Walsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1

Transcription

1 Walsh Diffusions Andrey Sarantsev University of California, Santa Barbara March 27, 2017 Andrey Sarantsev University of Washington, Seattle 1 / 1

2 Walsh Brownian Motion on R d Spinning measure µ: probability measure on unit sphere S in R d The process X = (X (t), t 0) diffuses along each ray R θ := {rθ r 0}, for θ S As it hits the origin, it chooses a new direction θ S according to measure µ Since it hits the origin and then instantaneously hits it infinitely many times, it is not trivial to make sense of this process Andrey Sarantsev University of Washington, Seattle 2 / 1

3 Historical Review Introduced by Walsh (1978) Barlow, Pitman, Yor (1989) Tsirelson (1997) Watanabe (1999) Picard (2005) Fitzsimmons, Kuter (2014) Andrey Sarantsev University of Washington, Seattle 3 / 1

4 Tree Topology Any point x R d \ {0} can be written in polar coordinates: x = rθ We also write it for x = 0, slightly abusing the notation Tree distance, or railway metric, between (r 1, θ 1 ) and (r 2, θ 2 ): { r 1 r 2, θ 1 = θ 2 r 1 + r 2, θ 1 θ 2 The corresponding topology is stronger than Euclidean topology Andrey Sarantsev University of Washington, Seattle 4 / 1

5 Properties Walsh Brownian motion exists in the weak sense and is unique in law, for any initial condition X (0) = x It is continuous in the tree topology Its radial component S( ) = X ( ) is a reflected BM on R + : S(t) = W (t) + L S (t) L S is continuous, nondecreasing, can increase only when S = 0; W is a Brownian motion on the real line L S is semimartingale local time at zero for S Andrey Sarantsev University of Washington, Seattle 5 / 1

6 Stochastic Calculus Let f : R d R be C 2 on each ray, f, f being radial derivatives Walsh Brownian motion satisfies Freidlin-Sheu formula: d f (X (t)) = 1 {X (t) 0} [f (X (t)) dw (t) + 1 ] 2 f (X (t)) dt + dl S (t) f (0+, θ)µ(dθ) S It is a Markov process with generator Lf := 1 2 f for f (0+, θ)µ(dθ) = 0 S Andrey Sarantsev University of Washington, Seattle 6 / 1

7 Local Time Partition Property For A S, the mapping R A : R d R, R A (r, θ) := r1 {θ A} cancels out all rays not in A Recall: S = X = R S (X ) Walsh BM X satisfies L R A(X ) (t) = µ(a)l S (t) Meaning: µ is indeed spinning measure Andrey Sarantsev University of Washington, Seattle 7 / 1

8 Walsh Diffusions A Walsh diffusion associated with triple (µ, g, σ) is a process X = (X (t), t 0): continuous in the tree topology with X ( ) a seminartingale satisfying local time partition property satisfying the Freudlin-Sheu formula below: [ d f (X (t)) = 1 {X (t) 0} f (X (t))σ(x (t)) dw (t) + L(X (t)) dt ] + dl S (t) f (0+, θ)µ(dθ) S Lf (x) = g(x)f (x) σ2 (x)f (x) Andrey Sarantsev University of Washington, Seattle 8 / 1

9 Existence and Uniqueness State space is I := {(r, θ) 0 r < l(θ)} l : S (0, ] is a given boundary function Boundary: I := {(r, θ) r = l(θ) < } Theorem (Karatzas, Yan, 2016) Assume g, σ, σ 1 are locally bounded on I. For any initial condition, there exists in the weak sense a unique in law Walsh diffusion associated with (µ, g, σ), at least until it hits I. Andrey Sarantsev University of Washington, Seattle 9 / 1

10 Markov Property Lf (x) = g(x)f (x) σ2 (x)f (x) Theorem (Karatzas, Yan, 2016) A Walsh diffusion associated with (µ, g, σ) is a strong Markov process with generator L, for f such that f (0+, θ)µ(dθ) = 0 S Andrey Sarantsev University of Washington, Seattle 10 / 1

11 Time-Change Walsh diffusion X associated with (µ, 0, σ), S( ) = X ( ) ds(t) = σ(x (t)) dw (t) + dl S (t) Dambis-Dubins-Schwartz decomposition: For some Walsh Brownian motion Z with spinning measure µ, (Karatzas, Yan, 2016) X (t) = Z ( S t ), S t := t 0 σ 2 (X (s)) ds Andrey Sarantsev University of Washington, Seattle 11 / 1

12 Scale Function Walsh diffusion X associated with (µ, g, σ) r ( y s(r, θ) = exp g(z, θ) σ 2 (z, θ) dz ) dy P : R d R d : P : (r, θ) (s(r, θ), θ) Then Y = P(X ) is a Walsh diffusion associated with (µ, 0, σ) (Karatzas, Yan, 2016) σ(r, θ) = s (s 1 (r, θ), θ)σ(s 1 (r, θ), θ) Andrey Sarantsev University of Washington, Seattle 12 / 1

13 Convergence Results Let X n be Walsh diffusion associated with (µ n, g, σ), starting from X n (0) = x n Assume g, σ are continuous in Euclidean topology (stronger assumption that g, σ, σ 1 being locally bounded) Theorem (Ichiba, S, 2016) If µ n µ 0 weakly in S, and x n x 0 in Euclidean topology on R d, then X n X 0 weakly in C([0, T ], R d ) with max-norm. Andrey Sarantsev University of Washington, Seattle 13 / 1

14 Wasserstein Distance Fix a p 1. The Wasserstein distance W p between two probability measures P, Q on a metric space (X, d) is defined as inf [E d(x, Y ) p ] 1/p, where the inf is taken over all couplings (X, Y ) of (P, Q) Andrey Sarantsev University of Washington, Seattle 14 / 1

15 Wasserstein Distance Estimates Denote the law on C([0, T ], R d ) of a Walsh Brownian motion with spinning measure µ, starting from the origin, by Q T (µ) Theorem (Ichiba, S, 2016) If 1 q < p and 0 < ρ < p/(p + 1), then there exists a C > 0 depending on T, p, q, ρ such that W q (Q T (µ), Q T (µ )) C [ W p (µ, µ ) ] ρ for all probability measures µ and µ on S This is a refinement of the convergence result above Andrey Sarantsev University of Washington, Seattle 15 / 1

16 Feller Property Let (P t ) t 0 be the transition semigroup: P t f (x) := E(f (X (t) X (0) = x) Feller Property: P t f is continuous for every t > 0 and every bounded continuous f : R d R Theorem (Ichiba, S, 2016) (a) If g, σ are continuous, then the Walsh diffusion is Feller continuous in Euclidean topology (b) If g, σ, σ 1 are locally bounded, then the Walsh diffusion is Feller continuous in tree topology Part (a) actually follows from weak convergence result Andrey Sarantsev University of Washington, Seattle 16 / 1

17 Positivity of Transition Kernel Let P t (x, A) = P(X (t) A X (0) = x) be the transition kernel We use reference measure ν(dr, dθ) = µ(dθ) dr on R d Theorem (Ichiba, S, 2016) For every subset A of positive measure ν, every x, and t > 0, we have: P t (x, A) > 0. Andrey Sarantsev University of Washington, Seattle 17 / 1

18 Stationary Measure A (possibly infinite) measure π on R d is called stationary for the Walsh diffusion X = (X (t), t 0), if for every bounded measurable function f, and every t > 0, we have: P t f (x) π(dx) = f (x) π(dx) If π is finite, then it can be normalized to become a stationary distribution: π(r d ) = 1 Andrey Sarantsev University of Washington, Seattle 18 / 1

19 Explicit Form of the Stationary Distribution (Ichiba, S, 2016) Assume one of the two conditions holds: (a) g, σ, σ 1 are locally bounded, and µ has finite support; or (b) g, σ are continuous in the Euclidean topology. Then the following measure is stationary: ( r ) π(dr, dθ) = σ 2 g(ρ, θ) (r, θ) exp 2 σ 2 (ρ, θ) dρ 0 dr µ(dθ) Andrey Sarantsev University of Washington, Seattle 19 / 1

20 Driftless Case For the case g 0, we have: π(dr, dθ) = σ 2 (r, θ) dr µ(dθ) g 0, σ 1: Walsh Brownian motion with spinning measure µ π(dr, dθ) = dr µ(dθ) Andrey Sarantsev University of Washington, Seattle 20 / 1

21 Heuristics Assume π is normalized to a probability measure Then π is a combination of stationary distributions π θ for reflected diffusions with drift g(, θ) and diffusion σ 2 (, θ) on each ray θ, weighted by the share of time the process spends on this ray This share of time is determined by the spinning measure µ, weighted by the average time duration l(θ) of its excursions at zero in this direction θ π(dr, dθ) = l(θ) µ(dθ) π θ (dr) Andrey Sarantsev University of Washington, Seattle 21 / 1

22 Measure Norms For a measure ν on R d and a function g : R d [1, ), let ν g := sup h(x)ν(dx) h g For g 1, we write ν TV and call it the total variation norm Andrey Sarantsev University of Washington, Seattle 22 / 1

23 Ergodicity Walsh diffusion is called ergodic if there exists a unique stationary distribution π, and P t (x, ) π( ) TV 0, t V -uniformly ergodic: if, in addition to that, for some k > 0, P t (x, ) π( ) V CV (x)e kt Here, V : R d [1, ) is some function Andrey Sarantsev University of Washington, Seattle 23 / 1

24 Ergodicity Result (Ichiba, S, 2016) Under technical conditions (a) or (b) above, if lim g(x) < 0, then the Walsh diffusion is ergodic x if, in addition, lim σ(x) <, then the Walsh diffusion is x V -uniformly ergodic for V (x) = e λ x with some λ > 0 Andrey Sarantsev University of Washington, Seattle 24 / 1

25 Method of Proof: Reflected Walsh Diffusions These are like Walsh diffusions, but with rays cut at R(θ) and reflected on each cutoff back to the origin These are always positive recurrent, and uniformly ergodic, because the state space is compact (in Euclidean topology: always, in the tree topology: when µ has finite support) We can check that π is the stationary measure for reflected Walsh Brownian motion, then pass to driftless reflected Walsh diffusions by time-change, and to the general case by scale function Then let the cutoff R(θ) go to infinity Andrey Sarantsev University of Washington, Seattle 25 / 1

26 Thanks! Andrey Sarantsev University of Washington, Seattle 26 / 1