On the symmetry of a reflecting Brownian motion defined by Skorohod's equation for a multi-dimensional domain (Q1098180)

The existence and uniqueness of solutions to Skorokhod's equation for a multi-dimensional domain were discussed by \textit{P. L. Lions} and \textit{A. S. Sznitman} [Commun. Pure Appl. Math. 37, 511-537 (1984; Zbl 0598.60060)] and by the first author [Probab. Theory Relat. Fields 74, 455-477 (1987; Zbl 0591.60049)]. In this paper we prove that a reflecting Brownian motion X obtained by solving Skorokhod's equation for a domain D in \({\mathbb{R}}^ d\) is symmetric in the sense that \(\int f T_ tg dx=\int g T_ tf dx\) holds for any \(L^ 2\)-functions f and g on \(\bar D,\) where \(T_ t\) is the semigroup of X. The proof is based on the construction of X by the penalty method, that is, we prove the above result by showing that X can be approximated by symmetric diffusions (with respect to the invariant measures) which are described by stochastic differential equations with smooth drift coefficients of gradient type.