Gauge theorem for the Neumann problem. (English) Zbl 0585.60064

Stochastic processes, Semin. Evanston/Ill. 1984, Prog. Probab. Stat. 9, 63-70 (1986).
[For the entire collection see Zbl 0575.00017.]
For q from a certain class the gauge function for the Neumann problem is defined as \[ G_ q(x)={\mathbb{E}}^ x(\int^{\infty}_{0}\exp [\int^{\tau_ D}_{0}q(X_ s)ds]dL(s)) \] where (L(s): \(s\geq 0)\) is the boundary local time for the reflected Brownian motion on the bounded domain \(D\subset {\mathbb{R}}^ 3\) with \(C^ 3\)-boundary. The main result is that when \(G_ q\not\equiv \infty\) then it is continuous on \(\bar D.\)
Reviewer: O.Enchev


60H25 Random operators and equations (aspects of stochastic analysis)
60J55 Local time and additive functionals
60J65 Brownian motion


Zbl 0575.00017