� − D�� L

�v�X�t�,D�X�t�� = 0, �3.12�

called the SEL equation is a N1�I�-critical process for the functional FI associated with L.We have not been able to prove the converse of this lemma for N1 variations.As an example of possible difficulties, we have not found a suitable deformation of the

process Z�1� and Z�2� which are Nelson differentiable and which allow to conclude.More generally, we can search for variations of the form

f�t,Re� �L

�x�X�t�,D�X�t�� − D−�� L

�v�X�t�,D�X�t�� ,

where f is a sufficiently smooth function, which are adapted Nelson-differentiable processes.However, the characterization of Nelson-differentiable processes given by Proposition 1.1 in-volves a PDE whose solutions are not known or even characterized. As a consequence, we cannotdecide if a given deformation of the stochastic process via a function f is an adapted Nelson-differentiable process or not.



5. The coherence lemma

It is not clear that the SEL equation obtained by the stochastization procedure and the N1�I�or C1�I� LAP coincide. One easily sees that this is not the case for �=C1�I�. Does there exist aSLAP which ensures that we obtain the same equations?

We tackle this question by studying if the following diagram commutes:

L�x�t�,x��t��→S

L�X�t�,DX�t��

LAP↓ ↓SLAP

�EL�→S

�SEL��3.13�

Definition 3.7: We say that a �-Stochastic Least-action Principle (SLP) induces

�a� a strong Lagrangian coherence lemma if the previous diagram commutes, i.e., a solution of(SEL) is a stationary point for (SLP), and conversely

�b� a weak Lagrangian coherence lemma if a solution of (SEL) is a stationary point for (SLP).

A very consequence of the previous section is the following.Lemma 3.4: (Weak Lagrangian coherence lemma) The N1�I�-stochastic least-action principle

induces a weak Lagrangian coherence lemma.When �=0, i.e., in the reversible case, the previous lemmas and theorems are true under C1�I�

variations.Theorem 3.2: (Reversible coherence lemma) For �=0, diagram (3.13) commutes.Note that when �=0, our stochastic derivatives coincides with the Misawa-Yasue �Ref. 44�

canonical formalism for stochastic mechanics.

C. The stochastic Noether theorem

A natural question arising from the stochastization procedure of classical dynamical systems,in particular, Lagrangian systems, is to understand what remains from classical first integrals ofmotion. First integrals play a central role in many problems like the n-body problem. In thissection, we obtain a stochastic analog of the Noether theorem. A stochastic generalization of theNoether theorem has also been studied by Thieullen and Zambrini in Ref. 58. They use action

functionals of the form S�z�·��=Et0�t0

t1L�z�s� ,Dz�s� ,s�ds and they consider Bernstein diffusionswhich are critical points of this action.

We then defined the notion of first integrals for stochastic dynamical systems. We also discussthe consequences of the existence of first integrals in the context of chaotic dynamical systems.

1. Tangent vector to a stochastic process

Let X�C1�I� be a stochastic process. We define the analog of a tangent vector to X at point t.Definition 3.B: Let X�C1�I�, I�R. The tangent vector to X at point t is the random variable

DX�t�.Remark 3.2: Of course, in order to define stochastic Lagrangian systems in an intrinsic way,

one must define the stochastic analog of the tangent bundle to a smooth manifold. In our case, itis not clear what is the adequate geometric object underlying stochastic Lagrangian dynamics.For example, we can think of multidimensional Brownian surfaces (Ref. 22, sec. 16.4). All thesequestions will be developed in a forthcoming paper.15

2. Canonical tangent map

In the sequel, we will need the following mapping called the canonical tangent mapDefinition 3.9: For all X�C1�I�, we define the canonical tangent map as



T:�C1�I� → C1�I� � PC

X � �X,DX� .� �3.14�

The mapping T will be used in the following section to define the analog of the linear tangentmap for a stochastic suspension of a one-parameter group of diffeomorphisms.

3. Stochastic suspension of one-parameter family of diffeomorphisms

We begin by introducing a useful notion of stochastic suspension of a diffeomorphism.Definition 3.10: Let � :Rd→Rd be a diffeomorphism. The stochastic suspension of � is the

mapping � : P→P defined by

∀X � P, ��X�t�� = ��Xt�� . �3.15�

In what follows, we will frequently use the same notation for the suspension of a givendiffeomorphism and the diffeomorphism.

Remark 3.3: It seems strange that we have not defined directly the notion of diffeomorphismon a subset �� P, i.e., mapping � : �→ � which are Fréchet differentiable with an inverse which isalso Fréchet differentiable. However, these objects do not always exist.

Using the stochastic suspension, we are able to define the notion of stochastic suspension fora one-parameter group of diffeomorphisms.

Definition 3.11: A one-parameter group of transformations �s :�→�, s�R, where �� P, iscalled a �-suspension group acting on � if there exist a one-parameter group of diffeomorphisms�s :Rd→Rd, s�R, such that for all s�R, we have

�i� �s is the stochastic suspension of �s; and�ii� for all X��, �s�X��.

This notion of suspension group comes from our framework. It relies on the fact that we wantto understand how symmetries of the underlying Lagrangian systems are transported via thestochastic embedding. The nontrivial condition on the stochastic suspension of a one-parametergroup of diffeomorphisms acting on � comes from condition �ii�. However, imposing someconditions on the underlying one-parameter group, we can obtain a stochastic one-parametergroup which acts on the set �d of good diffusion processes.

Precisely, let us introduce the following class of one-parameter groups:Definition 3.12: An admissible one parameter group of diffeomorphisms �= �s�s�R is a one

parameter group of C2 diffeomorphisms on Rd such that

�s,x� � �s�x� is C3, �3.16�

and such that for all s�R, all k� 1, . . . ,d�, the kth component �s�k� of �s belongs to T, where T

is the set of all f �C1,2�I�R� such that for all X�� formula �1.17� holds.The condition �s

�k��T may seem to be restrictive, but it is satisfied for affine diffeomorphismof Rd. Such examples turn out to be important in classical mechanics as regards the conservationof momentum and angular momentum. It will be treated in this chapter from the stochastic pointof view.

The main property of admissible one-parameter groups is the fact that they are well behavedon the set of good diffusions.

Lemma 3.4: Let �= ��s�s�R be a stochastic suspension of an admissible one-parametergroup of diffeomorphisms. Then, for all X��d, we have for all t� I, and all s�R.

�i� The mapping s�D��sX�t��C1�R� �a.s.�.�ii� We have �� /�s��D��s�X��=D��s�X� /�s� �a.s.�.

This lemma extends the classical case where X is a smooth function and D� is the classicalderivative with respect to time. In both cases, it reduces to the Schwarz lemma. This equality playsan essential role in the derivation of the classical Noether’s theorem �see, Ref. 5, p. 89�.



Proof: �s�k��T, so we write formula �1.17�,

D��s�k��X��t� = D�X�t� · ��s

�k��X�t� +i�

2aij�ij

2 �s�k�X�t� �a.s.� .

So

�sD��s�k��X��t� = D�X�t� · ��s � �s

�k��X�t� +i�

2aij��s�ij

2 �s�k��X�t� �a.s.� .

Since �s ,x��s�x� is C3, we have �s��s�k�=��s�s

�k� and �s�ij2 �s

�k�=�ij2 �s�s

�k� by the Schwarzlemma.

We deduce that �s�s�k��T, and we can conclude that

�

�s�D��s�X�� = D�� s�X�

�s� �a.s.� .

�

It must be pointed out that every extension of this lemma will lead to a substantial improve-ment of the following stochastic Noether theorem.

4. Linear tangent map

Let X�C1�I� and � :Rd→Rd be a diffeomorphism. The image of X under the stochasticsuspension of �, denoted by �, induces a natural map for tangent vectors denoted by �*, calledthe linear tangent map, and defined as in classical differential geometry by the following

Definition 3.12: Let � be a stochastic suspension of a diffeomorphism � such that its kthcomponent ��k� belongs to T. The linear tangent map associated with �, and denoted by �*, isdefined for all X�C1�I� by

�*�X� = T��X�� = ��X�,D��X�� . �3.17�

5. Invariance

We then obtain the following notion of invariance under a one-parameter group of diffeomor-phisms.

Definition 3.13: Let �= �s�s�R be an admissible one-parameter group of diffeomorphismsand let L be a Lagrangian L :C1�I�→CC

1�I�. The functional L is invariant under � if

L��*X� = L�X� for all � � � .

As a consequence, if L is invariant under �, we have

L��s�X�,D��s�X�� = L�X,DX�

for all s�R and X�C1�I�.Remark 3.3: We note that this notion of invariance under a one-parameter group of diffeo-

morphisms does not coincide with the same notion as defined by Yasue [Ref. 66, p. 332, formula(3.1)] which in our notation is given by

L��s�X�,�s�DX�� = L�X,DX� for all s � R and X � C1�I� .

In fact, Yasue’s definition of invariance does not reduce to the classical notion (see, for example,Ref. 5, p. 88) for differentiable deterministic stochastic processes.

Moreover, Yasue’s definition is not coherent with the invariance notion used in his proof of thestochastic Noether’s theorem (Ref. 66, Theorem 4, p. 332) (see the comment below).



6. The stochastic Noether’s theorem

Noether’s theorem has already been generalized a great number of times and covers some-times different statements.28 Here, we follow Arnold’s �Ref. 5, p. 88� presentation of the Noethertheorem for Lagrangian systems. We correct a previous work of Yasue �Ref. 66, Theorem 4, p.332–333�.

Theorem 3.3: Let L be an admissible Lagrangian with all second derivatives bounded andinvariant under the admissible one-parameter group of diffeomorphisms �= �s�s�R. Let FI be theassociated functional defined by Eq. (3.2) on �. Let X��L be a C1�I�-stationary point of FI.Then, we have

d

dtE��vL · �Y

�s

s=0� = 0,

where

Ys = �s�X� . �3.18�

Proof: Let Y�s , t�=�sX�t� for s�R and a tb.As L is invariant under �= �s�s�R, we have

�

�sL�Y�s,t�,D�Y�s,t�� = 0 �a.s.� .

As Y�· , t� and D�Y�· , t� belong to C1�R� for all t� �a ,b� by Definition 3.11, �ii�, we obtain

�xL ·�Y

�s+ �vL ·

�D�Y

�s= 0 �a.s.� . �3.19�

Using Lemma 3.4, �ii�, this equation is equivalent to

�xL ·�Y

�s+ �vL · D�� Y

�s� = 0 �a.s.� . �3.20�

As X=Y�s=0 is a stationary process for Ja,b, we have

�xL = D−��vL . �3.21�

As a consequence, we deduce that

��D��vL� ·�Y

�s+ �vL · D�� Y

�s��

s=0= 0 �a.s.� .

Taking the absolute expectation, we obtain

E� ��D��vL� ·�Y

�s+ �vL · D�� Y

�s��

s=0� = 0. �3.22�

Using the product rule, we obtain

d

dtE��vL · �Y

�s

s=0� = 0,

which concludes the proof�



7. Stochastic first integrals

The previous theorem leads us to the introduction of the notion of first integral for stochasticLagrangian systems.21

(a) Reminder about first integrals. Let X be a Ck vector field or Rd, k�1 �k could be � or �,i.e., analytic�. We denote by �x�t� the solution of the associated differential equation, such that�x�0�=x and by S the set of all these solutions.

A first integral of X is a real valued function f :Rd→R such that for all �x�t��S, we have

f��x�t�� = cx, �3.23�

where cx is a constant.We have not imposed any kind of regularity on the function f , so that f can be just C0. In this

case, the existence of a first integral does not impose many constraint on the dynamics.If f is at least C1, then we can characterize first integrals by the following constraint:

X · f = 0. �3.24�

(b) Stochastic first integrals. The previous paragraph leads us to searching for an analog of theclassical notion of first integrals as a functional defined on the set of solutions of a given SELequation22 and real valued. Looking for the stochastic Noether theorem, we choose the followingdefinition.

Definition 3.14: Let L be an admissible Lagrangian system. A functional I :L2��→C is afirst integral for the E-L equation associated with L if

d

dt�I�X�t�� = 0 �3.25�

for all X satisfying the E-L equation.We can now interpret the stochastic Noether theorem in terms of first integrals, i.e., the fact

that the invariance of the Lagrangian L under of a one-parameter group of diffeomorphisms �= ��s�s�R induces the existence of a first integral for the associated E-L equation, defined by

I�X�t�� = E��vL · ��sX�t��s

s=0� = 0. �3.26�

8. Examples

(a) Translations. We follow the first example given by Arnold �Ref. 5, p. 89� for the Noethertheorem. Let L be the Lagrangian defined by

L�X,V� =V2

2− U�X� where X � R3, �3.27�

V= �V1 ,V2 ,V3��C3, V2ªV1

2+V22+V3

2 and U is taken to be invariant under the one-parametergroup of translations

�s�x� = x + se1, �3.28�

where e1 ,e2 ,e3� is the canonical basis of R3.Then, by the Stochastic Noether’s theorem, the quantity

21Of course, one can extend this definition to general stochastic dynamical systems.22Of course, this definition will extend to arbitrary stochastic dynamical systems.



E�DX1� �3.29�

is a first integral since �VL=V and �s�s�X1��=e1.(b) Rotations. We keep the notations of the previous paragraph. We consider the Lagrangian of

the two-body problem in R3, i.e.,

L�X,V� = q�V� −1

�X�where q�V� =

V2

2, �3.30�

where �¯� denotes the classical norm on R3 defined for all X�R3, X= �X1 ,X2 ,X3� by �X�2=X12

+X22+X3

2.We already know that the classical Lagrangian L is invariant under rotations when X�R3 and

V�R3. Here, we must prove that the same is true for the extended object, i.e., for L defined overR3 \ 0��C3. This extension, as long as it is defined, is canonical. Indeed, we define q�z� for z�C3 as

q�z� =1

2�z1

2 + z22 + z3

2�, z = �z1,z2,z3� � C3. �3.31�

Note that our problem is not to discuss an analytic extension of the real valued kinetic energy butonly to look for the same function on C3 simply replacing real variables by complex one. As longas the new object is well defined this procedure is canonical, which is not the case if we search foran analytic extension of q over C3 which reduces to q on R3.

Our main result is then that this group of symmetry is preserved under stochastization, whichis, in fact, a general phenomenon that will be discuss elsewhere.

Lemma 3.5: The Lagrangian L defined over R3 \ 0��C3 is invariant under rotations ��,k

around the ek axis by the angle �, k=1,2 ,3.The proof is based on the two following facts.

�a� As ��,k is a linear map whose matrix coefficients do not depend on t, we have

D��,k�X�� = ��,k�D�X� , �3.32�

where ��,k is trivially extended to C3.�b� A simple calculation gives

∀z � C3, q��,k�z�� = q�z� . �3.33�

We easily deduce the ��,k invariance of L, i.e., that

L��,kX,D��,kX�� = L�X,DX� . �3.34�

We now compute ��,k�X��=0=ek∧X and

�VL�X,DX� · ��,k�X��=0 = �X ∧ DX�k.

Therefore the expectation of the “complex angular momentum” X∧DX is a conserved vector�∧ is extended in a natural way to complex vectors�.

9. About first integrals and chaotic systems

In this section, we discuss some consequences of the stochastic Noether’s theorem in thecontext of chaotic dynamical systems. The study of deterministic chaotic dynamical systems isdifficult.

Here again, we return to the classical n-body problem, n�3. In this case, in particular, forlarge n, the dynamics of the system is very complicated and only numerical results give a globalpicture of the phase space. Despite the existence of a chaotic behavior, there exist several wellknown first integrals of the system.



These integrals are used as constraints on the dynamics and can give interesting results, as, forexample, Laskar’s34 approach to the Titus-Bode law for the repartition of the planets in the solarsystems and extrasolar systems.

Using our approach, we can go further by claiming that such kind of integrals continue toexist even if one consider a more general class of perturbations including stochasticity. We notethat this result is fundamental as long as one wants to relate numerical computations on the n-bodyproblem with the real dynamical behavior of the solar systems, and in this particular example, thedynamics of the protoplanetary nebulas.

D. Natural Lagrangian systems and the Schrödinger equation

In this section, we explore in details the stochastization procedure for natural Lagrangiansystems. In particular, by introducing a suitable analog of the action functional, we prove that theSEL equation leads to a nonlinear Schrödinger equation, depending on a free parameter related toa normalization constraint. For a suitable choice of this parameter we then obtain the classicallinear Schrödinger equation.

1. Natural Lagrangian systems

In Ref. 5 �p. 84� Arnold introduces the following notion of natural Lagrangian systems.Definition 3.15: A Lagrangian system is called natural if the Lagrangian function is equal to

the difference between kinetic and potential energies,

L�x,v� = T�v� − U�x� .

As an example, we have the natural Lagrangian function associated with Newtonian mechanics,

L�x,v� =1

2v2 − U�x� ,

where U is of class C�.

2. Schrödinger equations

(a) Some notations and a reminder of the Nelson wave function. We recall that �2 is the set of“good” diffusion processes for which we can compute the stochastic derivative of second order.Let �g

2 be the subset of �2 whose elements have a smooth gradient drift, and let �g,�2 be the subset

of �g2 whose elements have a constant matrix diffusion �Id. We set

S = X � �2�D2X�t� = − �U�X�t�� .

For a diffusion X in �2 with drift b and density function pt�x�, we set

� = �R+ � Rd� \ �t,x�, �pt�x� = 0� . �3.35�

If X��g,�2 then there exist real valued functions R and S smooth on � such that

DX�t� = �b −�2

2� log�pt� + i

�2

2� log�pt��X�t�� = ��S + i � R��X�t�� , �3.36�

since b is a gradient. Obviously,

R�t,x� =�2

2log�pt�x�� . �3.37�

In this case, we introduce the function



��t,x� = e�R+iS��t,x�/K �3.38�

�where K is a positive constant� called the wave function.The wave function has the same form than that of Nelson one �see Ref. 47�. We then set A

=S− iR. So �=eiA/K and �A�t ,X�t��=DX�t�. For a suitable K, Nelson shows that if X satisfies itsstochastized Newton equation �which is the real part of ours� then � satisfies a Schrödingerequation. We show, by using our operator D, the same kind of result in the next section.

(b) Schrödinger equations as necessary conditions.Theorem 3.3: If X�S��g,�

2 , then the wave function �Eq. �3.3�� satisfies the following non-linear Schrödinger equation on the set �:

iK�t� +K�K − �2�

2

��x��2

�+

�2

2�� = U� , �3.39�

Proof: As U is a real valued function, X�S implies

D2X�t� = − �U�X�t�� .

The definition of � implies that on �

�A = − iK��

�.

Since �A�t ,X�t��= �DX��t�, we obtain

iKD�x�

��t,X�t�� = �U�t,X�t�� .

Therefore, considering the kth component of the last equation and using Lemma 1.4, we deduce

iK��t�k�

�+ DX�t� · �

�k�

�− i

�2

2�

�k�

��t,X�t�� = �kU�X�t�� .

Now DX�t�=−iK�� /��t ,X�t��. Thus, by the Schwarz lemma, we obtain

DX�t� · ��k�

�= − iK

j=1

d� j�

�� j

�k�

�= −

iK

2�k

j=1

d � � j�

��2

and

��k�

�=

j=1

d

� j2�k�

�= �k

j=1

d

� j� � j�

�� = �k

j=1

d� j

2�

�− � � j�

��2

.

Therefore

iK�k� �t�

�+ i

�2 − K

2�k

j=1

d � � j�

��2

− i�2

2

��

�� t,X�t�� = �kU�X�t�� .

By adding an appropriate function of t in S, we can arrange the constant in x of integration inequation to be zero, and formula �3.6� follows as claimed. �

In order to recover the classical linear Schrödinger equation, we must choose the normaliza-tion constant K. The main point is that in this case, we obtain a clear relation between the modulusof the wave function and the density of the underlying diffusion process. Precisely, we have thefollowing.

Corollary 3.1: We keep the notations and assumptions of Theorem (3.3). We assume that



K = �2.

Then the wave functional � satisfies the linear Schrödinger equation,

i�2�t� +�4

2�� = U� . �3.40�

Moreover, if pt�x� is the density of the process X�t� at point x, then we have

��t,x� = pt�x� .

Proof: K=�2 kills the nonlinearity in Eq. �3.6� and furthermore

log�� =2

KR =

2

�2R = log�p� ,

which concludes the proof. �

(c) Remarks and questions.

�a� Obviously, in dimension 1, ��2 ��g,�

2 since b is continuous.�b� A natural question is to know if the converse of Corollary 3.1 is true. More precisely, if �

satisfies a linear Schrödinger equation, can we construct a unique process X which belongsto S��g,�

2 and whose density is such that pt�x�= ��t ,x��2? One can show that this processmust have the form of the one discovered by Nelson.47

Carlen8 and Carmona9 tackled the problem of the existence of the so-called Nelsonprocesses and proved under some conditions the existence of a process X with a gradientdrift related to � and whose density is such that pt�x�= ��t ,x��2. However, we do not knowif the second order stochastic derivatives of such a process exist, but we can prove formally,i.e., even so assuming that the formulas of the stochastized derivative to a function of theprocess holds, that X satisfies the Newton stochastized equation. Therefore, this leads one toquestion the extension of the derivative operator and the way it acts on a large class ofprocesses. This problem will be treated in a forthcoming paper �see Ref. 16�. Finally, wepoint out that Wu65 analyzed the various notions of uniqueness of the Nelson processes.

�c� The fact that a process X satisfies the stochastized Newton equation of Nelson implies�D2−D*

2�X=0 �for the potential U is real�. This is a general fact for diffusion with gradientdrift. Indeed, we can prove the following.Lemma 3.6: Let X��d, b its drift, and p its density function. Let Gi be the ith column of thematrix �Gij�ª �� jbi−�ibj�. Then �D2−D*

2�X=0 if and only if for all t�0, div�ptGi�=0.Thus, if X��g,�

2 it is clear that �D2−D*2�X=0 since the form bk�k is closed and so G

=0. An interesting question is then to know if the converse is true. So we may wonderourselves if S��g,�

2 .The difficulty relies on the fact that p and b are related via the Fokker-Planck equation, so

the condition div�ptGi�=0 may not be the good formulation. In Ref. 18, Darses and Nourdingive a proof of the converse based on the results of the martingale problem of Stroock andVaradhan. Actually, one has a characterization of a gradient diffusion via the dynamicalequation �D2−D*

2�X=0. One may also find in the work of Roelly and Thieullen54 an inter-esting characterization of the gradient drift property via an integration by parts of MalliavinCalculus.

�d� A basic notion in mechanics is that of action �see Ref. 5, p. 60�. The action associated witha Lagrangian system is in general obtained via the action functional. In our framework, anatural definition for such an action functional is given by the following.

Definition 3.16: Let A be the functional defined on �a ,b��C1��a ,b�� by

∀t � �a,b�, ∀ X � C1�I�, A�t,X� = E��a

t

L�Xs,�DX�s�ds�Xt� . �3.41�



This functional is called the action functional.Using this action functional, we have some freedom to define the corresponding “action.” The

natural one is defined by

AX�t,x� = E��a

t

L�Xs,�DX�s�ds�Xt = x� . �3.42�

Usually, the wave function associated with AX and denoted by � is then defined as

�X�t,x� = eiAX�t,x�. �3.43�

However, it is not at all clear that such kind of function satisfies the gradient condition, i.e., that

�A�t,X�t�� = DX�t� , �3.44�

which is fundamental in our derivation of the Schrödinger equation.However, condition �3.44� is equivalent to prove that the real part of DX is a gradient, which

is not at all trivial in dimension greater than 2.

3. About quantum mechanics

Even if we look for dynamical systems, our work can be used in the context of the so-calledStochastic mechanics, developed by Nelson.47 The basic idea is to reexpress quantum mechanics�see Ref. 39� in terms of random trajectories. We refer to Ref. 10 for a review.

The stochastic embedding theory can be seen as a quantization procedure �see Refs. 40 and59�, i.e., a formal way to go from classical to quantum mechanics. This approach is alreadydifferent from Nelson’s approach, which do not define a rigid procedure to associate with a givenequation a stochastic analog. Moreover, the acceleration defined by Nelson as

a�X� =DD*�X� + D*D�X�

2�3.45�

is only a particular choice. Many authors have tried to justify this form �Refs. 52, 53, 68, and 69�or to try another one. In our context, the form of the acceleration is fixed and corresponds, as inthe usual case, to the second �stochastic� derivative of X. As a consequence, stochastic embeddingscan be used to provide a conceptual framework to stochastic mechanics. We refer to Ref. 52 wherea complex valued velocity for a stochastic process is introduced corresponding to the stochasticderivative of X.

However, stochastic mechanics as well as its variants have many drawbacks with respect tothe initial wish to describe quantum mechanical behaviors. We refer to Refs. 48 and 10 for details.This is the reason why we will not develop further this topic.

E. Stochastic Hamiltonian systems

In this part, we introduce the stochastic pendant of Hamiltonian systems for classical Lagrang-ian systems. The strategy is first to define the stochastic analog of the classical momentum. Wethen define a stochastic Hamiltonian. However, this Hamiltonian is not obtained by the classicalstochastic embedding procedure. This is due to the fact that the momentum process is complexvalued. As a consequence, we must modify the procedure in order to obtain a coherent picturebetween the classical formalism and the stochastic one. This leads us to define the stochasticHamiltonian embedding procedure which reflects, in fact, the nontrivial character of the underly-ing stochastic symplectic geometry to develop. Having the stochastic Hamiltonian we prove aHamilton LAP using our stochastic calculus of variations. We then obtain an analog of the La-grangian coherence lemma in this case up to the fact that the underlying stochastic embeddingprocedure is now the Hamiltonian one.



1. Reminder about Hamiltonian systems

We denote by I an open interval �a ,b�, ab.Let L :Rd�Rd�R→R be a convex Lagrangian. The Lagrangian functional over C1�R� is

defined by

L:�C1�R� →C1�R�

x �L�x, x,t� .� �3.46�

We can associate with L a Hamiltonian function using the Legendre transformation �Ref. 5, p.65�. From the functional side, this induces a change of point of view, as the functional is not seenas acting on x�t�, which is the so-called configuration space of classical mechanics, but on�x�t� , x�t�� which is associated with the phase space. This dichotomy between position and veloci-ties has of course many consequences, one of them being that the system is more symmetric �thesymplectic structure�.

Definition 3.17: Let L�x ,v� be an admissible Lagrangian system. For all x�C1, we denote by

p�x� =�L

�v�x, x� , �3.47�

the momentum variable.We now introduce an important class of Lagrangian systems.Definition 3.18: Let L�x ,v� be an admissible Lagrangian system. The Lagrangian L is said to

possess the Legendre property if there exists a function f :Rd→Rd, called the Legendre transform,such that

x = f�x,p� �3.48�

for all x�C1.Most classical examples in mechanics possess the Legendre property. This follows from the

convexity of L in the second variable �see Ref. 5, p. 61 and 62�.We can introduce the fundamental object of this section:Definition 3.19: Let L be an admissible Lagrangian system which possesses the Legendre

property. The Hamiltonian function associated with L is defined by

H�p,x� = pf�x,p� − L�x, f�x,p�� , �3.49�

where f is the Legendre transform.The Hamiltonian function plays a fundamental role in classical mechanics. We introduce the

stochastic analog in the next section.

2. The momentum process

A natural stochastic analog of the momentum variable is defined as follows.Definition 3.20: Let L�x ,v� be an admissible Lagrangian system. For all X�C1�I�, we define

the stochastic process P�t�, called the canonical momentum process, by

P�t� =�L

�v�X�t�,DX�t�� . �3.50�

This definition can be made more natural using the embedding defined from C0�I� on Pdet

and the linear tangent map introduced in Sec. III C. Indeed, the momentum process can be viewedas a functional on X�C1�I�, P :C1�I�→PC defined by Eq. �3.39�. We have for all X� Pdet

1

= �C1�I��,

P�X� = �p�x�� , �3.51�

where x�C1�I� is such that X= �x�. As by definition, we have



�p�x�� = p� �x�� = p�X� . �3.52�

As p keeps a sense for X�C1�I�, we extend formula �3.41� to C1�I� leading to Definition 3.20.If we assume that the Lagrangian possesses the Legendre property, then there exists a Leg-

endre transform f such that for all x�C1, x= f�x , p�. We can ask if such a property is conserved forthe momentum process. We have the following.

Lemma 3.6: Let L�x ,v� be an admissible Lagrangian system possessing the Legendre prop-erty. Let f be the Legendre transform associated with L. We have

DX�t� = f�X,P� �3.53�

for all X�C1�I�.We can now define the stochastic Hamiltonian associated with L.Definition 3.21: Let L�x ,v� be an admissible Lagrangian system possessing the Legendre

property. The stochastic Hamiltonian system associated with L is defined by

H:�PC � C1�I� →PC

�P,X� �Pf�X,P� − L�X, f�X,P�� .� �3.54�

3. The Hamiltonian stochastic embedding

As in the previous chapter, we want to use the stochastic embedding procedure to associate anatural stochastic analog of the Hamiltonian equations. However, we must be careful with such aprocedure, as already discussed in Sec. II A. Indeed, the embedding procedure does not allow usto fix the notion of embedding for systems of differential equations. Moreover, we must keep inmind that the principal idea behind the Hamiltonian formalism is to work not in the configurationspace, i.e., the space of positions, but in the phase space, i.e., the space of positions and momenta.As the stochastic speed is by definition complex, this induces a particular choice for the embed-ding procedure in the case of Hamiltonian differential equations.

Definition 3.22: Let F :Rd�Cd�C be a holomorphic function, real valued on real arguments.This function defines a real valued functional over C1�I��C1�I�, for I a given open interval of R.The Hamiltonian embedding of the functional F is the functional denoted by FS, defined onC1�I�� PC�I� by H, i.e.,

FS�X,P��t� = F�X�t�,P�t�� . �3.55�

We denote by SH the procedure associating the stochastic functional FS to F. This procedurereduces to change the functional spaces for F from C1�I��C1�I� to C1�I�� PC.

The main property of the Hamiltonian stochastic embedding procedure �and, in fact, it can beused as a definition� is to lead to a coherent definition with respect to the momentum process.Precisely, we have the following.

Lemma 3.6: �Legendre coherence lemma� Let L�x ,v� be an admissible Lagrangian systempossessing the Legendre property. The following diagram commutes:

�x,p� →H

H�x,p�

SH↓ ↓SH

�X,P� →HS

H�X,P�

. �3.56�

The proof follows essentially from the fact that the stochastic Hamiltonian embedding of thefunctional H, denoted by HS, coincides with Definition 5.4 of the stochastic Hamiltonian systemassociated with H via the Legendre transform and the definition of the momentum process.



4. The Hamiltonian least action principle

Using the stochastic Hamiltonian function, we can use the stochastic calculus of variations inorder to obtain the set of equations which characterize the stationary processes of the followingfunctional:

Ja,b�X,P� = E��a

b

�P�t�DX − H�X�t�,P�t��dt� , �3.57�

defined on a domain ��C1�I��PC.In order to apply our stochastic calculus of variations, we restrict our attention to J on

��C1�I��CC1�I�. The fundamental result of this section is the following.

Theorem 3.4: A sufficient condition for an L-adapted process �X , P� to be N1�I�-criticalprocess of the functional Ia,b is that it satisfies the stochastic Hamiltonian equations

DX =�H

�P�X�t�,P�t�� ,

DP = −�H

�X�X�t�,P�t�� . �3.58�

Proof: We must use the weak LAP using the process Z= �X , P��C1�I��PC and the Lagrang-ian denoted by L defined on Rd�Cd�Cd�Cd by

L�x,p,v,w� = pv − H�x,p� . �3.59�

As L�x , p ,v ,w�=L�x ,v� formally via the Legendre transform, and L is assumed to be admissible,we deduce that L is again admissible.

Let �Z be a N1�I� variation of the form Z+�Z= �X+X1 , P+ P1�, where X1 and P1 are N1

processes.Using Lemma 3.4 a sufficient condition for Z to be a N1-critical process is that it satisfies the

EL equation associated with L given by

�L�x

�Z�t�,DZ�t�� − D�� L�v

�Z�t�,DZ�t�� = 0,

�L�p

�Z�t�,DZ�t�� − D�� L�w

�Z�t�,DZ�t�� = 0. �3.60�

An easy computation leads to

−�H

�x�Z�t�,DZ�t�� − D�P�t� = 0,

DX�t� −�H

�p�Z�t�,DZ�t�� = 0. �3.61�

This concludes the proof. �

Remark 3.4: In this proof we do not need a uniform assumption on the set of variations as theLagrangian does not depend on the variable w. In fact, we can assume a variation in the directionP which belongs to C1�I�.



5. The Hamiltonian coherence lemma

In this section, we derive the Hamiltonian analogue of the Lagrangian coherence lemma.Lemma 3.7: �The Hamiltonian cohrence lemma� Let H :Rd�Rd→R be an admissible Hamil-

tonian system. Then, the following diagram commutes:

H�x�t�,p�t�� →SH

H�X�t�,P�t��

LAP↓ ↓SLAP

�HE� →SH

�SHE�

. �3.62�

The main point is that this result is not valid if one replaces the Hamiltonian stochasticembedding by the natural stochastic embedding that we have used up to now. We can keep theclassical embedding procedure only when dealing with real valued versions of the stochasticderivative. For example, if one deals with the reversible stochastic embedding procedure, weobtain a unified stochastic embedding procedure for both Lagrangian and Hamiltonian systems.We think, however, that as well as the complex nature of the stochastic derivative has a funda-mental influence on the form of the stochastic Lagrangian equations, i.e., that we obtain theNelson acceleration, the fact to move from S to SH reflects a basic properties of the underlyingstochastic symplectic geometry we must take into account this complex character of the speed.This problem will be studied in another paper.

Conclusion and perspectives. This part aims at discussing possible developments and appli-cations of the stochastic embedding procedure.

6. Mathematical developments

(a) Stochastic symplectic geometry. The Hamiltonian formalism developed in the last partsuggest the introduction of what can be called a stochastic symplectic geometry. An interestingconstruction of symplectic structures on Hilbert spaces is given in Ref. 29.

The main point here is to construct an analog of the geometrical structure which puts inevidence the very particular symmetries of the Lagrangian equations in classical mechanics. Thereexist already many attempts to construct a given notion of symplectic geometry or at least a givengeometry for stochastic processes, but they are as far as we know of a different nature. We referto the book of Elworthy et al.21 for an overview. These geometries are only associated withstochastic processes and translate into data of geometrical nature properties of the underlyingstochastic processes �such as the Riemannian or sub-Riemannian structure associated with Brown-ian motions and diffusions�.

A recent work of Zambrini and Lescot36,37 deals specifically with symplectic geometry and anotion of integrability by quadratures.

For a discussion of integrability in our context see Sec. III E 6 �b�.(b) PDE’s and the stochastic embedding. The stochastic embedding of Lagrangian systems

over diffusion processes lead to a PDE governing the density of the solutions of the SEL equation.Moreover, we have defined a stochastic Hamiltonian system naturally associated with the La-grangian. However, some classical PDEs as, for example, the Schrödinger equation, possess anHamiltonian formulation. This remark, which goes back to the work of Zakharov and Faddeev67 isnow an important subject in PDEs known as Hamiltonian PDEs �see, for example, Ref. 29�. As aconsequence, we have the following situation:

HS

↓PDE → H

. �3.63�

Of course the relation between the PDE and HS is not of the same nature as the relation with H.In the sequel, we list a number of problems and questions which naturally arise from the

previous diagram:



—There exists a notion of completely integrable Hamiltonian PDE �see Ref. 29�. What aboutout stochastic Hamiltonian systems?

Assuming that we have a good notion of integrability for HS, we have the following questions:—Are there any relations between the integrability of H and HS?—Is there a stochastic analog of the Arnold-Liouville theorem?—Is there a special set of “coordinates” similar to the action/angle variables?We note that there already exists such a notion for Hamiltonian PDEs �see Ref. 67�.—Is there a notion of integrability by “quadratures?”In that respect, we think about the Lax work35 on the integrability of PDEs.

7. Applications

(a) Long term behavior of chaotic Lagrangian systems. The dynamical behavior of unstable orchaotic dynamical systems is far from being understood, unless we restrict to a very particularclass of systems such as hyperbolic systems or weak version of hyperbolicity. This question arisesnaturally for small perturbations of Hamiltonian systems for which there exists a large family ofresults dealing with this problem as, for example, the Kolmogorov-Arnold-Moser �KAM� theo-rem, Nekhoroshev theorem, and special phenomena such as the Arnold diffusion related to theso-called quasiergodic hypothesis.

Unfortunately, these results are difficult to use in concrete situations and only direct numericalsimulations provide some understanding of the dynamics.20

There exists of course ergodic theory �see Ref. 27� which tries to look for weaker informationon the dynamics than a direct qualitative approach. However, this theory leads also to verydifficult problems when one tries to implement it, as, for example, in the case of Sinaï billiard.Moreover, there is a wide opinion in the applied community that the long term behavior of achaotic systems is more or less equivalent to a stochastic process �see Ref. 50�. One example ofsuch opinion is well expressed in the article of Laskar34 in the context of the chaotic behavior ofthe solar system: “Since the characteristic time scale for the divergence of nearby orbits in thesolar system is approximately 5 Myr, the orbital evolution of the planet becomes practicallyunpredictable after 100 Myr. Thus in the long term, the motion of the solar system may bedescribed by a random process, where orbits wander erratically in a chaotic zone.”

What are the arguments leading to this idea? The first point is that chaotic dynamical systemsare in general characterized by the so-called sensitivity to initial conditions, meaning that a smallerror on the initial condition leads to very different solutions. Of course, one must quantify thiskind of sentence, and we can do that, with more or less canonicity, by introducing Lyapounovexponents and Lyapounov time. Whatever we do, there are a noncanonical data in this, which isprecisely to what extent we consider that two solutions are different. This must be a matter ofchoice for a given system and cannot be fixed by any mathematical tool. In the sequel, we assumethat a system is sensitive to initial conditions in some region R of the phase space, and for a givenmetric, if for all x0�R and all ��0, the distance at time t between a trajectory starting at x0 andx0+�, denoted by d�t� is23 approximately given by

d�t� = �et/T, �3.64�

where T�0 is the so-called Lyapounov time or horizon of predictability for the system.24 For anexample of such an estimate, we refer to Laskar33 where he gives numerical evidences for thechaotic behavior of the solar system.

As a consequence, for t sufficiently large with respect to T, we have no prediction any more,or in other words, we cannot assign to a given prediction a precise initial condition. We then have

23As we already stress, we can in some situations give a precise meaning to all this point, for example, in the SmaleHorseshoes, but this is far to cover the wide variety of chaotic behavior which are studied in the applied literature.24In concrete systems, one must involve a macroscopic scale �see Ref. 19, p. 17�, which bound the admissible size of anerror on a prediction. Here, this quantity is arbitrary replaced by e.



lost the deterministic character of the equations of motions. An idea is then to say that one mustthen consider not a fixed initial condition x0, but a given random variable representing all thepossible behaviors �kind of trajectories� one is lead to after a fixed time t: for example, ��0 beingfixed, we consider all the intersections of trajectories starting in the disk D�x0 ,�� with the ballB�x0 ,��. We then obtain a family of directions. Assuming that we can compute an average over thefamily of such a quantity which obtain an averaged direction which select a given point of the ballB�x0 ,��. We then follow the selected trajectory during the time t and continue again this proce-dure. Such a construction is reminiscent of the classical construction of the Brownian motion �seeRef. 26, p. 66�. Of course, this programme can only be carried in some specific examples. We referto the article of Sinaï55 for a heuristic introduction to all these problems.

If we agree with the previous heuristic idea, one can then ask for the following: how is theunderlying stochastic process governed by the dynamical system?

We return again to the Hamiltonian/Lagrangian case. The stochastic embedding procedureanswers precisely this question. The SEL equation is the track of the underlying Lagrangiansystem on stochastic processes. As a consequence, we can think that we are able to capture eventhe desired long term behavior of the Lagrangian system using this procedure.

In order to support our point of view, we suggest the following strategy. Consider a pertur-bation of a completely integrable Hamiltonian system H��x�=h�x�+�f�x�, with x�R2n, for ex-ample. Let us assume that h�x� leads to a particular PDE under stochastic embedding, which canbe well understood and solved. The long term behavior of the completely integrable Hamiltoniansystem is trivial. This not the case for the stochastic analog. What about the long term behavior ofH�? We think that it is controlled by the stochastic analog of the unperturbed Hamiltonian. Thisresult is related to a kind of stochastic stability which we must define. However, this approach canbe tested on a wide variety of examples, in particular, celestial mechanical problems.

(b) Celestial mechanics. There exist many theories dealing with the problem of the formationof gravitational structures. For planetary systems, this question is related to a long standingproblem related to the “regular” spacing of planets in the solar system. This problem which goesback to Kepler �1595�, Kant �1755�, von Wolf �1726�, and Lambert �1761� and takes a mathemati-cal form under the Titius �1766� formulation of the so-called Titius-Bode law giving a geometricprogression of the distance of the planets from the sun. We refer to the book of Nieto49 for moredetails. Even if this empirical law fails to predict correctly the real distance for the planet Pluto,for example, its interest is that it suggests that the repartition of exoplanet orbital semimajor axescould satisfy a simple law. As a consequence, one searches for a possible physical/dynamicaltheory supporting the existence of such kind of law. Moreover, the discovery of many exoplan-etary systems can be used to test if the theory is based on universal phenomena and not related toour knowledge of the solar system.

All the actual theories about the origin of the solar system presuppose the formation of aprotoplanetary nebula, formed by some material �gas, dust, etc.� with a central body �a star or a bigplanet�. We refer to Lissauer38 for more details.

Instead, we use a simplified model consisting of a large central body of mass m0 with a largenumber of small bodies �mj� j=1,. . .,n, whose mass is assumed to be small with respect to m0. Themain problem is to understand the long term dynamics of this model.

Following the work of Albeverio et al.2 �see also Ref. 3�, we can modelize the motion of agiven grain in the protoplanetary nebula by a stochastic process �see Ref. 2, pp. 366–367�, moreprecisely a diffusion process. The problem is then to find what is the equation governing thedynamics of such a stochastic process. Using our stochastic embedding theory, we can use theclassical formulation in order to obtain the desired equation. This question will be detailed in aforthcoming article.

The main idea behind stochastic modelisation is the following. The motion of a given smallbody in a protoplanetary nebula is given by the Kepler model and a perturbation due to the largenumber of number of small bodies. In Ref. 2, this perturbation is replaced by a white noise. As aconsequence, the movement of a small body is assumed to be described by a diffusion process. Itmust be noted that this assumption is related to a number of arguments, one of them being that the



dynamics of the underlying classical system is unstable. We then return to our previous descriptionof the chaotic behavior of a dynamical system. However, using the stochastic embedding theory,we can try to justify the passage from a classical motion to a stochastic one looking at thefollowing problem.

Let L�=LKepler+ P� be the Lagrangian system describing the dynamics of our model. TheLagrangian LKepler is the classical Lagrangian of the Kepler problem and P� is the perturbation.Using the stochastic embedding theory, we can deduce two stochastic dynamical systems, oneassociated with L� and denoted by S� and one associated with LKepler denoted by SKepler. If theprevious strategy to replace the perturbative effect by a White noise is valid, then we must have akind of stochastic stability between SKepler and S�. The notion of stochastic stability must bedefined rigorously and be consistent with the stochastic embedding theory.25 Why such a stabilityresult is reasonable? The main thing is that we already look in SKepler for statistical properties ofthe set of trajectories of stochastic �diffusion� processes under the Kepler Lagrangian. There is noreason that the statistic of this trajectories really differs when adding a small perturbation. This isof course different if one look for the underlying deterministic system. All these questions will bestudied in a forthcoming paper.

(c) Strange attractors. Strange attractors play a fundamental role in turbulence and lead tomany difficult problems. Most of the time, one is currently interested in the geometrical propertiesof attractors �Hausdorf dimension, etc.� and special dynamical properties �existence of a Sibaï-Ruelle-Bowen �SRB� measure,64 stability under perturbations, etc.�. However, focusing on a givenattractor hides the fact that most of the time we cannot predict from the equation the existence ofsuch an attractor. This is, in particular, the case for the Lorenz attractor or the Henon attractor.These attractors are obtained numerically. In some models, we can construct a geometric modelfrom which we can prove the existence of such a structure �this is the case for the geometricLorenz model�.23 For example, Smale56 asks for an existence proof for the Lorenz equation of theattractor. This has been done recently by Tucker.60,61 However, no general strategy exists in orderto predict such an attractor.

Our idea is to use the stochastic embedding theory in order to predict the existence of such anobject. Let us consider the Lorenz equations. These equations are not a Lagrangian system.However, there exits a canonical embedding in a Lagrangian system �see the report of Audin6�.This Lagrangian can then be studied via the stochastic embedding procedure. The solutions arestochastic processes whose density is controlled by a PDE. As we already explain, we expect thatthe long term behavior of the system is coded by this PDE. As the long term dynamics of theLorenz system if precisely supported by the Lorenz attractor, we think that this structure can bedetected in the PDE �as a stationary state, for example�.

We can also take this problem as a first step toward understanding the existence of coherentstructures in chaotic dynamical systems. Moreover, the Lorenz attractor is widely studied andthere exists a great amount of results such as the existence of a unique SRB measure �see Ref. 61�.We can then take this example as a good system to compare classical methods of ergodic theoryand our approach. For more problems related to the Lorenz attractor, SRB measure, etc., see Refs.62 and 63.

NOTATIONS

d: dimension.�� ,A ,P�: probability space.

A. Stochastic processes

We denote by

25It must be noted that there exists already several notion of stochastic stability in the literature, as for example,Has’inskii,25 Kushner,30 and more recently Handel.24



dX = b�t,X�dt + ��t,X�dW ��

the stochastic differential equation where b is the drift, � the diffusion matrix, and W is ad-dimensional Wiener process defined on �� ,A ,P�.We denote by X�t� the solution of � *� and by pt�x� its density �when it exists� at point x.��Xs ,asb�: the � algebra generated by X between a and b.Ft: an increasing family of � algebras.Pt a decreasing family of � algebras.E�·�B�: the conditional expectation with respect to B�·�: norm on stochastic processes.

B. Functional spaces

PR: real valued stochastic processes.PC: complex valued stochastic processes.Pdet: the set of deterministic stochastic processes.Pdet

k : the set of deterministic stochastic processes of class Ck.�: good diffusion processes.�g: good diffusion processes with a gradient drift.Lp��: set of random variables which belongs to Lp.L2: the set of real valued processes which are Pt and Ft adapted and such that E��0

1Xt2dt�

�.C1,2��0,1��Rd� the set of function which are C1 in the first variable and C2 in the secondone.N1: the set of Nelson-differentiable processes.

C. Operators

�: the gradient operator.�: the Laplacian operator.Let f�x1 , . . . ,xn� be a given function. We denote by �xi

f the partial derivative of f withrespect to xi.Let f�x1 , . . . ,xn ,y1 , . . . ,ym� be a given function. We denote by �xf , x= �x1 , . . . ,xn� the partialdifferential of f in the direction x.D: Nelson forward derivative.D*: Nelson backward derivative.D: the stochastic derivative.Dn, D*

n, Dn: the nth iterate of D, D*, or D.d and d*: adapted forward and backward derivative.

k�1.

Ck: the set of real valued processes which are Pt and Ft adapted and such that Di exists,1 ik.CC

k : the set of complex valued processes which are Pt and Ft adapted and such that Di exists,1 ik.Re�z�: real part of z�C.Im�z�: imaginary part of z�C.

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