On the effect of a noise on the solutions of thefocusing supercritical nonlinear Schrodinger equation

A. de BouardCNRS et Universite Paris-Sud, UMR 8628,

Bat. 425, Universite de Paris-Sud, 91405 Orsay, France

A. DebusscheENS de Cachan, Antenne de Bretagne,

Campus de Ker Lann, Av. R. Schuman, 35170 Bruz, France

Abstract. We investigate the influence of a random perturbation of white noise typeon the finite time blow up of solutions of a focusing supercritical nonlinear Schrodingerequation. We prove that, contrary to the deterministic case, any initial data gives birthto a solution which develops singularities. Moreover, the singularities appear immediately.We use a stochastic generalization of the variance identity and a control argument.

Key-words. Stochastic partial differential equation, nonlinear Schrodinger equation,blow up, variance identity, white noise.

AMS subject classification. 35Q55, 60H15, 76B35.

Introduction

The nonlinear Schrodinger equation occurs as a basic model in many areas of physics :hydrodynamics, plasma physics, nonlinear optics, molecular biology,... It describes thepropagation of waves in media with both nonlinear and dispersive responses.

In this article, we investigate the influence of a noise on the qualitative behavior ofthe solutions of this equation. We are particularly interested in blow-up phenomena. This

1

happens for instance for solutions of the following nonlinear Schrodinger equation withfocusing nonlinearity :

(0.1) idu

dt−(∆u + |u|2σu

)= 0.

The unknown u = u(x, t) is complex valued and depends on x ∈ IRn and t ≥ 0. It iswell known that for an initial data with negative energy :∫

IRn

(12|∇u(x, 0)|2 − 1

2σ + 2|u(x, 0)|2σ+2

)dx < 0,

the solution of (0.1) with 2n ≤ σ < 2

n−2 ( 2n ≤ σ, for n = 1, 2) cannot be smooth for all

time. There exists a t∗ > 0 such that∫IRn

(|∇u(x, t)|2 + |u(x, t)|2

)dx →∞

as t t∗. It can be seen that this implies that the L∞-norm also blows up at t∗ (see [18],chapter 5 and the references therein).

Equation (0.1) is an idealized model and does not take into account many aspects :inhomogeneities, higher order terms, external forces,... It seems reasonable to model theseby a random excitation. For instance in [11], the authors study the influence of an additivenoise. In [2] a real valued additive noise is also mentioned in the context of crystals. Here,we study the following stochastic nonlinear Schrodinger equation

(0.2) idu

dt−(∆u + |u|2σu

)= η

where η is of white noise type and is complex valued.The case where the noise acts as a real potential – i.e. with ηu in the left-hand side

of (0.2), and η real valued – is also of great interest (see [1], [2], [12]) and it is believedthat such a noise has a strong influence on the solutions which blow up. It is expectedthat it may delay or even prevent the formation of a singularity. In [10], some numericalsimulations tend to show that this is the case for a very irregular noise : for a space-timewhite noise. However, for a noise which is correlated in space, it has been observed that, onthe contrary, any solution seems to blow up in a finite time. Recall that in the deterministiccase, only a restricted class of solutions blow up. Our aim is to prove rigorously such abehavior.

2

There are two kinds of difficulties when trying to consider space time white noise. Thefirst one is the spatial irregularity of the correlations (and the lack of smoothing effect inthe Schrodinger equation). The second one is the homogeneity of the noise, when workingon the whole space IRn (see [9] for more details in the case of the Korteweg-de Vriesequation). For those reasons, we restrict our attention to the study of correlated noise.

Even if we start our study with the case of an additive noise and the equation (0.2),most of the ideas used in this paper can be applied for a multiplicative noise, i.e. a noiseacting as a real potential, but some technical problems have to be overcome. This shallbe studied in a future work (see however [7] and [8] for the existence and uniqueness ofsolutions).

Before describing our results, we recall the idea of the proof of finite time blow-up inthe deterministic case. The key ingredient is the so-called variance identity [14], [20], [21].The variance in the context of the deterministic Schrodinger equation is defined by

V (u) =∫

IRn

|x|2|u(x)|2dx.

We do not use the variance in the probabilistic sense here. This should not cause anyconfusion. We also introduce the energy

H(u) =∫

IRn

(12|∇u(x)|2 − 1

2σ + 2|u(x)|2σ+2

)dx,

which is a conserved quantity for (0.1). Then if u is a solution of finite variance of (0.1),it may be proved (see [14]) that

(0.3)V (u(t)) = V (u(0)) +

d

dtV (u(t))

∣∣∣∣t=0

t + 8H(u(0))t2

+4(2− σn)

σ + 1

∫ t

0

(t− s)|u(s)|2σ+2L2σ+2(IRn)ds.

We notice that if σ ≥ 2n the last term is negative for all t ≥ 0, so that if the energy

H(u(0)) is negative, the right-hand side of (0.3) becomes negative at some time t which isimpossible since V (u(t)) is a non negative quantity.

For the stochastic equation (0.2) with a spatially smooth noise, we are able to derivea generalization of the variance identity. This allows us to prove that, if σ ≥ 2/n andunder some conditions on the initial data which are a little more complicated than in thedeterministic theory, blow-up occurs. Blow-up here means that for some time t > 0, either

(0.4) IE∫ t

0

(∫IRn

|∇u(x, s)|2dx +(∫

IRn

|u(x, s)|2σ+2dx

) 2σ+1σ+1 )

ds = +∞,

3

or

(0.5) IP(u(·, s) exists on [0, t]) < 1.

Moreover, under the stronger assumption 2/n < σ < 2/3, we show that necessarily (0.5)happens. Note that this assumption is compatible only when n ≥ 4.

In these results, the noise is considered as a perturbation of the deterministic equationand we deduce that it does not prevent blow-up.

We then use another idea. Roughly speaking, if the noise is non degenerate, equation(0.2) should have some irreducibility properties. Namely, for any time T > 0, final data uT

and initial data u0, the solution of (0.2) such that u(0) = u0 should be close to uT at timeT with positive probability. The idea is to take uT satisfying the assumptions ensuringblow-up before time T , so that the solution u starting from u0 has to blow-up between thetimes T and 2T .

We show that this scenario is correct under some suitable smoothness assumptions onthe initial data and on the noise. In many cases, the assumptions on u0 reduce to H(u0)and V (u0) to be finite, so that we obtain the result that for any initial data, blow-upoccurs in the sense of (0.4) or (0.5) for t > 0 arbitrary. Again, if σ < 2/3, only (0.5)happens. Thus the noise strongly influences this blow up phenomenon. This result is inperfect agreement with the numerical simulations.

We end this introduction by mentioning the articles [16] and [17] where the influenceof a noise on the heat equation is studied. It is shown there that the noise can createsolutions which blow up in the sense (0.5).

1. Notations and main results

We consider a stochastic version of the nonlinear Schrodinger equation. The noiseacts as a forcing term, i.e. it is additive. We introduce a probability space (Ω,F , IP), afiltration (Ft)t≥0 on this space, and a cylindrical Wiener process (W (t))t≥0 on L2 adaptedto this filtration. Alternatively :

(1.1) W (t) =∞∑

i=0

βi(t)ei, t ≥ 0,

where (βi)i∈IN is a sequence of independent Brownian motions adapted to (Ft)t≥0 and(ei)i∈IN is an orthonormal basis of L2. We have denoted by L2 the space of square integrablecomplex valued functions on IRn.

4

The space time white noise on IRn is formally given by the time derivative of W.

Unfortunately, this defines a very irregular process and we are not able to handle such anoise.

Let φ be a linear operator on L2. The noise we consider is formally given by the timederivative of φW :

η = φ∂W

∂t.

Its covariance operator is equal to φ∗φ. An example is provided by the case of an operatorφ defined through a kernel k :

(1.2) φf(x) =∫

IRn

k(x, y)f(y)dy, f ∈ L2, x ∈ IRn.

We can then describe the correlation more precisely :

(1.3) IE(φW (x, t)φW (y, s)) = c(x, y)(t ∧ s),

for x, y ∈ IRn, t, s ∈ IR+, with c(x, y) =∫IRn k(x, z)k(y, z)dz. From (1.3), we have formally

(1.4) IE (η(x, t)η(y, s)) = c(x, y)δt−s.

The Ito form of the stochastic nonlinear Schrodinger equation studied in this work is

(1.5) idu−(∆u + |u|2σu

)dt = φdW, x ∈ IRn, t ≥ 0,

in which ∆ is the Laplace operator and the unknown u is a complex valued function ofx ∈ IRn and t ≥ 0. Equation (1.5) is supplemented with an initial condition

(1.6) u(x, 0) = u0(x), x ∈ IRn.

We introduce the linear group (S(t))t∈IR defined by S(t) = e−it∆, t ∈ IR. We rewrite(1.5)-(1.6) in the mild form

(1.7) u(t) = S(t)u0 − i

∫ t

0

S(t− s)(|u(s)|2σu(s))ds + i

∫ t

0

S(t− s)φdW (s), t ≥ 0,

where the last term is a stochastic Ito integral (see e.g. [6]).We now set some notations used throughout the paper. A norm on a Banach space X

will be denoted by | · |X . The classical Lebesgue spaces of complex valued functions defined

5

on IRn will be denoted by Lp for p ≥ 1; if p = 2, we denote the inner product of the realHilbert space L2 by ( , ) :

(v, w) = Re∫

IRn

v(x)w(x)dx, v, w ∈ L2.

We will also use, for s ∈ IR, the Sobolev spaces Hs of tempered distributions v ∈ S′(IRn)whose Fourier transform v satisfies

(1 + |ξ|2

)s/2v(ξ) ∈ L2.

Note that the group of operators (S(t))t∈IR defined above is a unitary group in Hs

for any s ∈ IR. This is easily seen, using the spatial Fourier transform. It follows that thelinear part of the equation has no smoothing effect in the Sobolev spaces Hs.

For any integer p ≥ 1, W 1,p is the space of functions f in Lp such that any first orderpartial derivative of f belongs to Lp.

If I is an interval of IR, r ≥ 1 and X is a Banach space, Lr(I;X) is the space ofstrongly measurable functions v from I into X such that t 7→ |v(t)|X is in Lr(I). AlsoC(I;X) stands for the space of continuous functions from I into X.

Given two Hilbert spaces H and K, L02(H,K) is the space of Hilbert-Schmidt operators

from H to K with

|φ|2L02(H,K) =

∑i∈IN

|φεi|2K = Tr(φ∗φ) = Tr(φφ∗), φ ∈ L02(H,K),

where (εi)i∈IN is any orthonormal basis of H. We denote by L(H,K) the space of boundedlinear operators from H to K. Note that L0

2(H,K) is continuously embedded in L(H,K).We also use the notion of γ-radonifying operator. When H is a Hilbert and E a Ba-

nach space, φ a linear operator from H into E is γ-radonifying if the image of the canonicalGaussian distribution on H is a Gaussian measure on E. The space of γ-radonifying oper-ators from H into E is denoted by R(H,E) and its norm is

|φ|R(H,E) =

IE

∣∣∣∣∣∞∑

i=1

γiφεi

∣∣∣∣∣2

E

1/2

where (εi)i∈IN is any orthonormal basis of H,(Ω, F , IP

)any probability space and (γi)i∈IN

any sequence of independent normal random variables on this space (see [3],[4]). Note thatif K is a Hilbert space such that K ⊂ E with a continuous embedding then L0

2(H,K) ⊂R(H,E) with a continuous embedding.

6

Our aim in this work is to study the finite time blow-up of solutions and for thispurpose, it is natural to work in the framework of H1 valued solutions. Existence anduniqueness of solutions for equation (1.7) is given by the following result which is provedin [8].

Theorem 1.1. Assume that 0 ≤ σ < 2n−2 if n ≥ 3 or 0 ≤ σ for n = 1, 2, that

φ ∈ L02(L

2,H1) and that the initial data u0 is a F0−measurable random variable with

values in H1; then there exists a unique solution u(u0, ·) to (1.7) with continuous H1

valued paths. This solution is defined on a random interval [0, τ∗(u0, ω)), where τ∗(u0, ω)is a stopping time such that

τ∗(u0, ω) = +∞ or limtτ∗(u0,ω)

|u(u0, t)|H1 = +∞.

Furthermore, τ∗ is almost surely lower semicontinuous with respect to u0.

Since the linear part of the equation has no smoothing effect in the Sobolev spacesHs, the assumption on the covariance operator and on the initial data are necessary. Notethat the assumption on σ is the same as in the deterministic theory (see [5], [13], [15]).

Throughout the article, (ei)i∈IN denotes an arbitrary orthonormal basis of L2 and(βi)i∈IN is a sequence of independent Brownian motions such that W =

∑i∈IN βiei.

If σ < 2n , using Lemma 2.1 below it is not difficult to prove that τ∗(u0, ω) = +∞

almost surely for any u0. Moreover, if φ is only Hilbert-Schmidt from L2 into L2 and u0

is almost surely in L2, we can prove a global existence theorem in L2 provided σ < 2n (see

[19] for the deterministic case, and [7] for the case of a multiplicative noise).On the contrary, if σ ≥ 2

n , we have to work with H1 valued solutions and it is notexpected that τ∗(u0, ω) = +∞ almost surely for any initial data. Indeed it is known in thiscase that some solutions of the deterministic equation develop singularities in finite time(see [18], chapter 5, and the references therein). To study such phenomena, it is convenientto introduce the space

Σ =

v ∈ H1 :∫

IRn

|x|2|v(x)|2dx < ∞

,

endowed with the norm | · |Σ where

|v(x)|2Σ =∫

IRn

|x|2|v(x)|2dx + |v|2H1 .

7

In the deterministic theory, initial data in Σ with negative energy are known to blow up.We need to introduce other spaces. For k, ` ∈ IN we define

Σk,` =v ∈ Hk+` : |x|`v ∈ Hk

endowed with the norm | · |Σk,` :∣∣v∣∣2

Σk,` =∣∣v∣∣2

Hk+` +∣∣|x|`v∣∣2

Hk .

We also introduce the space Sn whose description depends on the space dimension n :

S1 = Σ, S2 =⋃α>0

Σα,1, Sn = Σ1,2 if n ≥ 3.

We can now state our main result.

Theorem 1.2. Assume that 2n ≤ σ < 2

n−2 if n ≥ 3, or 2n ≤ σ if n = 1, 2, that W is a

cylindrical Wiener process on L2, φ is Hilbert-Schmidt from L2 into Sn and kerφ∗ = 0.Then, for any u0 ∈ Σ and any t > 0, denoting by u(u0, ·) the solution given by Theorem

1.1, we have either

(1.8) IP(τ∗(u0) < t) > 0

or

(1.9) IE∫ t

0

(|u(u0, s)|2H1 + |u(u0, s)|4σ+2

L2σ+2

)ds = +∞.

Furthermore, if 2n < σ < min

(2

n−2 , 23

)and φ is γ-radonifying from L2 into L4σ+2, and

bounded from L2 into H2 ∩ L∞, then (1.8) occurs.

This result shows that, contrary to the deterministic case where only a restrictedclass of initial data gives birth to a solution which blows up, in the presence of noise,any initial data yields a singular solution and this happens immediately ; at least in theweaker sense (1.9). We believe that in fact, (1.8) always occurs. This is supported bynumerical experiments which will be presented elsewhere ([10]). We can prove this factonly if 2

n < σ < 23 and under stronger assumptions of the noise. Note that σ < 2

3 andσ > 2

n is possible only for n ≥ 4.

8

All the conclusions of Theorem are still valid in the case of a real valued noise, ifwe assume some extra smoothness on u0. The proof is almost the same, except for thecontrollability argument of Section 3.2, which is much more complicated in the real valuedcase. In order to keep the present paper in a reasonable length, the details of the realvalued case will appear elsewhere.

It follows easily from the arguments below that, under the conditions of Theorem 1.2,we also have for any T1, IP(τ∗(u0) < T1) < 1. The numerical experiments confirm this andindicate that IP(τ∗(u0) < T1) 1 when T1 → ∞. We are however not able to prove thisfact.

We end this section with a remark.

Remark 1.1. If f(x, t) is a complex valued function, then∫IRn

∫ t

0

fφdWdx =∫ t

0

∫IRn

fdxφdW =∑k∈IN

∫IRn

∫ t

0

f(x, s)dβk(s)φek(x)dx,

is well defined as soon as∫ t

0|φ∗f |2L2 ds < ∞ a.s. If moreover IE

∫ T

0|φ∗f |2L2 ds < ∞, then∫ t

0

∫IRn fdxφdW is a square integrable martingale with zero expectation and quadratic

variation∫ t

0|φ∗f |2L2 ds.

2. The stochastic variance identity

In this section, we generalize to the stochastic context an identity which is usuallycalled “the variance identity” concerning Σ-valued solutions of the deterministic nonlinearSchrodinger equation. This is the starting point of the proof of finite time blow up.

The identity is based on the evolution of the following three quantities :the energy

H(v) =12

∫IRn

|∇v(x)|2dx− 12σ + 2

∫IRn

|v(x)|2σ+2dx, v ∈ H1,

the variance1

V (v) =∫

IRn

|x|2|v(x)|2dx, v ∈ Σ,

andG(v) = Im

∫IRn

v(x)x · ∇v(x)dx, v ∈ Σ.

1 This quantity should not be confused with the probabilistic variance

9

We will also use the square L2 norm :

M(v) =∫

IRn

|v(x)|2dx, v ∈ L2.

We set a few lemmas concerning the time evolution of those four quantities whenv = u(t) is a solution of (1.7). The following is proved in [8]. The proof is based onIto formula; many cancellations appear since M and H are conserved quantities for thedeterministic equation.

Lemma 2.1. Let u0, σ and φ be as in Theorem 1.1. For any stopping time τ such that

τ < τ∗(u0) a.s., we have, setting u(·) = u(u0, ·),

(2.1) M(u(τ)) = M(u0)− 2Im∑`∈IN

∫ τ

0

∫IRn

u(x)φe`(x)dxdβ`(s) + |φ|2L02(L

2,L2)τ.

Moreover, for any k ∈ IN, t ≥ 0, IE[(

M(u(τ)))k] ≤ MkIE

[(M(u0)

)k]for a constant

Mk ≥ 0, and

(2.2)

H(u(τ)) = H(u0)− Im∫

IRn

∫ τ

0

(∆u + |u|2σu)φdWdx

+12

∑`∈IN

∫ τ

0

∫IRn

[|∇φe`|2 − |u|2σ |φe`|2 − 2σ|u|2σ−2(Re(uφe`))2

]dxds.

Lemma 2.2. Let u0, σ and φ be as in Theorem 1.1 and assume furthermore that φ is

Hilbert-Schmidt from L2 into Σ and that u0 ∈ Σ a.s. Then for any stopping time τ such

that τ < τ∗ a.s., u(·) = u(u0, ·) belongs to L∞(0, τ ; Σ). Moreover

(2.3) V (u(τ)) = V (u0) + 4∫ τ

0

G(u(s))ds + 2Im∫

IRn

∫ τ

0

|x|2uφdW dx + τcΣφ

with

cΣφ =

∑`∈IN

∫IRn

|x|2 |φe`|2 dx

and

(2.4)

G(u(τ)) = G(u0) + 4∫ τ

0

H(u(s))ds +2− σn

σ + 1

∫ τ

0

∫IRn

|u|2σ+2dxds

+ Re∑`∈IN

∫ τ

0

∫IRn

u(2x · ∇φe` + nφe`)dxdβ` + τc2φ

10

with

c2φ = Im

∑`∈IN

∫IRn

φe`x · ∇(φe`)dx.

Proof : We use a regularization procedure. Let ε > 0 and apply Ito’s formula in the formgiven in [6], Theorem 4.17 to

Vε(u) =∫

IRn

e−ε|x|2 |x|2|u(t, x)|2dx.

Integrating by parts, we easily get(V ′

ε (u),−i(∆u + |u|2σu))

= 4Im∫

IRn

e−ε|x|2(1− ε|x|2)(x.∇u)udx

and since(V ′

ε (u), iφdW ) = −2Im∫

IRn

e−ε|x|2 |x|2uφdW

andTr (V ′′(u)(iφ)(iφ)∗) = 2

∑`∈IN

∫ τ

0

∫IRn

e−ε|x|2 |x|2 |φe`|2 dx

we obtain

(2.5)

Vε(u(τ)) = Vε(u0) + 4Im∫ τ

0

∫IRn

e−ε|x|2(1− ε|x|2)(x · ∇u)udxds

− 2Im∫

IRn

∫ τ

0

e−ε|x|2 |x|2uφdWdx +∑`∈IN

∫ τ

0

∫IRn

e−ε|x|2 |x|2 |φe`|2 dxds.

Let us take in (2.5) the stopping time t ∧ τk with τk = inf t ∈ [0, T ]; |u(t)|H1 ≥ kfor some T ≥ 0, then using Cauchy Schwarz’s inequality in the second term, and sincee−ε|x|2/2|1− ε|x|2| ≤ 1, for all x ∈ IRn,

IE (Vε(u(t ∧ τk))) ≤ IE (Vε(u0)) + 4kIE∫ t∧τk

0

Vε(u(s))1/2ds + TcΣφ .

We easily deduce from Gronwall’s lemma that IE (Vε(t ∧ τk)) ≤ c1(T, k, φ, u0), wherec1(T, k, φ, u0) does not depend on ε. Moreover, by a classical martingale inequality (see[6], Theorem 3.14)

IE

(sup

t∈[0,T ]

Vε(u(t ∧ τk))

)≤ IE (Vε(u0)) + 4kIE

∫ T

0

Vε(u(s ∧ τk)1/2ds

+ 6IE

(∫ T

0

Vε(u(s ∧ τk))|φ|2L02(L

2;Σ)ds

)1/2+ TcΣ

φ .

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It follows that :

IE

(sup

t∈[0,T ]

Vε(u(t ∧ τk))

)≤ c2(T, k, φ, u0)

and letting ε → 0

IE

(sup

t∈[0,T ]

V (u(t ∧ τk))

)≤ c2(T, k, φ, u0).

Thus u has trajectories in L∞(0, τk; Σ) for any k ∈ IN. Since τk τ∗ a.s. and τ < τ∗ a.s.we deduce that u has trajectories in L∞(0, τ ; Σ).

Knowing this property of u, we can let ε → 0 in (2.5) and obtain (2.3). Finally,applying Ito’s formula to G(u) yields (2.4).

Now, collecting (2.2), (2.3) and (2.4), we get the following corollary.

Corollary 2.1. Under the same assumptions as in Lemma 2.2, we have

(2.6)

V (u(τ)) = V (u0) + 4G(u0)τ + 8H(u0)τ2 + 42− σn

σ + 1

∫ τ

0

(τ − s)|u|2σ+2L2σ+2ds

+ cΣφ τ + 2c2

φτ2 +43c1φτ3

− 4∑`∈IN

∫ τ

0

(τ − s)2∫

IRn

|u|2σ |φe`|2 + 2σ|u|2σ−2 (Re(uφe`))2dxds

+ 2Im∫

IRn

∫ τ

0

|x|2uφdWdx

− 16Im∫

IRn

∫ τ

0

∫ s2

0

∫ s1

0

(∆u + |u|2σu

)φdW (r)ds1ds2dx

+ 4Re∑`∈IN

∫ τ

0

∫ s

0

∫IRn

u(2x · ∇φe` + nφe`)dxdβ`ds,

with cΣφ and c2

φ given in Lemma 2.2, and c1φ =

∑`∈IN |∇φe`|2L2 .

The first four terms above arise in the deterministic identity. The others vanish in theabsence of noise. The last three terms are stochastic integrals. These are difficult to treatand are responsible for the technical difficulties below. In fact, we will take the expectationof (2.6) so that the stochastic integrals disappear. We need the integrands to be squareintegrable. The lemma below gives sufficient conditions for this.

12

Lemma 2.3. Assume that 2/n ≤ σ ≤ 2/(n− 2) and let u0, φ be as in Lemma 2.2; assume

furthermore that there exist Ω0, a F0 measurable set such that IP(Ω0) > 0 and T > 0 withT < τ∗(u0) a.s. such that

IE(|u0|2H1 11Ω0

)< ∞, IE

(|u0|2σ+2

L2σ+2 11Ω0

)< ∞, IE (V (u0)11Ω0) < ∞,

IE

(∫ T

0

(|∇u(u0, s)|2L2 + |u(u0, s)|4σ+2

L2σ+2

)ds 11Ω0

)< ∞.

Then, supt∈[0,T ]

IE (V (u(t)11Ω0) < ∞.

Proof : We use the notation IEΩ0(f) = IE(f11Ω0) for f ∈ L1(Ω0) or f measurable and nonnegative on Ω0. We take in Corollary 2.1 the stopping time t ∧ τk with

τk = infs ∈ [0, τ ], V (u(s)) ≥ k.

Then, we multiply (2.6) by 11Ω0 and take the expectation. The fourth and eighth termsare non positive. The ninth term has zero expectation since, by definition of τk, it is astochastic integral with a square integrable integrand. We majorize the last two terms asfollows :

(2.7)

IEΩ0

∣∣∣∣∫IRn

∫ t∧τk

0

∫ s2

0

∫ s1

0

(∆u + |u|2σu

)φdW (r)ds1ds2dx

∣∣∣∣≤∫ T

0

T IEΩ0

(∣∣∣∣∫IRn

∫ s

0

(∆u + |u|2σu

)φdW (r)dx

∣∣∣∣) ds

≤ T 2

(IEΩ0

∫ T

0

∣∣φ∗(∆u + |u|2σu)∣∣2L2 ds

)1/2

≤ 2T 2|φ|2L02(L

2,H1)

(IEΩ0

∫ T

0

(|∇u|2L2 + |u|4σ+2

L2σ+2

)ds

)1/2

.

We have used Cauchy-Schwarz’s inequality, Remark 1.1, the inequality

(2.8)|φ∗(∆u + |u|2σu)|L2 ≤ |φ|L(L2,H1)|∇u|L2 + |φ|L(L2,L2σ+2)|u|2σ+1

L2σ+2

≤ |φ|L02(L

2,H1)(|∇u|L2 + |u|2σ+1L2σ+2)

and the Sobolev imbedding H1 ⊂ Lp, for any p with 2 ≤ p < 2n/(n− 2) and in particularfor p = 2σ + 2.

13

Similarly,

(2.9)

IEΩ0

(∣∣∣∣∣∑`∈IN

∫ t∧τk

0

∫ s

0

∫IRn

u (2x · ∇φe` + nφe`) dxdβ`ds

∣∣∣∣∣)

≤∑`∈IN

∫ T

0

IEΩ0

(∣∣∣∣∫ s∧τk

0

∫IRn

(u(2x · ∇φe` + nφe`)) dxdβ`

∣∣∣∣) ds

≤∫ T

0

(IEΩ0

∫ s∧τk

0

8V (u)c1φ + 2n2M(u)|φ|2L0

2(L2,L2)

)1/2

ds

≤ IEΩ0

∫ t∧τk

0

V (u)ds + Λ1(φ, T, u0),

where Λ1 is finite by (2.1). We obtain from (2.6), (2.7) and (2.9)

IEΩ0 (V (u (t ∧ τk))) ≤ IEΩ0 (V (u0)) + 4T IEΩ0 (G (u0)) + 8T 2IEΩ0 (|H (u0)|)

+ cΣφT + 2c2

φT 2 +43c1φT 3 + Λ2 (φ, T, u0) + IEΩ0

∫ t∧τk

0

V (u)ds,

with Λ2 equal to the sum of the right hand side of (2.7) and of Λ1. We deduce fromGronwall’s lemma that IEΩ0 (V (u (t ∧ τk))) is bounded independently of k and t and obtainthe result by Fatou’s lemma with k →∞.

3. Proof of Theorem 1.2

We divide the proof in several steps.

3.1. Blow-up for a restricted class of initial data

Proposition 3.1. Assume that 2n ≤ σ < 2

n−2 , and that φ is Hilbert-Schmidt from L2

into Σ. For each T > 0, V > 0, G > 0, H1 > 0, there exists H2 > 0 such that if u0 is F0

measurable with values in Σ and Ω0 is a F0 measurable set such that IP(Ω0) > 0 and

(3.1)IE(|u0|2H1 11Ω0

)≤ H1, IE (|G(u0)| 11Ω0) ≤ G,

IE (V (u0) 11Ω0) ≤ V , and IE(|u0|2σ+2

L2σ+2 11Ω0

)≥ H2,

then, either

(3.2) IP(τ∗ (u0) < T

)> 0

14

or

(3.3) IE

((∫ T

0

(|∇u(u0, s)|2L2 + |u(u0, s)|4σ+2

L2σ+2

)ds

)11Ω0

)= ∞.

Proof : Let T > 0, V > 0, G > 0, H1 > 0. We choose H2 > 0 such that

(3.4) V +12(4G + cΣ

φ

) T

2+(

2H1 −2

σ + 1H2 + c2

φ

)T 2

2+

16c1φT 3 < 0

with c1φ, c2

φ and cΣφ as in Corollary 2.1.

Let u0,Ω0 be such that (3.1) holds but neither (3.2) nor (3.3) hold, so that τ∗(u0) isalmost surely greater than T and

(3.5) IEΩ0

(∫ T

0

(|∇u|2L2 + |u|4σ+2

L2σ+2

)ds

)< ∞.

We have used the shorter notation IEΩ0(·) = IE (·11Ω0) and u(·) = u(u0, ·). Since u0 is fixed,this does not cause any confusion.

For any t ∈ [0, T ], we take, in Corollary 2.1, τ to be the deterministic stopping timeequal to t. We then multiply (2.6) by 11Ω0 and take the expectation. We claim that thethree stochastic terms in (2.6) have a zero expectation. Indeed,∣∣φ∗ (|x|2u)∣∣

L2 ≤ |φ|L(L2,Σ)|u|Σ,

so that by Lemma 2.3,∫IRn

∫ t

0|x|2uφdWdx11Ω0 is a square integrable martingale and has

zero expectation. For the second one, we write

IEΩ0

(Im∫

IRn

∫ t

0

∫ s2

0

∫ s1

0

(∆u + |u|2σu

)φdWds1ds2dx

)= Im

∫ t

0

∫ s2

0

IEΩ0

(∫IRn

∫ s1

0

(∆u + |u|2σu

)φdWdx

)ds1ds2

.

and by Remark 1.1, (2.8) and (3.5), this term vanishes. The third one is treated similarly.Neglecting the fourth and eighth terms in the right-hand side of (2.6) which are non

positive, we obtain thanks to (3.1)

IEΩ0(V (u(t))) ≤ V +(4G + cΣ

φ

)t + 2

(2H1 −

2σ + 1

H2 + c2φ

)t2 +

43c1φt3

15

for t ∈ [0, T ]. This is clearly impossible because, due to (3.4), the right-hand side is negativefor t = T

2 . We deduce that either (3.2) or (3.3) holds.

If we consider a more regular noise and take more stringent assumptions on σ, we canstrengthen the conclusions of Proposition 3.1.

Proposition 3.2. Assume that 2n < σ < min

(23 , 2

n−2

), that φ is Hilbert-Schmidt from

L2 into Σ, γ-radonifying from L2 into L4σ+2 and bounded from L2 into H2∩L∞. Then for

each T > 0, V > 0, G > 0, H1 > 0 there exists H2 > 0 such that if u0 is F0 measurable, Ω0

is a F0 measurable set such that IP(Ω0) > 0, (3.1) holds and IE(M(u0)1/(1−σ)11Ω0

)< ∞,

then

IP(τ∗(u0) < T

)> 0.

Proof : We first note that in the proof of Proposition 3.1, we can replace (3.5) by

(3.6) IEΩ0

(∫ T

0

|u|4σ+2L2σ+1ds

)< ∞,

where as before we use the notations IEΩ0(·) = IE(·11Ω0) and u(·) = u(u0, ·). Indeed, (3.5)is used to state that the second stochastic integral in (2.6) has zero expectation but since

(3.7)∣∣φ∗ (∆u + |u|2σu

)∣∣L2 ≤ |φ|L(L2,H2)|u|L2 + |φ|L(L2,L∞)|u|2σ+1

L2σ+1 ,

it is clear that under the new assumptions on φ, we can replace (3.5) by (3.6).Thus, with T , V , G, H1, H2 as in Proposition 3.1 and u0,Ω0 satisfying (3.1) we deduce

that either

(3.8) IP(τ∗(u0) < T

)> 0

or

(3.9) IEΩ0

(∫ T

0

|u|4σ+2L2σ+1ds

)= ∞.

Note that if σ = 12 , we already know that (3.9) is impossible since the momenta of the L2

norm of u are bounded by constants depending only on their value at t = 0, by Lemma2.1. Hence (3.8) holds and the result is proved in this special case. A slight modificationof (3.7) and the above arguments can be used for σ < 1

2 .

16

The case σ > 12 is more difficult and is treated hereafter. The idea is to use the fourth

term – which is non positive – in (2.6) to control the second stochastic integral of thisinequality. This unfortunately involves tedious computations.

Assume that (3.8) fails so that τ∗(u0) ≥ T almost surely. For any k ∈ IN we set

τk = infs ∈ [0, T ] : V (u(s)) + |u(s)|H1 ≥ k

.

Then by Lemma 2.1

IEΩ0

(sup

s∈[0,t∧τk]

|H(u(s))|

)

≤ IEΩ0 (|H(u0)|) + IEΩ0

(sup

s∈[0,t∧τk]

∣∣∣∣∫ s

0

∫IRn

(∆u + |u|2σu

)φdxdW

∣∣∣∣)

+12IEΩ0

(∫ t∧τk

0

∫IRn

∑`∈IN

|∇φe`|2 + (2σ + 1)|u|2σ |φe`|2 dxds

).

A martingale inequality (see [6], Theorem 3.14) yields

IEΩ0

(sup

s∈[0,t∧τk]

|H(u(s))|

)≤ IEΩ0 (|H(u0)|)

+ 3IEΩ0

((∫ t∧τk

0

∣∣φ∗ (∆u + |u|2σu)∣∣2

L2 ds

)1/2)

+12c1φT

+2σ + 1

2|φ|2R(L2,L4σ+2)IEΩ0

(∫ t∧τk

0

|u|2σL2σ+1ds

).

Using (3.1), (3.7), Lemma 2.1 and Holder’s inequality in the last term above, we obtainafter easy manipulations

(3.10) IEΩ0

(sup

s∈[0,t∧τk]

|H(u(s))|

)≤ K1

(H1, H2, φ, σ, T

)+ IEΩ0

(∫ t∧τk

0

|u(s)|4σ+2L2σ+1ds

),

where, here and below, Ki(·, · · ·), i = 1, 2, · · · depend only on its arguments and is finiteunder our assumptions. Since σ ≥ 1

2 , Holder’s inequality gives |u|4σ+2L2σ+1 ≤ |u|2/σ

L2 |u|4σ+2−2/σL2σ+2

17

so that

IEΩ0

(∫ t∧τk

0

|u|4σ+2L2σ+1ds

)≤ IEΩ0

(sup

s∈[0,t∧τk]

|u|2/σL2

∫ t∧τk

0

|u|4σ+2−2/σL2σ+2 ds

)

≤ IEΩ0

(sup

s∈[0,t∧τk]

|u|2/σL2

(∫ t∧τk

0

(t ∧ τk − s)−λ

1−λ ds

)1−λ

×(∫ t∧τk

0

(t ∧ τk − s)|u|2σ+2L2σ+2ds

)λ)

with λ = 2σ+1−(1/σ)σ+1 . Note that λ

1−λ ∈ [0, 1) for σ < 23 . We use Holder’s inequality for the

expectation, Young’s inequality and Lemma 2.1 again to get

(3.11)IEΩ0

(∫ t∧τk

0

|u(s)|4σ+2L2σ+1ds

)≤ K2

(H1, H2, IEΩ0(M(u0)1/(1−σ)), φ, σ, T

)+

σn− 28(σ + 1)T 2

IEΩ0

(∫ t∧τk

0

(t ∧ τk − s)|u(s)|2σ+2L2σ+2ds

).

We now infer from Lemma 2.2 that

(3.12)

IEΩ0

(∫ t∧τk

0

G(u(s))ds

)+

σn− 2σ + 1

IEΩ0

(∫ t∧τk

0

(t ∧ τk − s)|u(s)|2σ+2L2σ+2ds

)= IEΩ0 (t ∧ τkG(u0)) + 4IEΩ0

(∫ t∧τk

0

(t ∧ τk − s)H(u(s))ds

)+ IEΩ0

(Re∑`∈IN

∫ t∧τk

0

∫ r

0

∫IRn

u(s)(2x · ∇φe` + nφe`)dxdβ`dr

).

The last term is majorized as in (2.9) :

IEΩ0

(Re∑`∈IN

∫ t∧τk

0

∫ r

0

∫IRn

u(s)(2x · ∇φe` + nφe`)dxdβ`dr

)

≤ IEΩ0

∫ t∧τk

0

V (u(s))ds + K3

(IEΩ0(M(u0)), T , φ

).

Then, combining (3.10), (3.11), (3.12) and this inequality gives

IEΩ0

∫ t∧τk

0

G(u(s))ds +σn− 2

2(σ + 1)IEΩ0

∫ t∧τk

0

(t ∧ τk − s)|u(s)|2σ+2L2σ+2ds

≤ K4

(IEΩ0

(M (u0)

1/(1−σ))

, H1, H2, φ, σ, T , G)

+ IEΩ0

∫ t∧τk

0

V (u(s))ds.

18

Again by Lemma 2.2, we have

IEΩ0 (V (u (t ∧ τk))) = IEΩ0 (V (u0)) + 4IEΩ0

∫ t∧τk

0

G(u(s))ds + cΣφ IEΩ0(t ∧ τk),

so that using (3.1) again,

IEΩ0 (V (u (t ∧ τk))) +2(σn− 2)

σ + 1IEΩ0

∫ t∧τk

0

(t ∧ τk − s)|u(s)|2σ+2L2σ+2ds

≤ K5

(IEΩ0

(M(u0)1/(1−σ)

), H1, H2, G, V , φ, σ, T

)+ 4IEΩ0

∫ t∧τk

0

V (u(s))ds.

By Gronwall’s lemma, we obtain

IEΩ0 (V (u (τk))) +2(σn− 2)

σ + 1IEΩ0

∫ τk

0

(τk − s)|u(s)|2σ+2L2σ+2ds

≤ K6

(IEΩ0

(M(u0)1/(1−σ)

), H1, H2, G, V , φ, σ, T

).

We now let k →∞ and use the monotone convergence theorem to get

IEΩ0

∫ T

0

(T − s

)|u(s)|2σ+2

L2σ+2ds ≤ K7

(IEΩ0

(M(u0)1/(1−σ)

), H1, H2, G, V , φ, σ, T

)and letting k → ∞ in (3.11), we obtain a contradiction with (3.9). We deduce that ourassumption that (3.8) fails is impossible.

3.2. Controllability and continuous dependence

In order to prove that any initial data reaches with positive probability a state which isknown to lead to a solution which blows up, we introduce the following perturbed nonlinearSchrodinger equation

(3.13)

idv

dt−(∆v + |v + z|2σ(v + z)

)= 0,

v(0) = u0,

whose solution at time t, when it exists, is denoted by v(z, u0, t), z being a given functionin C([0, T ];H1) ∩ Lr

(0, T ;W 1,2σ+2

)with r = 4(σ+1)

nσ .

19

We now construct a function z such that z(0) = 0 and u = z + v (z, u0, ·) satisfies theconditions which ensure blow up at some time T1 > 0. This is the only point where theresult would be different in the case of a real valued noise. Indeed, we would have in thiscase to control the equation by a forcing term which is the time derivative of a real valued

function ; it is of course more difficult to find such a function than if we only require thatit is complex valued. The details of the real valued case will appear elsewhere.

Here, we use the translated equation (3.13) instead of the forced equation

(3.14) i∂u

∂t− (∆u + |u|2σu) =

∂f

∂t

because we have better continuous dependence properties of v with respect to z than of u

with respect to f .

Proposition 3.3. For any T1 > 0, u0 ∈ Σ, u1 ∈ Σ, there exists a z in C ([0, T1] ; Σ) ∩Lr(0, T1;W 1,2σ+2

)∩ L1(0, T1; Σ1,2), r = 4(σ+1)

σ+1 , such that z(0) = 0, v (z, u0, ·) exists on

[0, T1] and

z(T1) + v (z, u0, T1) = u1.

Proof : Take k1 ∈ IN and denote by U(t) the semigroup on Σ associated to the linearequation

dw

dt+ (−∆)k1w + |x|2k1w = 0, x ∈ IRn,

w(0) = u0.

For t ∈ [0, T1] , we set

(3.16)

u(t) =T1 − t

T1U(t)u0 +

t

T1U(T1 − t)u1,

f(t) = i(u(t)− u(0))−∫ t

0

(∆u(s) + |u|2σ

u)

ds,

z(t) = −if(t)−∫ t

0

S(t− s)(i∆)f(s)ds.

Clearly, u(0) = u0 and u(T1) = u1. Moreover, it is not difficult to see that z satisfies idz

dt−∆z =

df

dt,

z(0) = 0,

20

so that u and f satisfy (3.14).Note that under stronger smoothness assumptions on u0 and u1, we could have avoided

the use of U(t). From the construction of u, we know that

u ∈ C ([0, T1] ; Σ) ∩ L2(0, T1; Σ0,k1

).

Thus, for k1 sufficiently large depending on n and σ we can prove by interpolation argu-ments and Sobolev embedding theorems that

u ∈ Lr(0, T1;W 1,2σ+2

)and |u|2σ

u ∈ L1 ([0, T1] ; Σ) .

We set v = u− z. Then

v(t) = S(t)u0 − i

∫ t

0

S(t− s)(|u|2σ

u)

ds,

and v = v(z, u0, ·).Since (S(t))t∈IR is a strongly continuous semigroup in Σ, (see for example [5]), we

deduce that v ∈ C ([0, T1] ; Σ) and z = u− v ∈ C ([0, T1] ; Σ) . By Strichartz estimates (see[13]), v ∈ Lr

(0, T1;W 1,2σ+2

)and z = u − v ∈ Lr

(0, T1;W 1,2σ+2

). It remains to check

that z belongs to L1(0, T1; Σ1,2

). This involves long and tedious computations and we do

not give details. We use the operator J2 defined by

J2(t)z = |x|2z + 4tinx · ∇z + 2in2tz − 4n2t2∆z,

and note that, thanks to several integration by parts, for any ε > 0, there exists c(ε) suchthat

|x.∇z|H1 ≤ c(ε)|z|H3 + ε∣∣|x|2z∣∣

H1 .

This implies that it suffices to prove that J2z ∈ L1(0, T1;H1

)and z ∈ L1

(0, T1;H3

).

The latter is easily obtained from (3.15) for k1 sufficiently large. Then, since J2(t)S(t) =S(t)J2(0), we have

J2(t)z(t) = −iJ2(t)f(t)−∫ t

0

S(t− s)J2(s)(i∆)f(s)ds,

so that again we can check that J2z ∈ L1(0, T1;H1

)provided k1 is taken large enough.

21

Once we have constructed this control z, we need some continuous dependence resultfor the solution of (3.13) with respect to z. The following proposition is proved in [8].

Proposition 3.4. Let z ∈ C([0, T ];H1

)∩ Lr

(0, T ;W 1,2σ+2

), with r = 4(σ+1)

σn , and

u0 ∈ H1 be such that v (z, u0, ·) exists on [0, T ] and is in C([0, T ],H1

). There exist

neighborhoods V of z in C([0, T ];H1

)∩Lr

(0, T ;W 1,2σ+2

)and W of u0 in H1 such that for

any (u0, z) ∈ W×V, the solution v(u0, z, ·) of (3.13) exists and is unique in C([0, T ];H1

)∩

Lr(0, T ;W 1,2σ+2

). Furthermore, the mapping (u0, z) 7→ v(z, u0, ·) is continuous from

W ×V into C([0, T ];H1

).

Next, we prove some further continuous dependence in the space Σ.

Proposition 3.5. Let z, u0 be as in Proposition 3.3 and assume furthermore that u0 ∈ Σ,

z ∈ C([0, T ]; Σ) ∩ L1 (0, T ;Sn) ; then for t ∈ [0, T ] the mapping (u0, z) 7→ v (z, u0, t) is

lower semicontinuous from W × V1, into Σ, where V1 = V ∩ C([0, T ],Σ) ∩ L1 (0, T ;Sn) ,

and V, W are given by Proposition 3.4.

Proof : Let (u0, z) ∈ W × V1 and set v = v (z, u0, ·) be the associated solution of (3.13).We derive formally some estimates. These could be easily justified, considering actuallyVε(v) as in Lemma 2.2, and letting afterwards ε go to zero.

We multiply (3.13) by |x|2v, integrate over IRn and take the imaginary part to obtain

(3.16)12

d

dtV (v) + 2 Im

∫IRn

x · ∇vvdx + Im∫

IRn

|x|2|v + z|2σ(v + z)vdx = 0.

We first treat the case n ≥ 3 so that Sn = Σ1,2. Thanks to Holder’s inequality and Sobolev’sembedding theorem, we have, since |x|2|v + z|2σ(v + z)(v + z) is real valued,

(3.17)

∣∣∣∣Im ∫IRn

|x|2|v + z|2σ(v + z)vdx

∣∣∣∣ = ∣∣∣∣Im ∫IRn

|x|2|v + z|2σ(v + z)zdx

∣∣∣∣≤ |v + z|2σ+1

L2σ+2

∣∣|x|2z∣∣L2σ+2 ≤ K2|v + z|2σ+1

H1

∣∣|x|2z∣∣H1

and ∣∣∣∣2Im∫

IRn

x · ∇vvdx

∣∣∣∣ ≤ 2|v|H1V (v)1/2 ≤ |v|2H1 + V (v).

We deduce thatd

dtV (v(t)) ≤ 2V (v(t)) + K3(v(t), z(t))

22

where, if n ≥ 3, under our assumptions∫ T

0K3(v(t), z(t))dt < ∞. By Gronwall’s lemma, it

follows that V (v) is bounded on [0, T ], and, by Proposition 3.4, v is in L∞(0, T ; Σ). Thelower semicontinuity is easily proved since the mapping is in fact weakly continuous withvalues in Σ.

The statement for n = 1 is obtained by replacing (3.17) by∣∣∣∣Im ∫IR

|x|2|v + z|2σ(v + z)zdx

∣∣∣∣ ≤ |v + z|2σL∞V (v + z)1/2V (z)1/2

≤ K4

(|v + z|4σ

L∞ + 1)V (z) + V (v)

and using the embedding of H1 into L∞.

For n = 2, we use∣∣∣∣Im ∫IR2|x|2|v + z|2σ(v + z)zdx

∣∣∣∣ ≤ |v + z|2σLpV (v + z)1/2||x|z|L2+β

with 2σp + 1

2+β = 12 , the embedding of H1 into Lp for any p ≥ 2, and the embedding of Hα

into L2+β with 12+β = 1

2 −α2 .

3.3. Proof of Theorem 1.2

Let T1, T > 0, u0 ∈ Σ. We choose any u1 ∈ Σ such that

(3.18) |∇u1|2L2 ≤12H1, |G(u1)| ≤

12G, V (u1) ≤

12V , and |u1|2σ+2

L2σ+2 ≥ 2H2,

with H1, G, V arbitrary but fixed, and H2 given by Proposition 3.1. The fact that such au1 ∈ Σ exists may be proved by a scaling argument.

Then by Proposition 3.3, there exists a z in C ([0, T1] ,Σ) ∩ Lr(0, T1;W 1,2σ+2

)∩

L1(0, T1; Σ1,2

)such that z(0) = 0 and z(T1) + v (z, u0, T1) = u1. By Propositions 3.4 and

3.5, we can find a ball B centered at z in C ([0, T1] ; Σ)∩Lr(0, T1;W 1,2σ+2

)∩L1 (0, T1;Sn)

such that for any z in B, u = z + v(z, u0, ·) exists on [0, T1] and satisfies

(3.19) |∇u(T1)|2L2 ≤ H1, |G(u(T1))| ≤ G, |V (u(T1))| ≤ V , and |u(T1)|2σ+2L2σ+2 ≥ H2.

It is not difficult to see that the solution of (1.7) is given by u (u0, t) = z(t) + v (z, u0, t) ,

with

z(t) =∫ t

0

S(t− s)φdW (s),

23

almost surely on [0, τ∗ (u0)) . Since φ is Hilbert-Schmidt from L2 to Sn, z is almost surelyin C ([0, T1] ; Sn) (see [6], Theorem 6.10). Moreover, it is shown in [8] that z is almostsurely in Lr

(0, T ;W 1,2σ+2

). Since kerφ∗ = 0, φ has dense range in Sn and in W 1,2σ+2,

we deduce that z is non-degenerate and IP(z ∈ B) > 0, therefore

IP τ∗ (u0) ≥ T1 and u (u0, T1) satisfies (3.20) > 0.

We now set

ΩT1 = ω ∈ Ω/τ∗ (u0) ≥ T1 and u (u0, T1) satisfies (3.20)

and note that u (u0, T1) ,ΩT1 , T , V , G, H1, H2 satisfy the condition ensuring that Proposi-tion 3.1, or Proposition 3.2 can be applied. The result follows.

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