Reflection Brownian motions: Quasimartingales and strong Caccioppoli sets
The paper deals mainly with the following problem: When is a reflecting Brownian function in a bounded domain a quasi-martingale? The answer depends on the smoothness of the topological boundary of \(D \subset R^ d\) and it is well-known that when the boundary of \(D\) is \(C^ 2\)-smooth, then for every given Dirichlet space \(\{H^ 1(D), \xi)\) there is a continuous Markov process \(X\) associated with \(\xi\) which admits the Skorokhod semi-martingale decomposition starting at any point in \(\overline D:=D\cup\partial D\). The authors introduce the notion of strong Caccioppoli sets which are nothing else than bounded domains \(D \subset R^ d\) such that \[ \int_ D {\partial g \over \partial x_ i} dm \leq C | g |_ \infty, \qquad i = 1,2, \dots, \;\forall g \in H^ 1(D) \cap C_ b(D), \] where \(|\;|_ \infty\) denotes the supremum norm, \(C_ b\) the class of continuous bounded functions and \(H^ 1\) the class of \(g \in L^ 2(D,dm)\) whose first distributional derivatives \(\partial g/ \partial x_ i\) belong also to \(L^ 2\), \(C\) a fixed constant. Let \(\{H^ 1, \xi, dm\}\) be a symmetric Dirichlet space (in the sense given by Fukushima) on a given strong Caccioppoli set \(D\subset R^ d\) and let \(X = (X_ t)\) be the stationary continuous Markov process on \(\overline D\) associated with \(\xi\) (called usually the stationary reflecting Brownian motion on \(\overline D)\), then the main result is the following Theorem 1.1. The stationary reflecting Brownian motion on \(\overline D\) is a quasi-martingale iff \(D\) is a strong Caccioppoli set. It is noted that the definition of a quasi-martingale given by the author here is slightly different from the known one introduced by Dellacherie- Meyer. In the last Sections 4 and 5 the authors extend the above mentioned result to the case of symmetric reflecting diffusion processes associated with some class of strongly elliptic operators with coefficients in \(H^ 1(D)\).
scientific article; zbMATH DE number 427947
scientific article