## 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

### Keywords:

gauge theorem; potential; Green function

Zbl 0659.00010