The Annals of Applied Probability

Characterization of stationary distributions of reflected diffusions

Weining Kang and Kavita Ramanan

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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.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 4 (2014), 1329-1374.

Dates
First available in Project Euclid: 14 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1400073651

Digital Object Identifier
doi:10.1214/13-AAP947

Mathematical Reviews number (MathSciNet)
MR3210998

Zentralblatt MATH identifier
1306.60111

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]
Secondary: 90B15: Network models, stochastic 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Reflected diffusions invariant distribution stationary density submartingale problem stochastic differential equations with reflection basic adjoint relation (BAR) adjoint partial differential equation skew-symmetry condition product-form solutions skew-transform gradient drift queueing networks

Citation

Kang, Weining; Ramanan, Kavita. Characterization of stationary distributions of reflected diffusions. Ann. Appl. Probab. 24 (2014), no. 4, 1329--1374. doi:10.1214/13-AAP947. https://projecteuclid.org/euclid.aoap/1400073651


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