## Reflection Brownian motions: Quasimartingales and strong Caccioppoli sets.(English)Zbl 0812.60065

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 $$H^ 1(D)$$.
Reviewer: X.L.Nguyen (Hanoi)

### MSC:

 60J65 Brownian motion 31C25 Dirichlet forms 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 60J60 Diffusion processes 60J55 Local time and additive functionals
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