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Multi-skewed Brownian motion and diffusion in layered media

Author: Jorge M. Ramirez
Journal: Proc. Amer. Math. Soc. 139 (2011), 3739-3752
MSC (2010): Primary 60J60; Secondary 60G17
Published electronically: March 3, 2011
MathSciNet review: 2813404
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Abstract: Multi-skewed Brownian motion $ B^{\alpha} = \{B^{\alpha}_t: t \geq 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) {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.

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Additional Information

Jorge M. Ramirez
Affiliation: Escuela de Matemáticas, Universidad Nacional de Colombia Sede Medellin, Calle 59A, No. 63-20, Medellin, Colombia

Keywords: Skew Brownian motion, diffusion, layered media
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: August 27, 2010
Published electronically: March 3, 2011
Additional Notes: The author’s research was partially supported by The University of Arizona and grants DMS-0073958 and CMG 03-27705 from the National Science Foundation to Oregon State University.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.