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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