Particle models for Wasserstein type diffusion

Transcription

1 Particle models for Wasserstein type diffusion Vitalii Konarovskyi University of Leipzig Bonn, 216 Joint work with Max von Renesse

2 Wasserstein diffusion and Varadhan s formula (von Renesse, Sturm 9) The Wesserstein diffusion is the process on P 2[, 1] (constructed by Dirichlet form methods) that is a weak solution of the SPDE with f, ˆΓ(ν) = dµ t = β µ tdt + ˆΓ(µ t)dt + div( µdw t), I gaps(ν) [ f (I +) + f (I ) 2 f ] (I +) f (I +). I Theorem For each Borel A, B P[, 1], where lim t ln pt(a, B) = d2 W(A, B), t 2 d W(A, B) = ess-inf {d W(ν 1, ν 2) : ν 1 A, ν 2 B}, p t(a, B) = P{µ A, µ t B}.

3 Goal of the talk We are going to propose other particles models on the real line which satisfy Varadhan formula for the short time with the quadratic Wasserstein distance as rate function. First we construct the process µ t in P 2(R) satisfying the SPDE dµ t = Γ(µ t)dt + div( µdw t), i.e. for each f Cb (R), f, µ t t f, Γ(µs) ds is a martingale with quadratic variation t (f ) 2, µ s ds. Then we state the Varadhan formula lim t ln pt(a, B) = d2 W(A, B), t 2

4 A system of coalescing diffusion particles

5 Arratia flow (Arratia 79) The Arratia flow (coalescing Brownian motions) is the mathematical model of interacting Brownian particles on the real line which start from all points of R; move independently up to the moment of collision; coalesce; the diffusion rate is unchangeable and equals 1. If x(u, t) is the position of particles at time t starting from u R then R u x(u, ) C[, T ] has a càdlág modification and 1 x(u, ) is a continuous square integrable martingale with respect to the joint filtration; 2 x(u, ) = u; 3 x(u, t) x(v, t), u < v; 4 x(u, ) t = t; 5 x(u, ), x(v, ) t =, t < τ u,v = inf{t : x(u, t) = x(v, t)}. Moreover, uniquely determine the distribution.

6 Arratia flow (Arratia 79) The Arratia flow (coalescing Brownian motions) is the mathematical model of interacting Brownian particles on the real line which start from all points of R; move independently up to the moment of collision; coalesce; the diffusion rate is unchangeable and equals 1. If x(u, t) is the position of particles at time t starting from u R then R u x(u, ) C[, T ] has a càdlág modification and 1 x(u, ) is a continuous square integrable martingale with respect to the joint filtration; 2 x(u, ) = u; 3 x(u, t) x(v, t), u < v; 4 x(u, ) t = t; 5 x(u, ), x(v, ) t =, t < τ u,v = inf{t : x(u, t) = x(v, t)}. Moreover, uniquely determine the distribution.

7 Arratia flow (Arratia 79) The Arratia flow (coalescing Brownian motions) is the mathematical model of interacting Brownian particles on the real line which start from all points of R; move independently up to the moment of collision; coalesce; the diffusion rate is unchangeable and equals 1. If x(u, t) is the position of particles at time t starting from u R then R u x(u, ) C[, T ] has a càdlág modification and 1 x(u, ) is a continuous square integrable martingale with respect to the joint filtration; 2 x(u, ) = u; 3 x(u, t) x(v, t), u < v; 4 x(u, ) t = t; 5 x(u, ), x(v, ) t =, t < τ u,v = inf{t : x(u, t) = x(v, t)}. Moreover, uniquely determine the distribution.

8 Physical modification of the Arratia flow Arratia flow Particles start from all points of R; move independently up to the moment of collision and then coalesce; the diffusion rate is unchangeable and equals 1. A modification Particles start from all points of [, 1]; move independently up to the moment of collision and then coalesce; transfer mass that obeys the conservation law; the mass distribution at start is the Lebesgue measure on [, 1] (the mass of particle is the size of the cluster); the diffusion rate is inversely proportional to the mass.

9 The main problem of description and investigation Particles must start with zero mass and, consequently, with infinite diffusion rate. Remedy: If particles has very high diffusion rate, then they quickly coalesce and form a particle with smaller diffusion rate. The motion of a fixed particle subsystem depends on the evolution of the whole system and, consequently, the methods created for the investigation of the Arratia flow do not work. Remedy:: It turns out that the process describing the evolution of particles is a martingale in an infinite dimensional space with good quadratic variation. It allows to use the inf.-dim. stochastic analysis for studying of the model.

10 Approximation by finite number of particles The system of particles {yk n (t)} starting from { k, k = 1,..., n} with masses 1 can be n n constructed by coalescing of trajectories of independent Wiener processes w k (t) as follows Setting y n (u, t) = yk n (t), for u [ k 1, k ), n n u [, 1], we have Theorem (K. 14) 1 ) y n (u, ) is a continuous square integrable martingale 2 ) y n (u, ) = un +1 n ; 3 ) y n (u, t) y n (v, t), u < v; 4 ) y n (u, ) t = t ds m n(u,s), m n(u, t) = Leb{v : y n (v, t) = y n (u, t)}; 5 ) y n (u, ), y n (v, ) t =, t τ n u,v = inf{t : y n (u, t) = y n (v, t)} T. y n y in D([, 1], C[, T ]) in distribution, where y satisfies 1 ) 5 ), with 2 ) replaced by y(u, ) = u, u [, 1].

11 Idea of the proof To prove the tightness (and, consequently, relatively compactness) of {y n } in D([, 1], C[, T ]) we need to control increments at three points P{ y n (u + h, ) y n (u, ) C[,T ] > λ, y n (u, ) y n (u h, ) C[,T ] > λ} 4h2 λ 2 ; to have a tightness of {y n (u, )} in C[, T ], for all u [, 1], following from: {y n (u, )} n 1 is a family of square integrable martingales with 1 + { E m β n(u, t) = P m n(u, t) < 1 } dr C β r 3 (,, β 3 ), 1/β t β 2 where P{m n(u, t) < r} 2 1 r 3/2 t 2π e x 2 2πt. 2 dx 2r r

12 Ito s formula and the induced random flow on P 2 (R) Ito s formula (K. 14) For each f C 2 b (R), 1 f(y(u, t))du = f(u)du + t 1 1 t f(y(u, s)) m(u, s) duds. f(y(u, s))dy(u, s)du Corollary. Let µ t = y(, t) # Leb, i.e. f, µ t = 1 f(y(u, t))du, then M f t = f, µ t is a martingale with the quadratic variation where f, Γ(µ s) = 1 2 [M f ] t = t 1 x supp(µ s) f(x). t f, Γ(µ s) ds f 2 (y(u, s))dsdu = t dµ t = Γ(µ t)dt + div( µdw t) f 2, µ s ds,

13 Ito s formula and the induced random flow on P 2 (R) Ito s formula (K. 14) For each f C 2 b (R), 1 f(y(u, t))du = f(u)du + t 1 1 t f(y(u, s)) m(u, s) duds. f(y(u, s))dy(u, s)du Corollary. Let µ t = y(, t) # Leb, i.e. f, µ t = 1 f(y(u, t))du, then M f t = f, µ t is a martingale with the quadratic variation where f, Γ(µ s) = 1 2 [M f ] t = t 1 x supp(µ s) f(x). t f, Γ(µ s) ds f 2 (y(u, s))dsdu = t dµ t = Γ(µ t)dt + div( µdw t) f 2, µ s ds,

14 y as a martingale in L 2 [, 1] For fixed h 1, h 2 L 2[, 1], (y(t), h i) = 1 y(u, t)h(u)du are square integrable martingales with the quadratic variation pr g h denotes the projection of h in L 2 on the subspace of σ(g)-measurable functions [(y( ), h 1), (y( ), h 2)] t = Moreover, t (pr y(s) h 1, h 2)ds. E y(t) id 2 L 2 = E t 1 1 dsdu <. m(u, s) Hence, L 2[, 1] y(t) = id + t pr y(s) dw (s), where E pr y(t) 2 HS = E{the number of distinct values of y(t)} = E 1 1 m(u,t) du C 3 t.

15 A couple words about dy(t) = pr y(t) dw (t) We consider the equation dy(t) = pr y(t) dw (t), y(t) L 2[, 1], (1) where W is a cylindrical Wiener process in L 2[, 1]. The map g pr g h is not continuous in L 2. For all y L 2+ε [, 1], (1) has a weak solution such that y() = y. Any solution of (1) (starting from y L 2+ε [, 1]) has a modification {ỹ(u, t)} from D((, 1), C[, T ]) sat satisfies the same martingale conditions. If { y (u ) u u α 1, α > 1. Then for all ε > 2 } { P { P lim t t m(u,t) 1 2α+1 (ln 1 t ) = = P 1+ɛ } { ỹ(u lim,t) y (u ) t α t 2α+1 (ln 1 t ) 1 2 +ɛ = = P m(u lim,t) t 1 t 2α+1 (ln 1 t ) ỹ(u lim,t) y (u ) t α t 2α+1 (ln 1 t ) ɛ = + } ɛ = + = 1, } = 1.

16 Large deviations for y ε ( ) = y(ε ) H = {ϕ C([, T ], L 2 [, 1]) : ϕ() = id and t ϕ(t) is absolutely continuous}, { 1 T 2 I(ϕ) = ϕ(t) 2 L 2 dt, ϕ H, +, otherwise. We extend the space L 2[, 1] to L 2([, 1], ν), where ν(du) = κ(u)du and κ : [, 1] [, 1] { u β, u [, 1/2] κ(u) = (1 u) β, u (1/2, 1], for some fixed β > 1. Theorem (K., von Renesse 15) The family of processes {y ε = y(ε )} ε> satisfies a large deviations principle in C([, T ], L 2([, 1], ν)) with the good rate function I (and weak LDP in C([, T ], L 2[, 1])). Corollary. The family {y ε (1) = y(ε)} ε> satisfies a large deviations principle in L 2([, 1], ν) with the good rate function L 2 (and weak LDP in L 2[, 1]).

17 Varadhan s formula Corollary For suitable measurable A L 2 [, 1], Corollary For suitable measurable A P(R), lim ε ln P{y(ε) A} = d2 L 2 (id, A). ε 2 lim ε ln ε P{µε A} = d2 W(Leb [,1], A). 2

18 A system of diffusion particles with sticky-reflected type interaction

19 Two interacting particles Let m 1, m 2 (, 1), m 1 + m 2 = 1, be fixed. We consider the evolution of two particles on R described by process X(t) = (X 1(t), X 2(t)), such that on the set {x 1 < x 2} the particles move as independent Brownian particles with 1 diffusions m 1 and 1 m 2 (with masses m 1 and m 2) respectively; if particles collide then they sticky-reflect from each other; on the set {x 1 = x 2} particles move as a Brownian particle with diffusion 1 (with mass m 1 + m 2 = 1) Remark. The sticky-reflected interaction implies t I {X 1 (s)=x 2 (s)}ds a.s., t +. The process X can be constructed by Dirichlet form method (Grothaus, Vosshall 15) or by SDE approach (Engelbert, Peskir 14)

20 The description of X (n = 2) Let E 2 = {(x 1, x 2) R 2 : x 1 < x 2} and Γ = E 2 = {(x 1, x 2) R 2 : x 1 = x 2}. For f, g Cb 2 (E 2 ) we set E {{1},{2}} (f, g) = 1 ( f 1 (x)g 1(x) + f ) 2(x)g 2(x) Leb {{1},{2}} (dx) 2 E 2 m 1 m 2 E {1,2} (f, g) = 1 f (s, s)g (s, s)ds = 1 f 2 R 2 {1,2}(x)g {1,2}(x)Leb {1,2} (dx) Γ {1,2} Let µ = Leb {{1},{2}} + βleb {1,2} and Then E(f, g) = E {{1},{2}} (f, g) + βe {1,2} (f, g) where Lf(x) = E(f, g) = ( Lf, g) L2 (E 2,µ), ( f 1 (x) m 1 + f 2 (x) m 2 )I E 2(x) +f (x 1, x m 1 m 2 1)I Γ(x) n f(x)i 2β Γ(x). ( ) ( dw1(t) d(m1w 1(t) + m 2w 2(t)) dx(t) = I E 2(X(t)) + I dw Γ(X(t)) 2(t) d(m 1w 1(t) + m 2w 2(t)) ( ) m1m 2 dt I Γ(X(t)) n 2β dt )

21 The description of X (n = 2) Let E 2 = {(x 1, x 2) R 2 : x 1 < x 2} and Γ = E 2 = {(x 1, x 2) R 2 : x 1 = x 2}. For f, g Cb 2 (E 2 ) we set E {{1},{2}} (f, g) = 1 ( f 1 (x)g 1(x) + f ) 2(x)g 2(x) Leb {{1},{2}} (dx) 2 E 2 m 1 m 2 E {1,2} (f, g) = 1 f (s, s)g (s, s)ds = 1 f 2 R 2 {1,2}(x)g {1,2}(x)Leb {1,2} (dx) Γ {1,2} Let µ = Leb {{1},{2}} + βleb {1,2} and Then E(f, g) = E {{1},{2}} (f, g) + βe {1,2} (f, g) where Lf(x) = E(f, g) = ( Lf, g) L2 (E 2,µ), ( f 1 (x) m 1 + f 2 (x) m 2 )I E 2(x) +f (x 1, x m 1 m 2 1)I Γ(x) n f(x)i 2β Γ(x). ( ) ( dw1(t) d(m1w 1(t) + m 2w 2(t)) dx(t) = I E 2(X(t)) + I dw Γ(X(t)) 2(t) d(m 1w 1(t) + m 2w 2(t)) ( ) m1m 2 dt I Γ(X(t)) n 2β dt )

22 The description of X (n = 2) Let E 2 = {(x 1, x 2) R 2 : x 1 < x 2} and Γ = E 2 = {(x 1, x 2) R 2 : x 1 = x 2}. For f, g Cb 2 (E 2 ) we set E {{1},{2}} (f, g) = 1 ( f 1 (x)g 1(x) + f ) 2(x)g 2(x) Leb {{1},{2}} (dx) 2 E 2 m 1 m 2 E {1,2} (f, g) = 1 f (s, s)g (s, s)ds = 1 f 2 R 2 {1,2}(x)g {1,2}(x)Leb {1,2} (dx) Γ {1,2} Let µ = Leb {{1},{2}} + βleb {1,2} and Then E(f, g) = E {{1},{2}} (f, g) + βe {1,2} (f, g) where Lf(x) = E(f, g) = ( Lf, g) L2 (E 2,µ), ( f 1 (x) m 1 + f 2 (x) m 2 )I E 2(x) +f (x 1, x m 1 m 2 1)I Γ(x) n f(x)i 2β Γ(x). ( ) ( dw1(t) d(m1w 1(t) + m 2w 2(t)) dx(t) = I E 2(X(t)) + I dw Γ(X(t)) 2(t) d(m 1w 1(t) + m 2w 2(t)) ( ) m1m 2 dt I Γ(X(t)) n 2β dt )

23 The case of n particles We fix m 1,..., m n. Let Θ be the set of all ordered partitions θ = {θ 1,..., θ p} of {1,..., n} E θ is corresponding subset of E n = {x 1... x n} Leb θ is the Lebesgue measure on E θ Let E θ is the Dirichlet form that corresponds the system of independent Brownian particles with masses m θ1,..., m θp We set E = θ Θ c θ E θ, µ n = θ Θ c θ Leb θ. Since the closure of the bilinear form E is recurrent, strongly local and regular on L 2(E n, µ n), there exist the process X(t) in E n that is properly associated with E.

24 An alternative description We replace the process X(t) by the process that takes values in L 2[, 1]. Let Y (t) = n X i(t)i πi L 2[, 1], ξ = i=1 where π i = m i and ξ 1 <... < ξ n. n ξ ii πi L 2[, 1], Let L 2 (ξ) denote the set of all non decreasing σ(ξ)-measurable functions from L2[, 1]. Theorem (K., von Renesse 16) For each ξ 1 <... < ξ n, there exists constants c θ (in the definition of the invariant measure µ n) such that Y (t) is the weak solution of the equation i=1 dy (t) = pr Y (t) dw (t) (ξ pr Y (t) ξ)dt (2) in L 2 (ξ). Moreover, the process µt = Y (t) #Leb satisfies dµ t = Γ(µ t)dt + div( µdw t) Question: Is that possible to construct a process Y on L 2 [, 1] (that can start from almost all points of L 2 [, 1]), e.g. for ξ = id, that satisfies (2)?

25 An alternative description We replace the process X(t) by the process that takes values in L 2[, 1]. Let Y (t) = n X i(t)i πi L 2[, 1], ξ = i=1 where π i = m i and ξ 1 <... < ξ n. n ξ ii πi L 2[, 1], Let L 2 (ξ) denote the set of all non decreasing σ(ξ)-measurable functions from L2[, 1]. Theorem (K., von Renesse 16) For each ξ 1 <... < ξ n, there exists constants c θ (in the definition of the invariant measure µ n) such that Y (t) is the weak solution of the equation i=1 dy (t) = pr Y (t) dw (t) (ξ pr Y (t) ξ)dt (2) in L 2 (ξ). Moreover, the process µt = Y (t) #Leb satisfies dµ t = Γ(µ t)dt + div( µdw t) Question: Is that possible to construct a process Y on L 2 [, 1] (that can start from almost all points of L 2 [, 1]), e.g. for ξ = id, that satisfies (2)?

26 The invariant measure on L 2 Let ξ is non-decreasing bounded càdlág function. For each n 1 let χ n(q, x) = n x ii [qi 1,q i ) L 2[, 1], i=1 where x E n = {x 1... x n}, q Q n = { < q 1 <... < q n 1}. Here x and q correspond for values and jumps of the step function χ n(q, x), respectively. Let Leb En is the Lebesgue measure on E n and νξ n is a measure on Q n defined by νξ n (A) = q 1(q 2 q 1)... (1 q n 1)dξ(q 1)... ξ(q n 1). A Let Ξ n is the push forward of the measure νξ n Leb En under the map χ n. Ξ = Ξ n. n=1 Properties: Ξ is σ-finite measure on L 2 [, 1]; If ξ = p i=1 ξiiπ i, then Ξ coincides with the push forward of the measure for finite number of particles under the map x p i=1 xiiπ i. supp Ξ = L 2 (ξ) (the set of all non decreasing σ(ξ)-measurable functions from L2[, 1])

27 Differential operator on L 2 and integration by parts formula Since supp Ξ = L 2 (ξ) we will consider the space L 2 (ξ) instead of L 2 [, 1]. { } FC = U = u((h 1, ),..., (h k, ))ϕ( 2 L 2 ), u Cb (R k ), ϕ C (R), h i L 2 (ξ). For g = n i=1 xii [q i 1,q i ), let D pru(g) = n i=1 U x i 1 q i q i 1 = ϕ( g 2 ) Integration by parts formula (K., von Renesse 16) For each U, V FC, (D pru(g), D prv (g)) L2 Ξ(dg) = k iu((h, g)) pr g h i + u((h, g))2ϕ ( g 2 )g i=1 LU(g)V (g)ξ(dg) [ ] k iu((h, g))(ξ pr g ξ, h i) L2 V (g)ξ(dg) where Lu((h, g)) = i,j i ju((h, g))(pr g hi, pr g hj)l 2, Lϕ( g 2 ) = 2ϕ ( g 2 ) g 2 + 2ϕ ( g 2 ) {number of values of g}. i=1

28 Dirichlet form on L 2 (L 2 (ξ), Ξ) We define E(U, V ) = 1 (D pru(g), D prv (g)) L2 Ξ(dg), 2 L 2 (ξ) U, V FC Theorem (K., von Renesse 16) E is a closable bilinear form on L 2(L 2 (ξ), Ξ) and its closure is quasi-regular local symmetric Dirichlet form. ( ) 1 Remark. Functions of the form U(g) = ϕ α(g(s))ds ψ( g 2 2) (ϕ, α C (R)) belongs to Dom(E) and D pru(g) = ϕ ( )α (g)ψ( ) + ϕ( )ψ ( )2g, [ 1 LU(g) = ϕ ( ) (α (g(s))) 2 ds + ϕ ( ) 1 ] α (g(s)) ds ψ( ) + ψ ( )[ ] m(s) Let X(t) be the process that corresponds to E and µ t = X(t) # Leb. Then dx(t) = pr X(t) dw (t) (ξ pr X(t)ξ)dt in L 2(ξ), dµ t = Γ(µ t)dt + div( µdw t) in P 2(R), with α, Γ(µ s) = 1 2 x supp(µ α(x). s)

29 Varadhan s formula Theorem (K., von Renesse 16) The L 2-metric on L 2 (ξ) is an intrinsic metric for E. Corollary For each A, B L 2 (ξ), < Ξ(A), Ξ(B) < where p t(a, B) = I AT ti BdΞ (by Ariyoshi, Hino 5) lim t ln pt(a, B) = d2 L 2 (A, B), t 2 Remark. Let ξ be strictly increasing function and the process X(t) corresponds to E. Then L 2 (ξ) = L 2 [, 1] and the measure-valued process µt = X(t) #Leb satisfies lim t ln t P(µ A, µt B) = d2 W(A, B). 2

30 Thank you for your attention!