Characterization of Stationary Distributions of Reflected Diffusions
Weining Kang, Kavita Ramanan · 2014
Given a domain G, a reflection vector field d(•) on ∂G, the boundary of G, and drift and dispersion coefficients b(•) and σ (•), let L be the usual second-order elliptic operator associated with b(•) and σ (•).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 π on Ḡ with π(∂G) = 0 is a stationary distribution for the corresponding reflected diffusion if and only iffor 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 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 C 2 function H . Finally, under a nondegeneracy condition on the diffusion coefficient, a boundary property is established that implies that the condition π(∂G) = 0 is necessary for π to be a stationary distribution.This boundary property is of independent interest.