Kai Lai Chung

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First Prev Next Go To Go Back Full Screen Close Quit 1 Kai Lai Chung 1917-29

1 First Prev Next Go To Go Back Full Screen Close Quit 1 Kai Lai Chung Mathematicians are more inclined to build fire stations than to put out fires.

2 Courses from Chung First Prev Next Go To Go Back Full Screen Close Quit 2

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Peter Kotelenez Also in memory of Peter Kotelenez and the fire stations he built.

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Particle representations for stochastic partial differential equations Exchangeability and de Finetti s theorem Convergence of exchangeable systems From particle approximation to particle representation Derivation of SPDE Stochastic Allen-Cahn equation Particle representation Boundary conditions Tightness in D of approximations Uniqueness References Abstract New material joint with Dan Crisan. Earlier work with Peter Donnelly, Phil Protter, Jie Xiong, Yoonjung Lee, Peter Kotelenez

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Exchangeability and de Finetti s theorem X 1, X 2,... is exchangeable if P {X 1 Γ 1,..., X m Γ m } = P {X s1 Γ 1,..., X sm Γ m } (s 1,..., s m ) any permutation of (1,..., m). Theorem 1 (de Finetti) Let X 1, X 2,... be exchangeable. Then there exists a random probability measure Ξ such that for every bounded, measurable g, g(x 1 ) g(x n ) lim = g(x)ξ(dx) n n almost surely, and E[ m g k (X k ) Ξ] = k=1 m k=1 g k dξ

6 First Prev Next Go To Go Back Full Screen Close Quit 6 Convergence of exchangeable systems Kotelenez and Kurtz (21) Lemma 2 For n = 1, 2,..., let {ξ1 n,..., ξn n n } be exchangeable (allowing N n =.) Let Ξ n be the empirical measure (defined as a limit if N n = ), Ξ n = 1 Nn N n i=1 δ ξi n. Assume N n For each m = 1, 2,..., (ξ n 1,..., ξ n m) (ξ 1,..., ξ m ) in S m. Then {ξ i } is exchangeable and setting ξ n i = s S for i > N n, {Ξ n, ξ n 1, ξ n 2...} {Ξ, ξ 1, ξ 2,...} in P(S) S, where Ξ is the definetti measure for {ξ i }. If for each m, {ξ n 1,..., ξ n m} {ξ 1,..., ξ m } in probability in S m, then Ξ n Ξ in probability in P(S).

7 First Prev Next Go To Go Back Full Screen Close Quit 7 Lemma 3 Let X n = (X n 1,..., X n N n ) be exchangeable families of D E [, )- valued random variables such that N n and X n X in D E [, ). Define Ξ n = 1 N n Nn i=1 δ X n i P(D E [, )) Ξ = lim m 1 m m i= δ X i V n (t) = 1 N n Nn i=1 δ X n i (t) P(E) V (t) = lim m 1 m m i=1 δ X i (t) Then a) For t 1,..., t l / {t : E[Ξ{x : x(t) x(t )}] > } (Ξ n, V n (t 1 ),..., V n (t l )) (Ξ, V (t 1 ),..., V (t l )). b) If X n X in D E [, ), then V n V in D P(E) [, ). If X n X in probability in D E [, ), then V n V in D P(E) [, ) in probability.

8 First Prev Next Go To Go Back Full Screen Close Quit 8 From particle approximation to particle representation Xi n (t) = Xi n () B i (t) W (t) 1 n = Xi n () B i (t) W (t) n j=1 R b(x n i (s) X n j (s))ds b(x n i (s) z)v n (s, dz)ds If b is bounded and continuous and {Xi n()} {X i()}, then relative compactness is immediate and any limit point satisfies X i (t) = X i () B i (t) W (t) b(x i (s) z)v (s, dz)ds Assuming uniqueness for the infinite system, V n V where V (t) is the de Finetti measure for {X i (t)}. R

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Derivation of SPDE Applying Itô s formula ϕ(x i (t)) = ϕ(x i ()) ϕ (X i (s))db i (s) ϕ (X i (s))dw (s) L(V (s))ϕ(x i (s))ds where L(v)ϕ(x) = ϕ (x) b(x z)v(dz)ϕ (x). Averaging gives V (t), ϕ = V (), ϕ V (s), ϕ dw (s) V (s), L(V (s))ϕ( ) ds

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Gaussian white noise W (du ds) will denote Gaussian white noise on U [, ) with mean zero and variance measure µ(du)ds. For example, W (C [, t]), t, is Brownian motion with mean zero and variance µ(c). For appropriately adapted and integrable Z, M Z (t) = Z(u, s)w (du ds) U [,t] is a square integrable martingale with quadratic variation [M Z ] t = Z(u, s) 2 µ(du)ds. U [,t]

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Coupling through the center of mass 1 Let X(t) = lim n n n i=1 X i(t) for X i (t) = X i () σ(x i (s), X(s))dB i (s) b(x i (s), X(s))ds α(x i (s), X(s), u)w (du ds) U [,t] Setting a(x, y) = σ(x, y)σ(x, y) T α(x, y, u)α(x, y, u)µ(du) and Lϕ(x, y) = 1 2 i,j a(x, y) i j ϕ(x) b(x, y) ϕ(x) V (t), ϕ = V (), ϕ V (s), α(, X(s), u) ϕ( ) W (du ds) U [,t] V (s), Lϕ(, X(s)) ds where V (t) = lim k 1 k k i=1 δ X i (t) and X(t) = zv (t, dz).

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Stochastic Allen-Cahn equation Consider a family of SPDEs of the form dv = vdt F (v)dt noise, v(, x) = h(x), x D, v(t, x) = g(x), x D, t >, where F (v) = G(v)v and G is bounded above. For example, F (v) = v v 3 = (1 v 2 )v. To be specific, in weak form the equation is V (t), ϕ = V (), ϕ U [,t] for ϕ Cc 2 (D). cf. Bertini, Brassesco, and Buttà (29) V (s), ϕ ds V (s), ϕg(v(s, )) ds ϕ(x)ρ(x, u)dxw (du ds), D

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Is it a nail? {X i } independent, stationary, reflecting Brownian motions in D. A i () = h(x i () da i (t) = G(v(t, X i (t))dt U ρ(x i (t), u)w (du dt) If X i hits the boundary at time t, A i (t) is reset to g(x i (t)). V (t) = lim k 1 k k i=1 A i(t)δ Xi (t) V (t), ϕ = D ϕ(x)v(t, x)π(dx) where π is the stationary distribution for X i (normalized Lebesgue measure on D).

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Particle representation More generally, let B be the generator of a reflecting diffusion X in D and assume that X is ergodic with stationary distribution π. Let {X i, i 1} be independent, stationary diffusion with generator B. Assume that the boundary of D is regular for both X i and the time reversal of X i. Let τ i (t) = sup{s < t : X i (s) D, and A i (t) = g(x i (τ i (t)))1 {τi (t)>} h(x i ())1 {τi (t)=} (1) where τ i (t) G(v(s, X i (s)), X i (s))a i (s)ds U (τ i (t),t] 1 V (t), ϕ = lim n n ρ(x i (s), u)w (du ds), n ϕ(x i (t))a i (t) = i=1 τ i (t) b(x i (s))ds ϕ(x)v(t, x)π(dx).

15 First Prev Next Go To Go Back Full Screen Close Quit 15 Corresponding SPDE Define M ϕ,i (t) = ϕ(x i (t)) Bϕ(X i(s))ds. Then ϕ(x i (t))a i (t) = ϕ(x i ())A i () = ϕ(x i ())A i () U [,t] A i (s)dm ϕ,i (s) ϕ(x i (s))da i (s) ϕ(x i (s))b(x i (s))ds Bϕ(X i (s))a i (s)ds ϕ(x i (s))g(v(s, X i (s)), X i (s))a i (s)ds ϕ(x i (s))ρ(x i (s), u)w (du ds) A i (s)dm ϕ,i (s) Bϕ(X i (s))a i (s)ds

16 First Prev Next Go To Go Back Full Screen Close Quit 16 Averaging V (t), ϕ = V (), ϕ V (s), ϕg(v(s, ), ) ds ϕ(x)ρ(x, u)π(dx)w (du ds) U [,t] D which is the weak form of v(t, x) = v(, x) (G(v(s, x), x)v(s, x) b(x))ds ρ(x, u)w (du ds) B v(x, s)ds, U [,t] where B is the adjoint determined by gbfdπ = fb gdπ. bϕdπds V (s), Bϕ ds

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Approximating systems Let ψ be an L 1 (π)-valued stochastic process that is compatible with W, and assume (W, ψ) is independent of {X i } Define A ψ i to be the solution of A ψ i (t) = g(x i(τ i (t)))1 {τi (t)>} h(x i ())1 {τi (t)=} D τ i (t) G(ψ(s, X i (s)), X i (s))a ψ i (s)ds U (τ i (t),t] ρ(x i (s), u)w (du ds). τ i (t) b(x i (s))ds The {A ψ i } will be exchangeable, so we can define Φψ(t, x) to be the density of the signed measure determined by 1 N ΦΨ(t), ϕ ϕ(x)φψ(t, x)π(dx) = lim A ψ i N N (t)ϕ(x i(t)). i=1

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Apriori bounds Assume Lemma 4 Let H i (t) = Then U [,t] K 1 K 2 sup x,d sup x D K 3 sup v R,x D b(x) < ρ(x, u) 2 µ(du) < G(v, x) <. ρ(x i (s), u)w (du ds) = B i ( ρ(x i (s), u) 2 µ(du)ds). U A ψ i (t) ( g h K 1(t τ i (t)) sup H i (t) H i (r) )e K 3(t τ i (t)) τ i (t) r t ( g h K 1 t sup H i (t) H i (s) )e K3t Γ i (t). s t

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Lemma 5 Suppose that (W, ψ) is independent of {X i }. Then Φψ is {F W,ψ t }- adapted and for each i, E[A ψ i (t) W, ψ, X i(t)] = Φψ(t, X i (t)) so Φψ(t, X i (t)) E[Γ i (t) W, ψ, X i (t)] Remark 6 Let G X i t = σ(x i (r) : r t). Then the Markov property and the independence of (W, ψ) and X i imply Φψ(t, X i (t)) = E[A ψ i (t) W, ψ, X i(t)] = E[A ψ i (t) σ(w, ψ) GX i t ]. The properties of reverse martingales and Doob s inequality give E[ sup Φψ(t, X i (t)) 2 ] 4E[ sup A ψ i (t) 2 ] 4E[ sup Γ i (t) 2 ] t T t T t T

20 First Prev Next Go To Go Back Full Screen Close Quit 2 Proof. By exchangeability, E[A ψ i (t)ϕ(x i(t))f (W, ψ)] = E[ = E[ ϕ(x)φψ(t, dx)f (W, ψ)] ϕ(x)φψ(t, x)π(dx)f (W, ψ)] = E[ϕ(X i (t))φψ(t, X i (t))f (W, ψ)]. The last equality follows by the independence of X i (t) and (W, ψ), and the lemma follows by the definition of conditional expectation.

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Boundary conditions Recall Φψ(t, x) is the density of the signed measure determined by 1 ΦΨ(t), ϕ = lim N N N A ψ i (t)ϕ(x i(t)). If X i (t) is close to D, then by the regularity assumption, with high probability t τ i (t) is small and A ψ i (t) g(x i(τ i (t))). Consequently, for y D ΦΨ(B ɛ (y)) Φψ(t, y) = lim = g(y). ɛ π(b ɛ (y)) i=1

22 First Prev Next Go To Go Back Full Screen Close Quit 22 Tightness in D of approximations A ψ i (t) = g(x i(τ i (t)))1 {τi (t)>} h(x i ())1 {τi (t)=} τ i (t) G(ψ(s, X i (s)), X i (s))a ψ i (s)ds U (τ i (t),t] ρ(x i (s), u)w (du ds). τ i (t) b(x i (s))ds Let Z ψ i (t) = g(x i(t)) G(ψ(s, X i (s)), X i (s))a ψ i (s)ds b(x i (s))ds ρ(x i (s), u)w (du ds). U [,t] Then A ψ i (t) = g(x i(t)) Z ψ i (t) Zψ i (τ i(t)) (h(x i ()) g(x i ()))1 {τi (t)=}.

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Estimates on modulus of continuity The Skorohod modulus of continuity of A ψ i can be bounded in terms of the ordinary modulus of continuity of Z ψ i. Lemma 7 Define γi = inf{t : X i (t) D}. Then with probability one, γi >, and for δ < γ i, w (A ψ i, δ, T ) w(g X i, 4δ, T ) 2w(Z ψ i, 4δ, T ). (2) The relative compactness for {A ψ i } for fixed i then follows from the relative compactness of {Z ψ i } in C Rd[, ).

24 First Prev Next Go To Go Back Full Screen Close Quit 24 Uniqueness L 1 sup v,x D G(v,x) 1 v 2 <. L 2 sup v1,v 2,x D G(v 1,x) G(v 2,x) v 1 v 2 ( v 1 v 2 ) <. A v 1 i (t) A v 2 i (t) τ i (t) τ i (t) G(v 1 (s, X i (s)), X i (s))a v 1 i (s) G(v 2 (s, X i (s)), X i (s))a v 2 i (s) ds L 1 (1 E[Γ i (s) W, X i (s)] 2 ) A v 1 i (s) A v 2 i (s) ds τ i (t) 2L 2 E[Γ i (s) W, X i (s)]γ i (s) v 1 (s, X i (s)) v 2 (s, X i (s)) ds L 1 (1 C 2 ) A v 1 i (s) A v 2 i (s) ds 2L 2 C 2 v 1 (s, X i (s)) v 2 (s, X i (s)) ds 1 {Γi (s)>c} {E[Γ i (s) W,X i (s)]>c}γ i (s)l 3 (1 E[Γ i (s) W, X i (s)] 2 )ds

25 First Prev Next Go To Go Back Full Screen Close Quit 25 References Lorenzo Bertini, Stella Brassesco, and Paolo Buttà. Boundary effects on the interface dynamics for the stochastic AllenCahn equation. In Vladas Sidoravičius, editor, New Trends in Mathematical Physics, pages Springer, 29. Peter M. Kotelenez and Thomas G. Kurtz. Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type. Probab. Theory Related Fields, 146(1-2): , 21. ISSN doi: 1.17/ s URL

26 First Prev Next Go To Go Back Full Screen Close Quit 26 Abstract Particle representations for stochastic partial differential equations Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. The talk will focus on situations where the particle locations are given by an iid family of diffusion processes, and the weights are chosen to obtain a nonlinear driving term and to match given boundary conditions for the SPDE. (Recent results are joint work with Dan Crisan.)

for all f satisfying E[ f(x) ] <.

for all f satisfying E[ f(x) ] <. . Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if