A Dirichlet process characterization of a class of reflected diffusions

Details A Dirichlet process characterization of a class of reflected diffusions A Dirichlet process characterization of a class of reflected diffusions

2010-10-11

arxiv.org/abs/1010.2106v1

Annals of Probability 2010, Vol. 38, No. 3, 1062-1105

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP487 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP487

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.

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