Brownian motion and diffusions

Multi-skewed Brownian motion Bα = {Bαt: t ≥ 0} with skewness sequence α = {αk: k ∈ ℤ} and interface set S = {xk: k ∈ ℤ} is the solution to Xt = X0 + Bt + ∫ℝ LX(t, x)dμ(x) with μ = ∑k∈ℤ(2αk - 1)δxk 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 Bα behave like Brownian motion in ℝ\S, and on Bα0 = xk the probability of reaching xk + δ before xk - δ is αk, for any δ small enough, and k ∈ ℤ. In this paper, a thorough analysis of the structure of Bα is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate Bα 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.