Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. (English) Zbl 0732.60090

The authors consider solutions of the Neumann boundary value problem for the Laplace and the heat equation on a bounded Hölder domain in Euclidean space. First, upper bounds for solutions of the heat equation are derived from Nash type inequalities. Then, a lower bound for solutions of the heat equation on Lipschitz domains is obtained from the representation of the solution in terms of expectation values with respect to reflecting Brownian motion. The following section is concerned with the proof of the existence of reflecting Brownian motion and boundary local time on a bounded Lipschitz domain.
An application to the Neumann boundary problem for the Laplace equation yields that a solution u can be represented as follows: \[ u(x)=\lim_{t\to \infty}{1\over 2} E^ x[\int^{t}_{0}f(X_ s)dL_ s]; \] \(L_ s\) is the boundary local time for reflecting Brownian motion. Finally, the authors show that the ideal boundary for the Neumann boundary value problem on a bounded Lipschitz domain coincides with the Euclidean boundary. A weaker statement remains true for Hölder domains.


60J65 Brownian motion
60J35 Transition functions, generators and resolvents
60J50 Boundary theory for Markov processes
35J25 Boundary value problems for second-order elliptic equations
31C35 Martin boundary theory
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