Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line

Transcription

1 Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line Vitalii Konarovskyi Leipzig university Berlin-Leipzig workshop in analysis and stochastics (017) Joint work with Max von Renesse

2 Motivation 1 The interacting particle system is physical modification of the Arratia flow: The particle system also appearers as a solution to the equation: dµ t = Γ(µ t)dt + div( µ tdw t) in P (R) i.e. ϕ C0 (R) ϕ, µ t t ϕ, Γ(µs) ds is a martingale with q.v. 0 t 0 ϕ, µ s ds E.g. Wasserstein diffusion (von Renesse, Sturm 09): [ ϕ ϕ, Γ(µ t) = β ϕ (I + ) + ϕ (I ), µ t + I gaps (µ t ) ϕ (I + ) ϕ ] (I ) I

3 Motivation 1 The interacting particle system is physical modification of the Arratia flow: The particle system also appearers as a solution to the equation: dµ t = Γ(µ t)dt + div( µ tdw t) in P (R) i.e. ϕ C0 (R) ϕ, µ t t ϕ, Γ(µs) ds is a martingale with q.v. 0 t 0 ϕ, µ s ds E.g. Wasserstein diffusion (von Renesse, Sturm 09): [ ϕ ϕ, Γ(µ t) = β ϕ (I + ) + ϕ (I ), µ t + I gaps (µ t ) ϕ (I + ) ϕ ] (I ) I

4 Motivation 1 The interacting particle system is physical modification of the Arratia flow: The particle system also appearers as a solution to the equation: dµ t = Γ(µ t)dt + div( µ tdw t) in P (R) i.e. ϕ C0 (R) ϕ, µ t t ϕ, Γ(µs) ds is a martingale with q.v. 0 t 0 ϕ, µ s ds E.g. Wasserstein diffusion (von Renesse, Sturm 09): [ ϕ ϕ, Γ(µ t) = β ϕ (I + ) + ϕ (I ), µ t + I gaps (µ t ) ϕ (I + ) ϕ ] (I ) I

5 Description of the particle system (K., von Renesse 17) Grayscale colour coding is for atom mass. Now, we assume that: 1 each particle has a mass that obeys the conservation law; diffusion rate of each particle inversely depends on its mass; 3 particles can reflect from each other.

6 Only two particles x 1(t) x (t) denote the positions of particles at time t 0 m 1 = m = 1 particle mass at start (the total mass always 1) Let w 1, w be two indep. Brownian motions with Var (w i(t)) = 1 m i t = t w1(t) + w(t) dx i(t) = I {x1 (t)<x (t)}dw i(t) + I {x1 (t)=x (t)}d +λ ii {x1 (t)=x (t)}dt, λ 1 λ

7 Only two particles x 1(t) x (t) denote the positions of particles at time t 0 m 1 = m = 1 particle mass at start (the total mass always 1) Sticky-reflected Brownian motion on R + Let w 1, w be two indep. Brownian motions with Var (w i(t)) = 1 m i t = t w1(t) + w(t) dx i(t) = I {x1 (t)<x (t)}dw i(t) + I {x1 (t)=x (t)}d dy(t) = I {y(t)>0} dw(t) + λi {y(t)=0} dt, λ 0 (Engelbert, Peskir 14) +λ ii {x1 (t)=x (t)}dt, λ 1 λ

8 Only two particles x 1(t) x (t) denote the positions of particles at time t 0 m 1 = m = 1 particle mass at start (the total mass always 1) Let w 1, w be two indep. Brownian motions with Var (w i(t)) = 1 m i t = t w1(t) + w(t) dx i(t) = I {x1 (t)<x (t)}dw i(t) + I {x1 (t)=x (t)}d +λ ii {x1 (t)=x (t)}dt, λ 1 λ

9 Three and more particles Can three or more particles occupy the same position and which therms should contain the equation? The equation should contain: where Var (w i(t)) = w(t) + w3(t) + w4(t) I {x (t)=x 3 (t)=x 4 (t)}d 3 1 m i t = 5t

10 Three and more particles Can three or more particles occupy the same position and which therms should contain the equation? The equation should contain: where Var (w i(t)) = w(t) + w3(t) + w4(t) I {x (t)=x 3 (t)=x 4 (t)}d 3 1 m i t = 5t

11 Uncountable number of particles. Diffusion term Let X(u, t) denote the position of the particle labeld by u [0, 1] at time t 0. X(u, t) X(v, t), u v 1 dx(u, t) = d where = {v : X(u, t) = X(v, t)} W t + drift term

12 Uncountable number of particles. Diffusion term Let X(u, t) denote the position of the particle labeld by u [0, 1] at time t 0. X(u, t) X(v, t), u v 1 dx(u, t) = d where = {v : X(u, t) = X(v, t)} W t + drift term

13 Uncountable number of particles. Drift term ξ(u) an interaction potential of the particle u, where ξ : [0, 1] R, ξ(u) ξ(v), u v. 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 ξ ) dt 1 If ξ(u) = ξ(v), then particles u and v can not split. If ξ = ξ 1I [0,1/) + ξ I [1/,1] and X(u, 0) = x 1(0)I [0,1/) + x (0)I [1/,1], then the equation corresponds its two dimensional analog: dx i (t) = I {x1 (t)<x (t)}dw i (t) + I {x1 (t)=x (t)}d w 1(t) + w (t) with λ i = ξ i ξ 1+ξ 3 If ξ(u) = u, then the the system has very complicated structure. + λ i I {x1 (t)=x (t)}dt

14 Uncountable number of particles. Drift term ξ(u) an interaction potential of the particle u, where ξ : [0, 1] R, ξ(u) ξ(v), u v. 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 ξ ) dt 1 If ξ(u) = ξ(v), then particles u and v can not split. If ξ = ξ 1I [0,1/) + ξ I [1/,1] and X(u, 0) = x 1(0)I [0,1/) + x (0)I [1/,1], then the equation corresponds its two dimensional analog: dx i (t) = I {x1 (t)<x (t)}dw i (t) + I {x1 (t)=x (t)}d w 1(t) + w (t) with λ i = ξ i ξ 1+ξ 3 If ξ(u) = u, then the the system has very complicated structure. + λ i I {x1 (t)=x (t)}dt

15 Uncountable number of particles. Drift term ξ(u) an interaction potential of the particle u, where ξ : [0, 1] R, ξ(u) ξ(v), u v. 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 ξ ) dt 1 If ξ(u) = ξ(v), then particles u and v can not split. If ξ = ξ 1I [0,1/) + ξ I [1/,1] and X(u, 0) = x 1(0)I [0,1/) + x (0)I [1/,1], then the equation corresponds its two dimensional analog: dx i (t) = I {x1 (t)<x (t)}dw i (t) + I {x1 (t)=x (t)}d w 1(t) + w (t) with λ i = ξ i ξ 1+ξ 3 If ξ(u) = u, then the the system has very complicated structure. + λ i I {x1 (t)=x (t)}dt

16 Uncountable number of particles. Drift term ξ(u) an interaction potential of the particle u, where ξ : [0, 1] R, ξ(u) ξ(v), u v. 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 ξ ) dt 1 If ξ(u) = ξ(v), then particles u and v can not split. If ξ = ξ 1I [0,1/) + ξ I [1/,1] and X(u, 0) = x 1(0)I [0,1/) + x (0)I [1/,1], then the equation corresponds its two dimensional analog: dx i (t) = I {x1 (t)<x (t)}dw i (t) + I {x1 (t)=x (t)}d w 1(t) + w (t) with λ i = ξ i ξ 1+ξ 3 If ξ(u) = u, then the the system has very complicated structure. + λ i I {x1 (t)=x (t)}dt

17 Uncountable number of particles. Drift term ξ(u) an interaction potential of the particle u, where ξ : [0, 1] R, ξ(u) ξ(v), u v. 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 ξ ) dt 1 If ξ(u) = ξ(v), then particles u and v can not split. If ξ = ξ 1I [0,1/) + ξ I [1/,1] and X(u, 0) = x 1(0)I [0,1/) + x (0)I [1/,1], then the equation corresponds its two dimensional analog: dx i (t) = I {x1 (t)<x (t)}dw i (t) + I {x1 (t)=x (t)}d w 1(t) + w (t) with λ i = ξ i ξ 1+ξ 3 If ξ(u) = u, then the the system has very complicated structure. + λ i I {x1 (t)=x (t)}dt

18 The equation in L [0, 1] 1 dx(u, t) = d = {v : X(u, t) = X(v, t)} W t + ( ξ(u) 1 This family of equations can be rewritten as a one equation but in infinite dimensional space L [0, 1]: ξ ) dt dx t = pr Xt dw t + (ξ pr Xt ξ)dt in L [0, 1] where X t := X(, t) and pr g is the projection in L [0, 1] on L (g) = {f : f is constant on intervals where g constant}

19 The main results. Reversible case Theorem (K., von Renesse 17) There exists a σ-finite measure Ξ on L [0, 1] and a Markov process Xt in L [0, 1] such that The measure Ξ is invariant for X t and supp Ξ = L (ξ) X t solves dx t = pr Xt dw t + (ξ pr Xt ξ)dt in L [0, 1] The process µ t = X(, t) # Leb [0,1], that describes the evolution of particle mass, solves the equation dµ t = 1 µ t dt + div( µ tdw t), in P (R), where µ t = x µ t δ x The Varadhan formula P{µ t = ν} e d W (µ 0,ν), t 1, holds, where d W denotes the Wasserstein distance

20 Other cases of drift therm Drift term: (ξ prxt ξ)dt, Invariant measure: Ξ Drift term: (ξ prxt ξ Xt )dt, Invariant measure: e kgk Ξ(dg) finite measure

21 Invariant measure and the Differential operator Invariant measure: Ξ = Ξ n, n=1 where Ξ n : n 1 jumps are distributed according dν n ξ = n (q k q k 1 )dξ(q 1)... ξ(q n 1), k=1 n-values are distributed according to Leb x1... x n Space of smooth functions: { } FC = U = u((h 1, ),..., (h k, ))ϕ( L ) ; Differential operator: DU(g) = pr g L U(g) L [0, 1]; (Ex. Du((h, g)) = u ((h, g))pr g h, D g L = g)

22 Integration by parts and Dirichlet form Integration by parts (K., von Renesse 17) Let U, V FC. Then (DU(g), DV (g))ξ(dg) = L L LU(g)V (g)ξ(dg) V (g)( L U(g), ξ pr g ξ)ξ(dg). L (Ex. Lu((h, g)) = u ((h, g)) pr g h L, L g L = #g) Dirichlet form: E(U, V ) = 1 (DU(g), DV (g))ξ(dg), L (ξ) U, V FC Theorem (K., von Renesse 17) E is a closable bilinear form on L (L, Ξ), its closure is a quasi-regular local symmetric Dirichlet form and L is its intrinsic metric.

23 Integration by parts and Dirichlet form Integration by parts (K., von Renesse 17) Let U, V FC. Then (DU(g), DV (g))ξ(dg) = L L LU(g)V (g)ξ(dg) V (g)( L U(g), ξ pr g ξ)ξ(dg). L (Ex. Lu((h, g)) = u ((h, g)) pr g h L, L g L = #g) Dirichlet form: E(U, V ) = 1 (DU(g), DV (g))ξ(dg), L (ξ) U, V FC Theorem (K., von Renesse 17) E is a closable bilinear form on L (L, Ξ), its closure is a quasi-regular local symmetric Dirichlet form and L is its intrinsic metric.

24 Thank you for your attention!