Convergence and Stationary Distributions for Walsh Diffusions

Abstract

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.

30 pages. Keywords: Walsh Brownian motion, Walsh diffusion, stochastic differential equation, stationary distribution, invariant measure, ergodic process, Lyapunov function, reflected diffusion

References in corpus (2)

• Inflationary Magnetogenesis in $R^2$-Inflation after Planck 2015
• Semimartingales on Rays, Walsh Diffusions, and Related Problems of Control and Stopping

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