The Annals of Probability

A Dirichlet process characterization of a class of reflected diffusions

Weining Kang and Kavita Ramanan

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Abstract

For a class of stochastic differential equations with reflection for which a certain ${\mathbb{L}}^{p}$ continuity condition holds with p>1, it is shown that any weak solution that is a strong Markov process can be decomposed into the sum of a local martingale and a continuous, adapted process of zero p-variation. When p=2, this implies that the reflected diffusion is a Dirichlet process. Two examples are provided to motivate such a characterization. The first example is a class of multidimensional reflected diffusions in polyhedral conical domains that arise as approximations of certain stochastic networks, and the second example is a family of two-dimensional reflected diffusions in curved domains. In both cases, the reflected diffusions are shown to be Dirichlet processes, but not semimartingales.

Article information

Source
Ann. Probab., Volume 38, Number 3 (2010), 1062-1105.

Dates
First available in Project Euclid: 2 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1275486188

Digital Object Identifier
doi:10.1214/09-AOP487

Mathematical Reviews number (MathSciNet)
MR2674994

Zentralblatt MATH identifier
1202.60059

Subjects
Primary: 60G17: Sample path properties 60J55: Local time and additive functionals
Secondary: 60J65: Brownian motion [See also 58J65]

Keywords
Reflected Brownian motion reflected diffusions rough paths Dirichlet processes zero energy semimartingales Skorokhod problem Skorokhod map extended Skorokhod problem generalized processor sharing diffusion approximations

Citation

Kang, Weining; Ramanan, Kavita. A Dirichlet process characterization of a class of reflected diffusions. Ann. Probab. 38 (2010), no. 3, 1062--1105. doi:10.1214/09-AOP487. https://projecteuclid.org/euclid.aop/1275486188


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