On reflecting Brownian motion - a weak convergence approach. (English) Zbl 0704.60081

Williams, R. J.; Zheng, W. A.

Ann. Inst. Henri Poincaré, Probab. Stat. 26, No. 3, 461-488 (1990).

Consider a d-dimensional domain D of finite Lebesgue measure. The authors define a certain sequence of stationary diffusion processes with drifts that tend to infinity at the boundary in such a way as to keep the sample paths in D. It is proved that this sequence is tight and any limit process is a continuous, stationary Markov process in \(\bar D\) that can be identified with the stationary reflecting Brownian motion defined by M. Fukushima [see Osaka J. Math. 4, 183-215 (1967; Zbl 0317.60033)] using the Dirichlet form that is proportional to \(\int_{D}| \nabla g|^ 2 dx,\) \(g\in H^ 1(D).\)
Furthermore, under a mild condition on the boundary of D, which is easily satisfied when D is a Lipschitz domain, it is shown that this process has a Skorokhod-like semimartingale representation.

Reviewer: R.Wu

MSC:

60J65	Brownian motion
60J60	Diffusion processes
60J45	Probabilistic potential theory

Keywords:

normal reflection; Skorokhod representation; stationary reflecting Brownian motion; Dirichlet form; semimartingale representation

Citations:

Zbl 0317.60033