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Propagation of chaos for interacting particles subject to environmental noise

Michele Coghi 23/10/2014

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1

Introduction

2

The model

3

Existence and limit theorem

4

Propagation of Chaos

5

Summary

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Part I Introduction

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Filtered pobability space, (Ω, F, (Ft)t≥0, P).

Interacting particle system

• dX i,N

t

= 1

N

N

j=1 K

• X i,N

t

− X j,N

t

• dt + ∞

k=1 σk

• X i,N

t

• dW k

t

i = 1, ..., N Empirical Measure SN

t = 1

N

N

• i=1

δX i,N

t

→ µt

Limit Equation

• dµt + div (bµtµt) dt + ∞

k=1 div (σk (x) µt) ◦ dW k t = 0

bµt = K ∗ µt

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Independent noise [Sznitman]

dX i,N

t

= 1 N

N

• j=1

K

• X i,N

t

− X j,N

t

• dt + dW i

t

i = 1, ..., N ∂tµt + div (bµtµt) − 1 2∆µt = 0

Deterministic model [Dobrushin]

d dt X i,N

t

= 1 N

N

• j=1

K

• X i,N

t

− X j,N

t

• i = 1, ..., N

∂tµt + div (bµtµt) = 0

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Part II The Model

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Lipschitz continuity of coefficients

K, σk : Rd → Rd, k ∈ N |K(x) − K(y)| ≤ LK|x − y|

∞

• k=1

|σk(x) − σk(y)|2 ≤ Lσ|x − y|2

Covariance

∞

k=1 |σk (x)|2 < ∞,

Q : Rd × Rd → Rd×d Qij (x, y) = ∞

k=1 σi k (x) σj k (y) .

Q : Rd → Rd×d Q (x, y) = Q (x − y) Q (0) = Id.

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From Stratonovich to Ito, t σk

• X i,N

s

• dBk

s =

t σk

• X i,N

s

• dBk

s + 1

2 t

∞

• k=1

(Dσk · σk)

• X i,N

s

• ds

where (Dσk · σk)i (x) = d

j=1 σj k (x) ∂jσi k (x). Further assume

div σk = 0 for each k ∈ N Along with the previous assumptions on Q, it implies 0 =

∞

• k=1

d

• j=1

σj

k (x) ∂jσi k (x) .

Stratonovich and Itô formulations coincide.

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SDE - Itô formulation

dX i,N

t

= 1 N

N

• j=1

K

• X i,N

t

− X j,N

t

• dt +

∞

• k=1

σk

• X i,N

t

• dBk

t

i = 1, ..., N.

SPDE - Itô formulation

dµt + div (bµtµt) dt +

∞

• k=1

div (σk (x) µt) dBk

t = 1

2∆µt.

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P1(Rd) =

• µ

probability on Rd

• Rd |x| dµ(x) < +∞
• W1(ν, µ) =

inf

m∈Γ(µ,ν)

• R2d |x − y|m( dx, dy),

µ, ν ∈ P1(Rd) m ∈ Γ(µ, ν) iff, Γ(µ, ν) = {m ∈ P1(R2d) : m(A×Rd) = µ(A), m(Rd ×A) = ν(A), ∀A ∈ B(Rd)}

Remark

The metric space (P1(Rd), W1) is complete and separable.

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X∞ space of the stochastic processes, µ : [0, T] × Ω → P1(Rd) such that E

• sup

t∈[0,T]

• Rd |x| dµt(x)
• < +∞

d∞(µ, ν) := E

• supt∈[0,T] W1(µt, νt)
• Remark

It follows by the completeness of (P1(Rd), W1) that (X∞, d∞) is a complete metric space.

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Initial Condition

Concerning the initial condition µ0 : Ω → P1(Rd) of the SPDE we shall always assume that i) µ0 is F0-measurable; ii) E

• Rd |x| dµ0(x)
• < ∞.

Definition

µ ∈ X∞ is a solution of the SPDE with initial condition µ0 if, for all φ ∈ C2

b

• Rd

, µt, φ is Ft-adapted for every test function φ ∈ C∞

b (Rd)

weak formulation µt, φ = µ0, φ + t µs, bµs · ∇φ ds + 1 2 t µs, ∆φ ds +

∞

• k=1

t µs, σk · ∇φ dBk

s .

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Part III Existence and limit Theorem

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Theorem

Given T ≥ 0 and µ0 : Ω → P1(Rd), there exists a solution µ = (µt)t∈[0,T]

• f the SPDE starting from µ0 and defined up to time T.

Moreover, if µN

0 → µ0, as N → ∞, then

µN → µ, as N → ∞, in the metric d∞.

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Linear SPDE

dµt + div (bµt) dt +

∞

• k=1

div (σk (x) µt) dBk

t = 1

2∆µt. b = b(x, t, ω), Ft-adapted process, continuous in t, Lipschitz continuous in x. dXt = b(t, Xt) dt + ∞

k σk(Xt) dBk t

X0 = x ∈ Rd

Proposition

Given µ0, the push forward µt(ω) = X(t, ., ω)#µ0(ω) is in X∞ and solves the linear SPDE.

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Given µ = (µt)t∈[0,T] ∈ X ∞, bµ(t, x, ω) :=

• Rd K(x − y)µt(ω, dy).

dX µ

t

= bµ(X µ

t ) dt + k σk(X µ t ) dBk t

X µ = x Φµ0 : X∞ → X∞ (Φµ0µ)t(ω) := X µ(t, ., ω)#µ0(ω) ω-a.s..

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Theorem

The operator Φµ0 has a unique fixed point µ = {µt} in X∞, this fixed point is a solution of the SPDE. Φµ0 is a contraction, d∞(Φµ0µ, Φµ0ν) ≤ γT d∞(µ, ν) ∀ µ, ν ∈ X ∞

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Let ω ∈ Ω and t ∈ [0, T] be fixed. m = (X µ(t, ., ω), X ν(t, ., ω))#µ0 ∈ Γ((Φµ0µ)t(ω), (Φµ0ν)t(ω)). Indeed, A ∈ B(Rd), m(A × Rd) = µ0{x ∈ Rd : X µ

t ∈ A} = (X µ t )#µ0(A) = (Φµ0µ)t(A).

In the same way, m(Rd × A) = (Φµ0ν)t(ω)(A). From the definition of the Wasserstein metric W1, d∞(Φµ0µ, Φµ0ν) ≤ E

• sup

t∈[0,T]

• Rd |X µ(t, x) − X ν(t, x)| dµ0
• .

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Given µ = {µt}t≥0, ν = {νt}t≥0 ∈ X ∞, E

• sup

t∈[0,T]

|X µ(t, x) − X ν(t, x)|

• F0
• ≤ γTE
• sup

t∈[0,T]

W1(µt, νt)

• F0
• By the definition of bµ:

|bµ(t, x) − bν(t, x)| =

• Rd K(x − y) dµt(y) −
• Rd K(x − y′) dνt(y′)
• Given ω ∈ Ω a.s. and t ∈ [0, T], for every m ∈ Γ(µt(ω), νt(ω))

|bµ(s, x) − bν(s, x)| =

• Rd×Rd K(x − y) dm(y, y′) −
• Rd×Rd K(x − y′) dm(y, y′)
• ≤
• Rd×Rd |K(x − y) − K(x − y′)| dm(y, y′)

≤LK

• Rd×Rd |y − y′| dm(y, y′)

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Lemma

Let T > 0. Let µ0, ν0 : Ω → P1(Rd) be two initial conditions, and let µ, ν ∈ X∞ be the respective solutions of the SPDE given by the contraction method described before, then there exists a constant CT > 0, such that d∞(µ, ν) ≤ CTE[W1(µ0, ν0)] SN

t (ω) = X SN

t

t

(., ω)#SN

0 (ω),

ω-a.s.

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Part IV Propagation of Chaos

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(Ω, F, Ft, P) filtered probability space, (Xi)i∈N, sequence of i.i.d., Rd-valued, F0-measurable r.v.s, Bk

t , k ≥ 1, independent Brownian motions, which are independent from

the Xi, (FB

t )t≥0 the filtration generated by (Bk t )k≥1.

SN

0 := 1 N

N

i=0 δXi → µ0, in the metric E [W1(·, ·)], as N → ∞.

Theorem

Given r ∈ N and φ1, ..., φr ∈ Cb

• Rd

, we have lim

N→∞ E

• φ1
• X 1,N

t

• · . . . · φr
• X r,N

t

• FB

t

• =

r

• i=1

µt, φi

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correlated stochastic particles existence result for a non linear SPDE convergence of the empirical measure SN

t to the solution of the SPDE µt

propagation of chaos

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Thank you for your attention

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