Multi-skewed Brownian motion and diffusion in layered media

Abstract

Abstract. Multi-skewed Brownian motion B α = {B α t : t 0} with skewness sequence α = {α k : k ∈ Z} and interface set S = {x k : k ∈ Z} is the solution to} and that S has no accumulation points. The process B α generalizes skew Brownian motion to the case of an infinite set of interfaces. Namely, the paths of B α behave like Brownian motion in R S, and on B α 0 = x k , the probability of reaching x k + δ before x k − δ is α k , for any δ small enough, and k ∈ Z. In this paper, a thorough analysis of the structure of B α is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate B α to a diffusion process with piecewise constant diffusion coefficient. As an application, we compute the asymptotic behavior of a diffusion process corresponding to a parabolic partial differential equation in a two-dimensional periodic layered geometry.

Random walks in a sparse random environment

“…A direct discrete-time analogue of the multi-skewed Brownian motion is a multi-skewed random walk, which can be introduced as a quenched variant of our model when (λ n ) n∈Z is a certain deterministic sequence of constants. In accordance to the physical motivation of the model in [24], the author refers to the marked sites (i.e., the elements of A in the author's notation) as interfaces, while the long stretches of "regular" sites between interfaces are referred to by the author as layers.…”

Occupation and local times for skew Brownian motion with applications to dispersion across an interface

“…This spawned further research leading to a number of subsequent foundational probability papers that highlight interesting and sometimes surprising special structure of skew Brownian motion, see, for example, Walsh (1978), Harrison and Shepp (1981), Ouknine (1990), Le Gall (1984, Barlow, Pitman and Yor (1989), Barlow et al (2001), Burdzy and Chen (2001), Ramirez (2010). This spawned further research leading to a number of subsequent foundational probability papers that highlight interesting and sometimes surprising special structure of skew Brownian motion, see, for example, Walsh (1978), Harrison and Shepp (1981), Ouknine (1990), Le Gall (1984, Barlow, Pitman and Yor (1989), Barlow et al (2001), Burdzy and Chen (2001), Ramirez (2010).…”

Continuity of Local Time: An Applied Perspective

“…Here, we "read off" the speed and scale measures from the backward operator written in the form (30), and construct the appropriate process via a stochastic differential equation. A similar approach was carried out in the case of piecewise constant coefficients by Ramirez (2011), and will be extended here to the present framework.…”