• Bernoulli
  • Volume 25, Number 4A (2019), 2439-2478.

Stationary distributions and convergence for Walsh diffusions

Tomoyuki Ichiba and Andrey Sarantsev

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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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Bernoulli, Volume 25, Number 4A (2019), 2439-2478.

Received: October 2017
Revised: June 2018
First available in Project Euclid: 13 September 2019

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ergodic process invariant measure Lyapunov function reflected diffusion stationary distribution stochastic differential equation Walsh Brownian motion Walsh diffusion


Ichiba, Tomoyuki; Sarantsev, Andrey. Stationary distributions and convergence for Walsh diffusions. Bernoulli 25 (2019), no. 4A, 2439--2478. doi:10.3150/18-BEJ1059.

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