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August 2014 Characterization of stationary distributions of reflected diffusions
Weining Kang, Kavita Ramanan
Ann. Appl. Probab. 24(4): 1329-1374 (August 2014). DOI: 10.1214/13-AAP947

Abstract

Given a domain $G$, a reflection vector field $d(\cdot)$ on $\partial G$, the boundary of $G$, and drift and dispersion coefficients $b(\cdot)$ and $\sigma(\cdot)$, let $\mathcal{L}$ be the usual second-order elliptic operator associated with $b(\cdot)$ and $\sigma(\cdot)$. Under mild assumptions on the coefficients and reflection vector field, it is shown that when the associated submartingale problem is well posed, a probability measure $\pi$ on $\bar{G}$ with $\pi(\partial G)=0$ is a stationary distribution for the corresponding reflected diffusion if and only if

\[\int_{\bar{G}}\mathcal{L}f(x)\pi(dx)\leq0\]

for every $f$ in a certain class of test functions. The assumptions are verified for a large class of obliquely reflected diffusions in piecewise smooth domains, including those that are not semimartingales. In addition, it is shown that any nonnegative solution to a certain adjoint partial differential equation with boundary conditions is an invariant density for the reflected diffusion. As a corollary, for bounded smooth domains and a class of polyhedral domains that satisfy a skew-symmetry condition, it is shown that if a certain skew-transform of the drift is conservative and of class $\mathcal{C}^{1}$, and the covariance matrix is nondegenerate, then the corresponding reflected diffusion has an invariant density $p$ of Gibbs form, that is, $p(x)=e^{H(x)}$ for some $\mathcal{C}^{2}$ function $H$. Finally, under a nondegeneracy condition on the diffusion coefficient, a boundary property is established that implies that the condition $\pi(\partial G)=0$ is necessary for $\pi$ to be a stationary distribution. This boundary property is of independent interest.

Citation

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Weining Kang. Kavita Ramanan. "Characterization of stationary distributions of reflected diffusions." Ann. Appl. Probab. 24 (4) 1329 - 1374, August 2014. https://doi.org/10.1214/13-AAP947

Information

Published: August 2014
First available in Project Euclid: 14 May 2014

zbMATH: 1306.60111
MathSciNet: MR3210998
Digital Object Identifier: 10.1214/13-AAP947

Subjects:
Primary: 60H10 , 60J60 , 60J65
Secondary: 90B15 , 90B22

Keywords: adjoint partial differential equation , basic adjoint relation (BAR) , gradient drift , Invariant distribution , product-form solutions , Queueing networks , Reflected diffusions , skew-symmetry condition , skew-transform , stationary density , Stochastic differential equations with reflection , submartingale problem

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 4 • August 2014
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