Open Access
November 2019 Stationary distributions and convergence for Walsh diffusions
Tomoyuki Ichiba, Andrey Sarantsev
Bernoulli 25(4A): 2439-2478 (November 2019). DOI: 10.3150/18-BEJ1059


A Walsh diffusion on Euclidean space moves along each ray from the origin, as a solution to a stochastic differential equation with certain drift and diffusion coefficients, as long as it stays away from the origin. As it hits the origin, it instantaneously chooses a new direction according to a given probability law, called the spinning measure. A special example is a real-valued diffusion with skew reflections at the origin. This process continuously (in the weak sense) depends on the spinning measure. We determine a stationary measure for such process, explore long-term convergence to this distribution and establish an explicit rate of exponential convergence.


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Tomoyuki Ichiba. Andrey Sarantsev. "Stationary distributions and convergence for Walsh diffusions." Bernoulli 25 (4A) 2439 - 2478, November 2019.


Received: 1 October 2017; Revised: 1 June 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110101
MathSciNet: MR4003554
Digital Object Identifier: 10.3150/18-BEJ1059

Keywords: ergodic process , invariant measure , Lyapunov function , Reflected diffusion , stationary distribution , Stochastic differential equation , Walsh Brownian motion , Walsh diffusion

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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