Multi-skewed brownian motion and diffusion in layered media

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 X_t = X₀ + B_t + ∫_ℝ L^X(t, x)dμ(x) with μ = ∑_k∈ℤ(2α_k - 1)δ_xk We assume that αk ∈ (0, 1)\{1/2} and that S has no accumulation points. The process Bα 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^α₀ = 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.