Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous N-particle systems with singular interactions

摘要

Assuming that Omega subset of R-n, n >= 2, is an open, relatively compact set with boundary delta Omega of Lebesgue measure zero we prove strong Feller properties for a class of distorted Brownian motions in (Omega) over bar with reflecting boundary condition. Dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem. Combining this result with the strong Feller properties we can construct a weak solution for specified starting points. If Omega has C-2-boundary the construction works for all starting points, where the drift term is not singular, even on the boundary. But also for a certain class of sets with less smooth boundary our approach works for all points in Omega, where the drift term is not singular, and at least some points from a delta Omega. Our techniques allow very singular drift terms. This enables us to construct continuous N-particle gradient stochastic dynamics in cuboids Lambda subset of R-d, d is an element of N, with reflecting boundary condition and singular interactions for dN >=, 2. We can start the stochastic dynamics in all initial configurations having at most one particle in delta Lambda, provided delta Lambda is locally smooth there. (c) 2007 Elsevier Inc. All rights reserved.