Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles...
Transcript of Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles...
![Page 1: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/1.jpg)
Branching Brownian particles with spatial selectionand the KPP equation
Julián Martínezjoint work with Pablo Groisman y Matthieu Jonckheere.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 1 / 20
![Page 2: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/2.jpg)
Fisher - Kolmogorov, Petrovskii, Piskunov equation
1937: R.A.Fisher - "The way of advance of advantageous genes"Spread of advantegeous gene in a population.
Difussion
∂tu = ∂2x u
Birth and death∂tu = u(1− u) = u − u2
u(x , t) /
∂tu = ∂2
x u + u2 − u
u(x ,0) = f (x)
Infinitely many travelling waves
u(x , t) = w(x − ct), with c = c(f ).
But only one has physical meaning, c =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 2 / 20
![Page 3: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/3.jpg)
Fisher - Kolmogorov, Petrovskii, Piskunov equation
1937: R.A.Fisher - "The way of advance of advantageous genes"Spread of advantegeous gene in a population.
Difussion
∂tu = ∂2x u
Birth and death∂tu = u(1− u) = u − u2
u(x , t) /
∂tu = ∂2
x u + u2 − u
u(x ,0) = f (x)
Infinitely many travelling waves
u(x , t) = w(x − ct), with c = c(f ).
But only one has physical meaning, c =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 2 / 20
![Page 4: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/4.jpg)
Fisher - Kolmogorov, Petrovskii, Piskunov equation
1937: R.A.Fisher - "The way of advance of advantageous genes"Spread of advantegeous gene in a population.
Difussion
∂tu = ∂2x u
Birth and death∂tu = u(1− u) = u − u2
u(x , t) /
∂tu = ∂2
x u + u2 − u
u(x ,0) = f (x)
Infinitely many travelling waves
u(x , t) = w(x − ct), with c = c(f ).
But only one has physical meaning, c =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 2 / 20
![Page 5: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/5.jpg)
Fisher - Kolmogorov, Petrovskii, Piskunov equation
1937: R.A.Fisher - "The way of advance of advantageous genes"Spread of advantegeous gene in a population.
Difussion
∂tu = ∂2x u
Birth and death∂tu = u(1− u) = u − u2
u(x , t) /
∂tu = ∂2
x u + u2 − u
u(x ,0) = f (x)
Infinitely many travelling waves
u(x , t) = w(x − ct), with c = c(f ).
But only one has physical meaning, c =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 2 / 20
![Page 6: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/6.jpg)
Fisher - Kolmogorov, Petrovskii, Piskunov equation
1937: R.A.Fisher - "The way of advance of advantageous genes"Spread of advantegeous gene in a population.
Difussion
∂tu = ∂2x u
Birth and death∂tu = u(1− u) = u − u2
u(x , t) /
∂tu = ∂2
x u + u2 − u
u(x ,0) = f (x)
Infinitely many travelling waves
u(x , t) = w(x − ct), with c = c(f ).
But only one has physical meaning, c =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 2 / 20
![Page 7: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/7.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 3 / 20
![Page 8: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/8.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 3 / 20
![Page 9: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/9.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 3 / 20
![Page 10: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/10.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 3 / 20
![Page 11: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/11.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 4 / 20
![Page 12: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/12.jpg)
Branching Brownian motion
Fisher: The velocity of advance should not depend on the initial cond.
Heat equation - Brownian motion
u(x , t) = Ex [f (Bt )] = G(·, t) ∗ f [x ], ←→ ∂tu = ∂2x u.
McKean, 1975: F-KPP - BBM
u(x , t) = Ex [f (X 1t ) . . . f (XNt
t )].
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 5 / 20
![Page 13: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/13.jpg)
Si f (x) = 1[0,∞](x), u(x , t) = P(
maxi=1,...,Nt
X it ≤ x
).
Bramson, Ferrari, Lebowitz,...; 1986.
Selection Principle
The microscopic model has a unique velocity vN ,
vN ↗ vmin.
Brunet, Derrida; 1997,2001,...:Efect of microscopic noise in front propagation,mainly numerical and heuristic arguments.
vN − vmin ' −Klog2 N
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 6 / 20
![Page 14: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/14.jpg)
Si f (x) = 1[0,∞](x), u(x , t) = P(
maxi=1,...,Nt
X it ≤ x
).
Bramson, Ferrari, Lebowitz,...; 1986.
Selection Principle
The microscopic model has a unique velocity vN ,
vN ↗ vmin.
Brunet, Derrida; 1997,2001,...:Efect of microscopic noise in front propagation,mainly numerical and heuristic arguments.
vN − vmin ' −Klog2 N
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 6 / 20
![Page 15: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/15.jpg)
Si f (x) = 1[0,∞](x), u(x , t) = P(
maxi=1,...,Nt
X it ≤ x
).
Bramson, Ferrari, Lebowitz,...; 1986.
Selection Principle
The microscopic model has a unique velocity vN ,
vN ↗ vmin.
Brunet, Derrida; 1997,2001,...:Efect of microscopic noise in front propagation,mainly numerical and heuristic arguments.
vN − vmin ' −Klog2 N
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 6 / 20
![Page 16: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/16.jpg)
Si f (x) = 1[0,∞](x), u(x , t) = P(
maxi=1,...,Nt
X it ≤ x
).
Bramson, Ferrari, Lebowitz,...; 1986.
Selection Principle
The microscopic model has a unique velocity vN ,
vN ↗ vmin.
Brunet, Derrida; 1997,2001,...:Efect of microscopic noise in front propagation,mainly numerical and heuristic arguments.
vN − vmin ' −Klog2 N
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 6 / 20
![Page 17: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/17.jpg)
Our model
N particles in R: ξ̄Nt = (ξ1
t , . . . , ξNt ),
each particle performs a Brownian motion independently of anyother,At rate 1
N−1 two particles are chosen at random, the rightmostbranches into two particles and the leftmost dies.
We are interested in FN(x, t) =
N∑i=1
1ξit≤x
N.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 7 / 20
![Page 18: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/18.jpg)
Our model
N particles in R: ξ̄Nt = (ξ1
t , . . . , ξNt ),
each particle performs a Brownian motion independently of anyother,
At rate 1N−1 two particles are chosen at random, the rightmost
branches into two particles and the leftmost dies.
We are interested in FN(x, t) =
N∑i=1
1ξit≤x
N.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 7 / 20
![Page 19: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/19.jpg)
Our model
N particles in R: ξ̄Nt = (ξ1
t , . . . , ξNt ),
each particle performs a Brownian motion independently of anyother,At rate 1
N−1 two particles are chosen at random, the rightmostbranches into two particles and the leftmost dies.
We are interested in FN(x, t) =
N∑i=1
1ξit≤x
N.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 7 / 20
![Page 20: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/20.jpg)
Our model
N particles in R: ξ̄Nt = (ξ1
t , . . . , ξNt ),
each particle performs a Brownian motion independently of anyother,At rate 1
N−1 two particles are chosen at random, the rightmostbranches into two particles and the leftmost dies.
We are interested in FN(x, t) =
N∑i=1
1ξit≤x
N.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 7 / 20
![Page 21: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/21.jpg)
LGN: (ξi)i≥1, ξi ∼ X , i .i .d =⇒ FN(x)P−−−−→
N→∞P(X ≤ x) ∀x .
Hydrodynamic limit
If ξ̄N0 ∼ ρN
0 such that FN(x ,0)P−−−−→
N→∞u0(x) ∀x , then
FN(x , t) P−−−−→N→∞
u(x , t), ∀x ,
where u is the solution of the KPP equation with IC u0.
Selection Principle
1 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.2
limN→∞
vN =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 8 / 20
![Page 22: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/22.jpg)
LGN: (ξi)i≥1, ξi ∼ X , i .i .d =⇒ FN(x)P−−−−→
N→∞P(X ≤ x) ∀x .
Hydrodynamic limit
If ξ̄N0 ∼ ρN
0 such that FN(x ,0)P−−−−→
N→∞u0(x) ∀x , then
FN(x , t) P−−−−→N→∞
u(x , t), ∀x ,
where u is the solution of the KPP equation with IC u0.
Selection Principle
1 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.2
limN→∞
vN =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 8 / 20
![Page 23: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/23.jpg)
LGN: (ξi)i≥1, ξi ∼ X , i .i .d =⇒ FN(x)P−−−−→
N→∞P(X ≤ x) ∀x .
Hydrodynamic limit
If ξ̄N0 ∼ ρN
0 such that FN(x ,0)P−−−−→
N→∞u0(x) ∀x , then
FN(x , t) P−−−−→N→∞
u(x , t), ∀x ,
where u is the solution of the KPP equation with IC u0.
Selection Principle
1 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.2
limN→∞
vN =√
2.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 8 / 20
![Page 24: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/24.jpg)
Derivation of the equation
uN(x , t) := Eξ̄N0
[FN(x , t)].
Yt y Yt − 1 at rate g(Yt ) ⇒ ∂tE [Yt ] = −E [g(Yt )].
Ey [Yh − y ]
h= 0 P
(no mark or1 mark and NC
)− 1 P(1 mark and C )
h + o(h) = −he−hg(y)h
∂tuN =
Difussion
∂xxuN −jumps
NN−1E [FN(1− FN)]
= ∂xxuN − uN(1− uN) + NN−1V (FN(x , t)) + 1
N−1(uN(1− uN)).
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 9 / 20
![Page 25: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/25.jpg)
Derivation of the equation
uN(x , t) := Eξ̄N0
[FN(x , t)].
Yt y Yt − 1 at rate g(Yt ) ⇒ ∂tE [Yt ] = −E [g(Yt )].
Ey [Yh − y ]
h= 0 P
(no mark or1 mark and NC
)− 1 P(1 mark and C )
h + o(h) = −he−hg(y)h
∂tuN =
Difussion
∂xxuN −jumps
NN−1E [FN(1− FN)]
= ∂xxuN − uN(1− uN) + NN−1V (FN(x , t)) + 1
N−1(uN(1− uN)).
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 9 / 20
![Page 26: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/26.jpg)
Derivation of the equation
uN(x , t) := Eξ̄N0
[FN(x , t)].
Yt y Yt − 1 at rate g(Yt ) ⇒ ∂tE [Yt ] = −E [g(Yt )].
Ey [Yh − y ]
h= 0 P
(no mark or1 mark and NC
)− 1 P(1 mark and C )
h + o(h) = −he−hg(y)h
∂tuN =
Difussion
∂xxuN −jumps
NN−1E [FN(1− FN)]
= ∂xxuN − uN(1− uN) + NN−1V (FN(x , t)) + 1
N−1(uN(1− uN)).
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 9 / 20
![Page 27: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/27.jpg)
Derivation of the equation
uN(x , t) := Eξ̄N0
[FN(x , t)].
Yt y Yt − 1 at rate g(Yt ) ⇒ ∂tE [Yt ] = −E [g(Yt )].
Ey [Yh − y ]
h= 0 P
(no mark or1 mark and NC
)− 1 P(1 mark and C )
h + o(h) = −he−hg(y)h
∂tuN =
Difussion
∂xxuN −jumps
NN−1E [FN(1− FN)]
= ∂xxuN − uN(1− uN) + NN−1V (FN(x , t)) + 1
N−1(uN(1− uN)).
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 9 / 20
![Page 28: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/28.jpg)
Derivation of the equation
uN(x , t) := Eξ̄N0
[FN(x , t)].
Yt y Yt − 1 at rate g(Yt ) ⇒ ∂tE [Yt ] = −E [g(Yt )].
Ey [Yh − y ]
h= 0 P
(no mark or1 mark and NC
)− 1 P(1 mark and C )
h + o(h) = −he−hg(y)h
∂tuN =
Difussion
∂xxuN −jumps
NN−1E [FN(1− FN)]
= ∂xxuN − uN(1− uN) + NN−1V (FN(x , t)) + 1
N−1(uN(1− uN)).
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 9 / 20
![Page 29: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/29.jpg)
Variance / Propagation of chaos
FN(x , t)2 = 1N2
N∑i,j=1
1ξit≤x1
ξjt≤x
V (FN) = E [F 2N ]−E [FN ]2 = 1
N2
N∑i,j=1
P(ξit ≤ x , ξj
t ≤ x)−P(ξit ≤ x)P(ξj
t ≤ x)
Lemma
supξ̄N
0
1N2
∣∣∣∣∣∣N∑
i=1
∑j
P(ξit ≤ x , ξj
t ≤ x)− P(ξit ≤ x)P(ξj
t ≤ x)
∣∣∣∣∣∣ ≤ 2et
N .
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 10 / 20
![Page 30: Branching Brownian particles with spatial selection and ...€¦ · Branching Brownian particles with spatial selection and the KPP equation Julián Martínez joint work with Pablo](https://reader035.fdocuments.us/reader035/viewer/2022071016/5fcfd78abadbbf2224597377/html5/thumbnails/30.jpg)
Variance / Propagation of chaos
FN(x , t)2 = 1N2
N∑i,j=1
1ξit≤x1
ξjt≤x
V (FN) = E [F 2N ]−E [FN ]2 = 1
N2
N∑i,j=1
P(ξit ≤ x , ξj
t ≤ x)−P(ξit ≤ x)P(ξj
t ≤ x)
Lemma
supξ̄N
0
1N2
∣∣∣∣∣∣N∑
i=1
∑j
P(ξit ≤ x , ξj
t ≤ x)− P(ξit ≤ x)P(ξj
t ≤ x)
∣∣∣∣∣∣ ≤ 2et
N .
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 10 / 20
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Variance / Propagation of chaos
FN(x , t)2 = 1N2
N∑i,j=1
1ξit≤x1
ξjt≤x
V (FN) = E [F 2N ]−E [FN ]2 = 1
N2
N∑i,j=1
P(ξit ≤ x , ξj
t ≤ x)−P(ξit ≤ x)P(ξj
t ≤ x)
Lemma
supξ̄N
0
1N2
∣∣∣∣∣∣N∑
i=1
∑j
P(ξit ≤ x , ξj
t ≤ x)− P(ξit ≤ x)P(ξj
t ≤ x)
∣∣∣∣∣∣ ≤ 2et
N .
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 10 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 11 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 11 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 12 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 13 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 14 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 15 / 20
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Velocity - Upper bound
Idea: Embed our process in N independent BBM
Maximum of the BBM
Let Mt := max{X 1t , . . . ,X
Ntt }. Then
limt→∞
Mtt =
√2, c.s.
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 16 / 20
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Velocity - Lower bound
ηit := ξ
(i+1)t − ξ(1)
t , η̄Nt = (η1
t , . . . , ηN−1t ).
Finite N1 There exists a unique equilibrium πN for (ηN
t )t≥0.
2 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.
mNt :=
N∑i=1
ξit
N
EπN [mNt ] = EπN [mN
t − ξ(1)t ] + EπN [ξ
(1)t ] =
(1)c1 + c2 t .
⇒(2)
c2 = vN
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 17 / 20
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Velocity - Lower bound
ηit := ξ
(i+1)t − ξ(1)
t , η̄Nt = (η1
t , . . . , ηN−1t ).
Finite N1 There exists a unique equilibrium πN for (ηN
t )t≥0.
2 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.
mNt :=
N∑i=1
ξit
N
EπN [mNt ] = EπN [mN
t − ξ(1)t ] + EπN [ξ
(1)t ] =
(1)c1 + c2 t .
⇒(2)
c2 = vN
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 17 / 20
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Velocity - Lower bound
ηit := ξ
(i+1)t − ξ(1)
t , η̄Nt = (η1
t , . . . , ηN−1t ).
Finite N1 There exists a unique equilibrium πN for (ηN
t )t≥0.
2 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.
mNt :=
N∑i=1
ξit
N
EπN [mNt ] = EπN [mN
t − ξ(1)t ] + EπN [ξ
(1)t ] =
(1)c1 + c2 t .
⇒(2)
c2 = vN
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 17 / 20
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Velocity - Lower bound
ηit := ξ
(i+1)t − ξ(1)
t , η̄Nt = (η1
t , . . . , ηN−1t ).
Finite N1 There exists a unique equilibrium πN for (ηN
t )t≥0.
2 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.
mNt :=
N∑i=1
ξit
N
EπN [mNt ] = EπN [mN
t − ξ(1)t ] + EπN [ξ
(1)t ] =
(1)c1 + c2 t .
⇒(2)
c2 = vN
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 17 / 20
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Velocity - Lower bound
ηit := ξ
(i+1)t − ξ(1)
t , η̄Nt = (η1
t , . . . , ηN−1t ).
Finite N1 There exists a unique equilibrium πN for (ηN
t )t≥0.
2 ∃ limt→∞ξ(1)
t = limt→∞ξ(N)
t = vN , a.s. and in L1.
mNt :=
N∑i=1
ξit
N
EπN [mNt ] = EπN [mN
t − ξ(1)t ] + EπN [ξ
(1)t ] =
(1)c1 + c2 t .
⇒(2)
c2 = vN
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 17 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx
=
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx
≥∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx
∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Velocity - Lower bound
limt→∞
limb→∞
∫ m(t)+b
m(t)−bu0(x , t)(1− u0(x , t))dx =
√2
mNt+h −mN
t =
∫ +∞
−∞FN(x , t)− FN(x , t + h) dx
∂tEπN [mNt ] = −
∫ +∞
−∞∂tEπN [FN(x , t)] dx =
∫ +∞
−∞EπN [FN(x , t)(1− FN(x , t))] dx
≥∫ +∞
−∞E0[FN(x , t)(1− FN(x , t))] dx ≥
∫ m(t)+b
m(t)−bE0[FN(x , t)(1− FN(x , t))] dx
∼=∫
E0[FN(x , t)]E0[1− FN(x , t)] dx ∼=N→∞
∫u0(x, t)(1− u0(x, t))dx
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 18 / 20
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Thanks for your attention!
J. Martínez (IMAS - UBA) ICAS, UNSAM - Junio 2016 19 / 20