## Conditional gauges.(English)Zbl 0552.60073

Stochastic processes, Semin. Gainesville/Fla. 1983, Prog. Probab. Stat. 7, 17-22 (1984).
[For the entire collection see Zbl 0546.00025.]
The conditional gauge $$u_ h$$ of a bounded open set $$D\subset {\mathbb{R}}^ d$$, $$d\geq 1$$ is defined by $$u_ h(x)=E^ x_ h[\exp [\int^{\tau_ D}_{0}q(B^ h_ s)ds]]$$, where q is a bounded Borel function and $$E^ x_ h$$ denotes the expectation wrt the probability law $$P^ x_ h$$ of h-conditional Brownian motion $$B^ h_ t$$ starting at x, $$\tau_ D$$ is the first exit time from D of $$B^ h_ t$$ and $$h>0$$ is harmonic in D. N. Falkner [Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 19-33 (1983; Zbl 0496.60078)] has proved the following: Suppose D is Green- smooth and that for some $$x\in D$$ and some h we have $$u_ h(x)<\infty$$. Then $$\sup_{h;h(x_ 0)=1}\sup_{x\in D}u_ h(x)<\infty.$$
In the present article a simpler proof of this result is given.
Reviewer: B.Øksendal

### MSC:

 60J45 Probabilistic potential theory 60J65 Brownian motion

### Keywords:

conditional gauge; first exit time

### Citations:

Zbl 0546.00025; Zbl 0496.60078