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Gauge theorem for unbounded domains. (English) Zbl 0689.60076

Stochastic processes, Proc. 8th Semin., Gainesville/Florida 1988, Prog. Probab. 17, 87-98 (1989).
[For the entire collection see Zbl 0659.00010.]
Let \(\{X_ t;t>0\}\) be the Brownian motion in \({\mathbb{R}}^ n\), \(n\geq 1\), D be a domain (open, connected) contained in \({\mathbb{R}}^ n\) and q be a Borel function on D. Put \[ T(\omega)=\inf \{t>0;X_ t(\omega)\not\in D\}\quad and \]
\[ e_ q(\omega)=\exp (\int^{T(\omega)}_{0}q(X_ t(\omega))dt),\quad u(x)=\int_{T<\infty}e_ q(\omega)d{\mathbb{P}}_ x(\omega). \] One says that the gauge theorem holds for the couple (D,q) if either \(u\equiv +\infty\) in D or u is bounded in D. It has been proved by the author and Rao that the gauge theorem holds for a domain D of finite Lebesgue measure and a function q bounded on D. Aizenman and Simon proved the same result for a potential q belongig to the “Kato’s class” and a bounded domain D. The gauge theorem has also been extended to unbounded domains in \({\mathbb{R}}^ n\), \(n\geq 3\), by the author and Zhao, assuming that the function \(q1_ D\) is in the Kato’s class and is integrable on D. Unfortunately this last result fails in dimensional 1 or 2.
In this paper the gauge theorem is proved to hold in any dimension for a domain D of finite Lebesgue measure and a potential q such that \(q1_ D\) is in the Kato’s class. The analytic definition of the Green function for an unbounded domain in \({\mathbb{R}}^ 2\) is not veery tractable, hence the author uses a purely probabilistic approach which ignores the Green function. The classical definition of the Kato’s class is then replaced by a probabilistic characterization due to Aizenman and Simon.
Reviewer: J.Lacroix

MSC:

60J65 Brownian motion
60J45 Probabilistic potential theory

Citations:

Zbl 0659.00010