Reflected Brownian motion with skew symmetric data in a polyhedral domain
This paper is concerned with the characterization and invariant measures of certain reflected Brownian motions (RBM's) in polyhedral domains. The kind of RBM studied here behaves like d-dimensional Brownian motion with constant drift \(\mu\) in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit. Under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced by \textit{J. M. Harrison} and the author [Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab., to appear], it is shown that such an RBM can be characterized in terms of a family of submartingales and that it reaches non-smooth parts of the boundary with probability zero. In the quoted paper, a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained there is shown to be the density of an invariant measure for the RBM. Furthermore, if the density is integrable over the polyhedral state space, then it yields the unique stationary distribution for the RBM. In the proofs of these results, a key role is played by a dual process for the RBM and by results obtained in the above mentioned paper, for reflected Brownian motions on smooth approximating domains.
characterization and invariant measures of certain reflected Brownian motions
scientific article; zbMATH DE number 3984262
scientific article