Discretization of Stochastic Differential Systems With Singular - - PowerPoint PPT Presentation

Discretization of Stochastic Differential Systems With Singular Coefficients Part I

Denis Talay

INRIA Sophia Antipolis, France TOSCA Project-team

ICERM - Brown – November 2012

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Why is Probability useful?

THE WORLD IS COMPLEX

◮ The physical model is badly calibrated (e.g., MEG or electrical

neuronal activity: few sensors),

◮ The physical law is not completely known (e.g., turbulence,

meteorology,. . . ),

◮ There is no physical law (e.g., finance).

THE PARTIAL DIFFERENTIAL EQUATIONS ARE COMPLEX

◮ Mathematical analysis (existence, uniqueness, smoothness), ◮ Probabilistic analysis of deterministic numerical methods (cf.

Kushner, or domain decompositions, or artificial boundary conditions),

◮ Probabilistic numerical methods for high dimensional problems

and/or equations in domains with possibly complex geometries and/or small viscosities (high Reynolds numbers),. . . ).

SUMMARY:

◮ Probability theory (in particuler, stochastic integration theory) is

used to solve problems which, by nature, are deterministic or ’stochastic’,

◮ Probabilistic models and numerical methods are used when

deterministic ones are unefficient.

◮ In all cases, one seeks a statistical information on the model:

classical numerical analysis needs to be deeply adapted. Remarks:

◮ For physicists, Stochastic PDEs often are PDEs with random

coefficients,

◮ Stochastic collocation methods are not stochastic.

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

General parabolic PDEs

Let b : Rd → Rd and σj : Rd → Rd, (1 ≤ j ≤ r). Consider the elliptic

• perator

Lψ(x) :=

d

• i=1

bi(x) ∂iψ(x) + 1 2

d

• i,j=1

ai

j(x) ∂ijψ(x),

where a(x) := σ(x) σ(x)t, and the evolution problem    ∂u ∂t (t, x) = Lu(t, x), t > 0, x ∈ Rd, u(0, x) = f(x), x ∈ Rd.

The Euler scheme for SDEs

Let (Gj

p) be i.i.d. N(0, 1) and h > 0 be the discretization step.

       ¯ X h

0 (x)

= x, ¯ X h

(p+1)h(x)

= ¯ X h

ph(x) + b

• ¯

X h

ph(x)

• h

+ r

j=1 σj

• ¯

X h

ph(x)

√ h Gj

p+1. ◮ Easy to simulate (even for Lévy driven SDEs). ◮ Discretizes the stochastic differential equation

Xt(x) = x + t b(Xs(x)) ds + t σ(Xs(x)) dWs.

Moments of the Euler scheme

E {¯ X h

(p+1)h(x) − ¯

X h

ph} = E b

¯ X h

ph(x)

• h,

E {(¯ X h

(p+1)h(x) − ¯

X h

ph) · (¯

X h

(p+1)h(x) − ¯

X h

ph)t} = E a(¯

X h

ph) h + O

• h2

.

Moments of the Euler scheme

E {¯ X h

(p+1)h(x) − ¯

X h

ph} = E b

¯ X h

ph(x)

• h,

E {(¯ X h

(p+1)h(x) − ¯

X h

ph) · (¯

X h

(p+1)h(x) − ¯

X h

ph)t} = E a(¯

X h

ph) h + O

• h2

.

Probabilistic interpretation of parabolic PDEs

E f(¯ X h

T(x)) − u(T, x)

=

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

(p+1)h(x)

• − u
• T − ph, ¯

X h

ph(x)

• =

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

ph(x)

• − u
• T − ph, ¯

X h

ph(x)

• + h

T/h−1

• p=0

E

• Lu
• T − (p + 1)h, ¯

X h

ph(x)

• +

T/h−1

• p=0

O

• h2

= h

T/h−1

• p=0

E

• Lu
• T − ph, ¯

X h

ph(x)

• − ∂u

∂t

• T − ph, ¯

X h

ph(x)

• + O (h)

= O (h) , since ∂u ∂t (t, x) = Lu(t, x).

Probabilistic interpretation of parabolic PDEs

E f(¯ X h

T(x)) − u(T, x)

=

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

(p+1)h(x)

• − u
• T − ph, ¯

X h

ph(x)

• =

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

ph(x)

• − u
• T − ph, ¯

X h

ph(x)

• + h

T/h−1

• p=0

E

• Lu
• T − (p + 1)h, ¯

X h

ph(x)

• +

T/h−1

• p=0

O

• h2

= h

T/h−1

• p=0

E

• Lu
• T − ph, ¯

X h

ph(x)

• − ∂u

∂t

• T − ph, ¯

X h

ph(x)

• + O (h)

= O (h) , since ∂u ∂t (t, x) = Lu(t, x).

Probabilistic interpretation of parabolic PDEs

E f(¯ X h

T(x)) − u(T, x)

=

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

(p+1)h(x)

• − u
• T − ph, ¯

X h

ph(x)

• =

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

ph(x)

• − u
• T − ph, ¯

X h

ph(x)

• + h

T/h−1

• p=0

E

• Lu
• T − (p + 1)h, ¯

X h

ph(x)

• +

T/h−1

• p=0

O

• h2

= h

T/h−1

• p=0

E

• Lu
• T − ph, ¯

X h

ph(x)

• − ∂u

∂t

• T − ph, ¯

X h

ph(x)

• + O (h)

= O (h) , since ∂u ∂t (t, x) = Lu(t, x).

Probabilistic interpretation of parabolic PDEs

E f(¯ X h

T(x)) − u(T, x)

=

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

(p+1)h(x)

• − u
• T − ph, ¯

X h

ph(x)

• =

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

ph(x)

• − u
• T − ph, ¯

X h

ph(x)

• + h

T/h−1

• p=0

E

• Lu
• T − (p + 1)h, ¯

X h

ph(x)

• +

T/h−1

• p=0

O

• h2

= h

T/h−1

• p=0

E

• Lu
• T − ph, ¯

X h

ph(x)

• − ∂u

∂t

• T − ph, ¯

X h

ph(x)

• + O (h)

= O (h) , since ∂u ∂t (t, x) = Lu(t, x).

Probabilistic interpretation of parabolic PDEs

E f(¯ X h

T(x)) − u(T, x)

=

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

(p+1)h(x)

• − u
• T − ph, ¯

X h

ph(x)

• =

T/h−1

• p=0

E

• u
• T − (p + 1)h, ¯

X h

ph(x)

• − u
• T − ph, ¯

X h

ph(x)

• + h

T/h−1

• p=0

E

• Lu
• T − (p + 1)h, ¯

X h

ph(x)

• +

T/h−1

• p=0

O

• h2

= h

T/h−1

• p=0

E

• Lu
• T − ph, ¯

X h

ph(x)

• − ∂u

∂t

• T − ph, ¯

X h

ph(x)

• + O (h)

= O (h) , since ∂u ∂t (t, x) = Lu(t, x).

Convergence rate

Let F(·) be a functional on the path space. The global error of a Monte Carlo method is E F(X·) − 1 N

N

• k=1

E F ¯ X h,k

·

• = E F(X·) − E F(¯

X h

· )

• =:ǫd(h)

+ E F(¯ X h

· ) − 1

N

N

• k=1

F ¯ X h,k

·

• =:ǫs(h,N)

. The statistical error satisfies ∃C > 0, E |ǫs(h)| ≤ C √ N for all h.

Concerning the discretization error : Suppose that f has a polynomial growth at infinity. Under hypoellipticity conditions, or when all the functions of the problem are smooth, one has (T.-Tubaro, Bally-T. etc.) ed(h) = Cf(T, x) h + Qh(f, T, x) h2, where |Cf(T, x)| + suph|Qh(f, T, x)| ≤ C (1 + xQ)1 + K(T) T q Thus, Romberg extrapolation techniques can be used: E

• 2

N

N

• k=1

f

• ¯

X h/2,k

T

• − 1

N

N

• k=1

f

• ¯

X h,k

T

• = O
• h2

. Remark: The technique used in the proofs is purely probabilistic (stochastic flows of diffeomorphisms, Malliavin variations calculus).

Dirichlet boundary conditions

For        ∂u ∂t (t, x) = Lu(t, x), t > 0, x ∈ D, u(0, x) = f(x), x ∈ D, u(t, x) = g(x), x ∈ ∂D,

• ne has

u(t, x) = E f(Xt(x)) It<τ + E g(Xτ(x)) It≥τ, where τ:= ‘first boundary hitting time of (Xt)’. The stopped Euler scheme is defined as ¯ X h

ph∧τ h(x),

where τ h:= ‘first boundary hitting time of the Euler scheme’. For a convergence rate analysis, see Gobet, Menozzi, etc.

Neumann boundary conditions

For        ∂u ∂t (t, x) = Lu(t, x), t > 0, x ∈ D, u(0, x) = f(x), x ∈ D, ∇u(t, x) · n(x) = 0, x ∈ ∂D,

• ne has

u(t, x) = E f(Xt(x)) where X:= ‘reflected diffusion process’: Xt(x) = x + t b(Xs(x)) ds + t σ(Xs(x)) dWs + t n(Xs)dLs(X). Here, (Lt(X)) is an increasing process, namely the local time of X at the boundary . The reflected Euler scheme is defined in such a way that the simulation of the local time is avoided.

Consider a domain D ⊂ Rd, with smooth boundary. Let n(s) denote the unit inward normal vector at s ∈ D. Suppose that the vector field γ defining the reflection direction is uniformly non tangent to the boundary. Consider the reflected S.D.E. with smooth coefficients and strictly uniformly elliptic generator Xt = x + t b(Xs)ds + t σ(Xs)dWs + t γ(Xs) dks, where kt = t IXs∈D dks. To discretize the above reflected SDE , start at x ∈ D at time 0, and assume that one has obtained ˜ X h

ph ∈ D. Then observe that for all x in

a neighborhood of D, there exist a unique pair of functions πγ

∂D taking

values in ∂D and F γ taking values in R such that x = πγ

∂D(x) + F γ(x)γ(πγ ∂D(x)).

Then,

◮ For t ∈ [tn i , tn i+1], set

˜ Y i

t := ˜

X h

ph + b(˜

X h

ph)(t − tn i ) + σ(˜

X h

ph)(Wt − Wtn

i ).

◮

i) If ˜ Y i

(p+1)h /

∈ D, set ˜ X h

(p+1)h = πγ ∂D(˜

Y i

(p+1)h) − F γ(˜

Y i

(p+1)h)γ(˜

Y i

(p+1)h).

ii) If ˜ Y i

(p+1)h ∈ D, ˜

X h

(p+1)h = ˜

Y i

(p+1)h.

Let f be a function of class C5

b(D, R) which satisfies the compatibility

condition ∀z ∈ ∂D, [∇f γ](z) = [∇(Lf) γ](z) = 0.

• Theorem. (Bossy–Gobet–T.) One has

|E (f(˜ X h

T)) − E (f(XT))| ≤ K(T)

n

• α:|α|≤5

∂α

x f∞

for some constant K(T) uniform w.r.t. x and f.

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Stochastic Hamiltonian dynamics

• dQt = ∂pH(Qt, Pt)dt,

dPt = −∂qH(Qt, Pt)dt − F1(H(Pt, Qt))∂pH(Qt, Pt)dt + F2(H(Pt, Qt))dWt, where H : Rd × Rd → R, and F1, F2 : R → R. Problems to solve:

◮ Existence, uniqueness of an invariant probability measure µ; ◮ The measure µ has a continuous and strictly positivite density; ◮ Construction of an approximate ergodic process ( ¯

Qn, ¯ Pn);

◮ Precise estimate on the global error

• f(q, p)dµ − 1

K

K

• k=1

f( ¯ Qn

k/n, ¯

Pn

k/n)

• In,K

.

Our main assumptions

◮ H, F1, F2 are smooth functions; ◮ A convexity type assumption on D2H; ◮ ∂ppH is bounded; ◮ ∃R > 0, ∃C0 > 0, F1(x) ≥ C0 for x ≥ R; ◮ ∃C0 > 0, F2(x) ≥ C0; ◮ Boundedness conditions on the derivatives of F 2.

The implicit Euler scheme

The explicit Euler scheme may have moments not uniformly bounded in time (counterexamples. . . ). The implicit Euler scheme:

Choose an arbitrary 0 < ρ < 1.

       ¯ Qn

k+1 = ¯

Qn

k + ∂pH( ¯

Qn

k+1, ¯

Pn

k+1)ρ

n, ¯ Pn

k+1 = ¯

Pn

k − ∂qH( ¯

Qn

k+1, ¯

Pn

k+1)ρ

n − F1 ◦ H( ¯ Qn

k+1, ¯

Pn

k+1)ρ

n + F2 ◦ H( ¯ Qn

k, ¯

Pn

k )

• W(k+1)ρ/n − Wkρ/n
• .

Remark: using H + 1 as a Lyapunov function, one can prove that the implicit Euler scheme has moments uniformly bounded in time.

Ergodicity of the implicit Euler scheme

Uniform in time upper bounds for the moments = ⇒ existence of an invariant probability measure ¯ µn for the implicit Euler scheme. To get uniqueness, prove that the chain is positive Harris recurrentowing to sufficient conditions found in Meyn and Tweedie:

◮ Prove that the chain is forward accessible and 0 is a globalling

attracting state, which provides the irreducibilityof the chain;

◮ Check that the chain is a T-chain; for an irreductible T-chain,

every compact set is a petite set; thus, there obviously exists a petite set K such that E

• H( ¯

Qn

1, ¯

Pn

1)

• ( ¯

Qn

0, ¯

Ph

0) = (q, p)

• − H(q, p)

≤ −1 + b IK(q, p), ∀(q, p) ∈ R2d. For similar results: see Shardlow & Stuart (1999), Higham & Mattingly & Stuart (1999).

Ergodicity of the Hamiltonian process

Uniform in time upper bounds for the moments = ⇒ existence of an invariant probability measure µ for (Qt, Pt). To get uniqueness, prove:

◮ The law of (Qt, Pt) has a smooth density π(t, q, p) for all t > 0:

this results, e.g., from hypoellipticityand a localization technique (the latter argument is used because of the possible unboundedness of ∂pqH, ∂qqH);

◮ The density π(t, q, p) is strictly positive everywhere: this results

from Michel & Pardoux’s controllability argument, since the reachibility set of the system

• dQu

t = ∂pH(Qu t , Pu t )dt,

dPu

t = −∂qH(Qu t , Pu t )dt − F1(H(Pu t , Qu t ))∂pH(Qu t , Pu t )dt + F2(H(Pu t , Qu t

is the whole space. Remark: the measure µ has finite moments of all order.

Exponential decay of moments of (Qt, Pt): the statement

Set u(t, x, v) := E

• f(Xt, Vt)
• (X0, V0) = (x, v)
• −
• R2d f dµ.

Theorem 1 Suppose that f is a smooth function, and that all its derivatives have a growth at most polynomial at infinity. Let Dmu(t) denote the vector of the derivatives of order m of the mapping (q, p) → u(t, q, p). For all integer m there exist an integer sm and Cm > 0, γm > 0 such that |Dmu(t)| ≤ Cm(1 + |q|sm + |p|sm) exp(−γmt), ∀t > 0, ∀(q, p) ∈ R2d.

Sketch of the proof of Theorem 1

• 1. Prove that, for any ball B in R2d, there exist C > 0 and

λ > 0 such that

• B

|u(t)|2dµ ≤ C exp(−γt), ∀t > 0.

• 2. Show that the preceding inequality also holds for any

spatial derivative of u(t) (possibly with different real numbers C and γ). As µ has a smooth and strictly positive density w.r.t. Lebesgue’s measure, deduce from the Sobolev imbedding Theorem that, for any ball B in R2d, there exist C > 0 and γ > 0 such that ∀(x, v) ∈ B, |u(t, q, p)| ≤ C exp(−γt), ∀t > 0.

Sketch of the proof of Theorem 1 (cont.)

• 3. Then show that there exist C > 0 and γ > 0 such that
• |u(t)|2πs(q, p) dq dp ≤ C exp(−γt), ∀t > 0,

where πs(q, p) := 1 (H(q, p) + 1)s for some integer s.

• 4. Finally, prove that the prededing inequality also holds

for any spatial derivative of u(t) (possibly with different real numbers s, C and γ). Then conclude by using the Sobolev imbedding Theorem again.

Sketch of the proof of Theorem 1 (end)

Main step:in spite of the degeneracy of the generator L of (Qt, Pt),

• ne has

A.∃C > 0, ∃γ0 > 0,

• |u(t)|2dµ ≤ C exp(−γ0t), ∀t ≥ 0,

B.∃Ckl > 0, ∃γkl > 0,

• |u(t)|2(|q|k + |p|ℓ)dµ ≤ Ckℓ exp(−γkℓt), ∀t ≥ 0,
• C. exp(γT)
• |u(T)|2 dµ +

T exp(γt)

• ∂u

∂p (t)

• 2

dµ dt ≤ C, ∀T > 0,

• D. −

T exp(γ1t) ∂u ∂q (t)∂u ∂p (t)|q|2dµ dt ≤ C, ∀T > 0, E.

• ∂u

∂q (T)

• 2

dµ ≤ C exp(−γ2T), ∀T > 0.

Convergence rate of the implicit Euler scheme

Decomposition of the global error:

• f(q, p)dµ − In,K =
• f(q, p)dµ −
• R2d f(q, p)¯

µn(dq, dp)

• ed(n)

+

• R2d f(q, p)¯

µn(dq, dp) − In,K

• es(n,K)

.

◮ The term ed(n) is a discretization error: we expand it in terms of

1ρ/n, from which we justify Romberg-Richardson extrapolation techniques to accelerate the convergence rate.

◮ The term es(n, K) is a statistical error: we provide estimates by

using classical results on the weak convergence of normalized martingales.

Convergence rate of the implicit Euler scheme (cont.)

Theorem 2 Suppose that f is a smooth function, and that all its derivatives have a growth at most polynomial at infinity. Then ed(n) = C1 n + . . . + Cm nm + O

• 1

nm+1

• , ∀m ∈ N − {0},

for some real numbers Cj uniform w.r.t. n, and es(n, K) − − − − − →

K→+∞ 0 a.s.

⊞

• n

K es(n, K) = ⇒ N(0, Σn), with Σn uniformly bounded w.r.t. n.

Sketch of the proof of Theorem 2

Set ¯ Y n

k := ( ¯

Qn

k, ¯

Pn

k ).

One has u(jρ/n, ¯ Y n

k+1) E

= u(jρ/n, ¯ Y n

k )+Lu(jρ/n, ¯

Y n

k )ρ

n+C0(jρ/n, ¯ Y n

k )ρ2

n2 +r n

j,k+1

1 n3 . As u(t, q, p) solves du/dt = Lu, one deduces u(jρ/n, ¯ Y n

k+1) E

= u((j + 1)ρ/n, ¯ Y n

kρ/n) + C(jρ/n, ¯

Y n

k )ρ2

n2 + Rn

j,k+1

1 n3 . The function C(t, y) is a sum of terms of the type φ(q, p)∂Ju(t, q, p). The remainder term Rn

j,k+1 is a sum of terms of the type

E

• P( ¯

Y n

k )∂Ju

• jρ/n, ¯

Y n

k + θ( ¯

Y n

(k+1)ρ/n − ¯

Y n

k )

• , θ ∈ (0, 1).

Sketch of the proof of Theorem 2 (cont.)

Observe that 1 K

K

• k=1

f( ¯ Y n

k ) = 1

K

K

• k=1

u(0, ¯ Y n

k ) +

• R2d f dµ.

Thus 1 K

K

• k=1

f( ¯ Y n

k ) E

=

• R2d f dµ+ 1

K

K

• k=1

u(jρ/n, ¯ Y0)+ 1 K

K

• k=1

k−1

• j=0

C(jρ/n, ¯ Y n

k )ρ2

n2 + 1 K

K

• k=1

By ergodicity, lim

K→∞

1 K

K

• k=1

E u(kρ/n, ¯ Y0) = 0 and lim

K→∞ E f( ¯

Y n

k ) =

• R2 f d ¯

µn.

Sketch of the proof of Theorem 2 (end)

In view of the estimates of Theorem 1,

+∞

• j=0

|Rn

j,k+1| ≤

C0 1 − exp(−γρ/n)E (1 + | ¯ Y n

k |s + | ¯

Y n

k+1|s),

from which

+∞

• j=0

|Rn

j,k+1| ≤ Cn(1 + E | ¯

Y n

0 |s).

Moreover, in view of Theorem 1 again, ρ n lim

K→∞

1 K

K

• k=1

k−1

• j=0

E C(jρ/n, ¯ Y n

k ) =

∞

• R2d C(t, q, p)µ(dq, dp) dt+O

ρ n

• .

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

A simplified stochastic Lagrangian model

Consider a d-dimensional standard Brownian motion (Wt; t ∈ [0, T]), and ((Xt, Ut); t ∈ [0, T]) solution of              Xt = X0 + t Us ds, Ut = U0 + t B [Xs, Us; ρs] ds + t σ(s, Xs, Us) dWs, ρt is the density distribution of (Xt, Ut) for all t ∈ (0, T]. (1)

Here, B is the mapping from Rd × Rd × L1(R2d) to Rd defined by B [x, u; γ] =           

• Rd b(v, u)γ(x, v) dv
• Rd γ(x, v) dv

if

• Rd γ(x, v) dv = 0,

0 elsewhere, (2) where b : Rd × Rd → Rd is bounded .

Formally, the drift component of (1) involves (x, u) → E

• b (Ut, u)
• Xt = x
• .

(3) Such nonlinearity is typical of Lagrangian stochastic models for the position Xt and the velocity Ut of a generic fluid-particle in a turbulent flow: see the dramatically complex Probability Density Function (PDF) methods for turbulent flows and S. Pope’s models which aim to be alternative approaches to the Navier-Stokes equations for turbulent flows. Our objective is to show existence and uniqueness of a solution to

• ur simplified Lagrangian model.

Lagrangian stochastic models for monophasic turbulent flows

Statistical solutions to the Navier–Stokes equation: the Reynolds decomposition of the Eulerian velocity U of a turbulent flow is U(t, x, ω) = U(t, x) + u(t, x, ω), where U is the (ensemble) averaged part, and u is the fluctuating part. Reynolds Averaged Navier-Stokes (RANS) equations:        (∇x · U) = 0, ∂tU(i) +

• U · ∇xU(i)
• = −1

̺∇xP(i) + ν△xU(i) − ∂xju(i)u(j), U(0, x) = U0(x).

The gradient pressure ∇xP solves the Poisson equation: − 1 ̺△xP = ∂2

xi,xjU(i)U(j) + ∂2 xixju(i)u(j).

The Reynolds stress tensor stands for the covariance of velocity components: u(i)u(j) = U(i)U(j) − U(i)U(j).

The RANS equation is not closed. In Pope’s model, Lagrangian and Eulerian quantities are related as follows: for all suitable measurable function g : Rd → Rd, g(U)(t, x) = E [g(Ut)/Xt = x] . (4) The simplest model proposed by Pope (2003) is the simplified Langevin model            Xt = X0 + t

0 Us ds,

Ut = U0 − 1 ̺ t

0 ∇xP(s, Xs) ds + ν

t

0 △xU(s, Xs) ds

+C1 t E (s, Xs) k(s, Xs) (U(s, Xs) − Us) ds + t

• C2E (s, Xs) dWs.

Technical difficulties in the analysis

Difficulties come from the dependency of the drift coefficient on the conditional expectation. Related situations: Sznitman (1986): dζt = pt(ζt) dt + dWt, where pt is the Lebesgue density of ζt. Oelschlager (1985): dζt = F (ζt, pt(ζt)) dt + dWt, where F : Rd × R is a bounded function, and dζt = ∇pt(ζt) dt + dWt. Dermoune (2003): dζt = E (v(ζ0) / ζt) dt + dWt, where v : Rd → Rd is a bounded continuous function.

Our situation drastically differs from the above:

◮ our drift coefficient depends on conditional distributions rather

than the density ρt itself;

◮ the infinitesimal generator of (Xt, U) is not strongly elliptic .

A stochastic particle system

Consider                        X i,ǫ,N

t

= X i

0 +

t

0 Ui,ǫ,N s

ds, Ui,ǫ,N

t

= Ui

0 +

t

N

• j=1

b(Uj,ǫ,N

s

, Ui,ǫ,N

s

)φǫ(X i,ǫ,N

s

− X j,ǫ,N

s

)

N

• j=1
• φǫ(X i,ǫ,N

s

− X j,ǫ,N

s

) + ǫ

• ds

+ t

0 σ(s, Xs, Ui,ǫ,N s

) dW i

s, i = 1, · · · , N,

where {φǫ; ǫ > 0} is a family of mollifiers.

We prove that the particles propagate chaos: as N tends to infinity, (X 1,ǫ,N, U1,ǫ,N) converges weakly to the solution of              X ǫ

t = X0 +

t Uǫ

s ds,

Uǫ

t = U0 +

t Bǫ [X ǫ

s , Uǫ s; ρǫ s] ds +

t σ(s, Xs, Us) dWs, ρǫ

t is the density of (X ǫ t , Uǫ t ) for all t ∈ (0, T],

(5)

where the kernel Bǫ [x, u; γ] is defined by: for all nonnegative γ ∈ L1(R2d), (x, u) ∈ R2d, Bǫ [x, u; γ] =

• Rd b(v, u)φǫ ∗ γ(x, v) dv
• Rd φǫ ∗ γ(x, v) dv + ǫ

, (6) where φǫ ∗ γ(x, u) =

• Rd φǫ(x − y)γ(y, u) dy.

Our main theorem

Our assumptions.

◮ b is a bounded continuous function. ◮ The velocity diffusion coefficient σ is bounded and strongly

elliptic.

◮ For all 1 ≤ i, j ≤ d, σ(i,j) is Hölder continuous (in a reinforced

sense). Theorem. (i) For all ǫ > 0, the sequence {(X 1,ǫ,N, U1,ǫ,N); N ≥ 1} converges weakly to a weak solution (X ǫ, Uǫ) of (5). This solution is unique and, if Pǫ denotes the law by (X ǫ, Uǫ) (5), the interacting particle system is Pǫ-chaotic; that is, for every integer k ≥ 2 and every finite family {ψl; l = 1, · · · , k} of Cb(C([0, T] ; R2d)), Pǫ,N, ψ1 ⊗ ...ψk ⊗ . . . − →

k

• l=1

Pǫ, ψl, when N − → +∞. (7) (ii) When ǫ tends to 0, (X ǫ, Uǫ) converges weakly to the unique solution (X, U) of (1).

The non-linear martingale problems

• Definition. A probability measure P on C([0, T] ; R2d) is said a weak

solutionto (1) or a solution to the martingale problem (MP) if (i) P ◦ (x0, u0)−1 = µ0. (ii) For all t ∈ (0, T], the time-marginal P ◦ (xt, ut)−1 has a positive density ρt w.r.t. Lebesgue measure on R2d. (iii) For all f ∈ C2

b(R2d), the process

f(xt, ut) − f(x0, u0) − t Aρs(f)(s, xs, us) ds (8) is a P-martingale, where, for each positive γ ∈ L1(R2d), Aγ is defined as Aγ(f)(t, x, u) = (u · ∇xf(x, u)) + (B [x, u; γ] · ∇uf(x, u)) + 1 2

d

• i,j=1

(σσ∗)(i,j)(t, x, u)∂2

ui,ujf(x, u).

(9)

• Definition. A probability measure Pǫ on C([0, T]; R2d) is said a weak

solution to (5) or a solution to the martingale problem (MPǫ) if (i) Pǫ ◦ (x0, u0)−1 = µ0. (ii) For all t ∈ (0, T], the time–marginal Pǫ ◦ (xt, ut)−1 has a density ρǫ

t w.r.t. Lebesgue measure on R2d.

(iii) For all f ∈ C2

b(R2d), the process

f(xt, ut) − f(x0, u0) − t Aǫ

ρǫ

s (f)(s, xs, us) ds

is a Pǫ-martingale where, for all γ ∈ L1(R2d), Aǫ

γ is

defined as Aǫ

γ(f)(t, x, u) = (u · ∇xf(x, u)) + (Bǫ [x, u; γ] · ∇uf(x, u))

+ 1 2

d

• i,j=1

(σσ∗)(i,j)(t, x, u)∂2

ui,ujf(x, u).

A uniqueness result

• Proposition. There is at most one weak solution to Equation (1) and
• ne weak solution to Equation (5).

Sktech of the proof: One can easily prove existence and weak uniqueness for

• Y s,y,v

t

= y + t

s V s,y,v θ

dθ, V s,y,v

t

= v + t

s σ(θ, Y s,y,v, V s,y,v θ

) dWθ. In addition, the transition density Γ(s, y, v; t, x, u) of the solution satisfies the following estimate (see Francesco and Pascucci (2006)): sup

(y,v)∈R2d

• Rd |∇vΓ(s, y, v; t, x, u)| dx du ≤

C √ t − s, ∀ 0 ≤ s < t ≤ T.

A uniqueness result

Set S∗

t,s(f)(x, u) =

• R2d Γ(s, y, v; t, x, u)f(y, v) dy dv.

We then prove that the densities ρt and ρǫ

t are the unique solutions (in

appropriate spaces) to mild equations: for example, ∀ t ∈ (0, T], ρt = S∗

t,0(µ0) +

t S′

t,s (ρs(·)B [· ; ρs]) ds in L1(R2d).

An existence result

• Proposition. The martingale problem (MPǫ) has a unique solution Pǫ

and, when ǫ tends to 0, Pǫ converges to a solution of (MP). The proof proceeds in two steps:

◮ We show that {¯

Pǫ,N; N ≥ 1} is relatively compact and that any weakly convergent subsequence assigns full measure to the set

• f the solutions to the martingale problem (MPǫ).

◮ The probability measure Pǫ, solution of the martingale problem

(MPǫ), converges to the solution of the martingale problem (MP).

First step

Let ¯ µǫ,N be the empirical measure defined on M(C([0, T]; R2d)) by ¯ µǫ,N = 1 N

N

• i=1

δ{X i,ǫ,N,Ui,ǫ,N}. Let ¯ Pǫ,N = Q ◦ (¯ µǫ,N)−1 be the probability law of ¯ µǫ,N. Easy: The sequence {¯ Pǫ,N; N ≥ 1} is tight on M(C([0, T]; R2d)). Let ¯ Pǫ,∞ be the limit of a weakly converging sequence.

• Lemma. ¯

Pǫ,∞ assigns full measure to the set of the solutions to the martingale problem (MPǫ).

Second step

The sequence

• Pǫ := Pǫ ◦ ((xt, ut, ut − u0 −

t Bǫ[xs, us; ρǫ

s] ds); t ∈ [0, T])−1

is tight. The support of any accumulation point has full measure on the set of continuous functions ( x, u, D) of C([0, T]; R3d) satisfying

• x(t) =

x(0) + t

• u(s) ds, ∀ t ∈ [0, T],

and there exists a bounded function a such that sup

t∈[0,T]

• a(t)
• ≤ b∞,

and

• u(t) −

u(0) − D(t) = t

• a(s) ds, ∀ t ∈ [0, T].

Consider the following marginal distribution P of P on C([0, T]; R2d): P = P ◦ ((xt, ut); t ∈ [0, T])−1.

• Proposition. P solves the martingale problem (MP).

Sketch of the proof:

◮ Weak convergence (but we have to make fractions converge. . . ) ◮ Estimates by Francesco and Pascucci (2006) for ultraparabolic

PDEs .

◮ To overcome the difficulty due to the fact that Pǫ and Bǫ[· ; ρǫ]

depend on ǫ, we adapt a method designed by Stroock and Varadhan for the case of strongly elliptic diffusion processes:

Key lemma. For all 0 < t ≤ T, ρǫ

t converges to ρt in L1(R2d) when

ǫ → 0+. Key result (Stroock & Varadhan (1979)). Let {fn; n ≥ 1} be a sequence of non–negative measurable functions such that

• Rq fn(z) dz = 1, for all n ≥ 1. Suppose

◮ There exists a density function f such that, for all ψ ∈ Cc(Rq),

lim

n→+∞

• Rq fn(z)ψ(z) dz =
• Rq f(z)ψ(z) dz.

◮ For all h ∈ Rq,

lim

|h|→0 sup n∈N

• Rq |fn(z + h) − fn(z)| dz = 0.

Then {fn} converges towards f in L1(Rq).

Complements and perspectives

◮ Extensions to (simplified) stochastic Lagrangian models with

specular boundary conditions : see Bossy & Jabir (2008-2012).

◮ For numerical issues and an application to meteorology,

see Bossy et al. (2008-2012)).

◮ For the study of the Poisson equation, see Bossy and Fontbona

(in progress).

◮ For the full treatment of Pope’s model: a challenging problem!

Outline

Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Many other problems:

◮ McKean–Vlasov systems for Navier–Stokes equations, ◮ Discretization of (reflected) backward SDEs for variational

inequalities (American options, Stefan problems),

◮ Stochastic PDEs (Gyöngy, Walsh, de Bouard–Debussche), ◮ Discretization of Hamilton–Jacobi–Belman equations for

stochastic control (Krtlov, Barles–Jakobsen),

◮ Bessel–type SDEs, ◮ Variance reduction methods (Arouna–Lapeyre,

Kebaier–Kohatsu-Higa,. . . ).