Multi-skewed Brownian motion and diffusion in layered media

Abstract:

Multi-skewed Brownian motion $B^{\alpha } = \{B^{\alpha }_t: t \geqslant 0\}$ with skewness sequence $\alpha = \{\alpha _{k}: k\in \mathbb {Z}\}$ and interface set $S= \{x_{k}: k \in \mathbb {Z}\}$ is the solution to $X_t = X_0 + B_t + \int _{\mathbb {R}} L^X(t,x) \mathrm {d} \mu (x)$ with $\mu = \sum _{k \in \mathbb {Z}}(2 \alpha _{k}-1) \delta _{x_{k}}$. We assume that $\alpha _{k} \in (0,1) \setminus \{\frac {1}{2}\}$ and that $S$ has no accumulation points. The process $B^{\alpha }$ generalizes skew Brownian motion to the case of an infinite set of interfaces. Namely, the paths of $B^{\alpha }$ behave like Brownian motion in $\mathbb {R} \smallsetminus S$, and on $B^{\alpha }_0 = x_k$, the probability of reaching $x_k+\delta$ before $x_k-\delta$ is $\alpha _k$, for any $\delta$ small enough, and $k \in \mathbb {Z}$. In this paper, a thorough analysis of the structure of $B^{\alpha }$ is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate $B^{\alpha }$ 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.