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Conditional Brownian motion, Whitney squares and the conditional gauge theorem. (English) Zbl 0689.60077

Stochastic processes, Proc. 8th Semin., Gainesville/Florida 1988, Prog. Probab. 17, 109-119 (1989).
[For the entire collection see Zbl 0659.00010.]
Let \(\{X_ t,{\mathbb{P}}^ x;t>0\}\) be the Brownian motion in \({\mathbb{R}}^ 2\), killed at the exit time \[ T(\omega)=\inf \{t>0;\quad X_ t(\omega)\not\in D\} \] of a bounded domain D in \({\mathbb{R}}^ 2\) and q be a Borel function on D. Put \[ e_ q(\omega)=\exp (\int^{T(\omega)}_{0}q(X_ t(\omega))dt),\quad u_ z(x)=\int e_ q(\omega)d{\mathbb{P}}^ x_ z(\omega) \] where \({\mathbb{P}}^ x_ z\) is the distribution of the motion conditioned on \(X_ T=z\). For Kato’s class of potentials q it is proved that \(u_ z\) is bounded from zero and infinity with little or no assumptions on the smoothness of the boundary of D. This is done by revising a result of B. Davis on the occupation time of conditional Brownian motion in Whitney squares. The rather technical condition imposed on D involves the logarithmic capacity of \(D^ c\) near a Withney square which is certainly satisfied when D is simply connected but may hold in more general situations (Salisbury’s maze for example). The author suggests that actually the conditional gauge theorem seems to hold for any planar bounded domain.
Reviewer: J.Lacroix

MSC:

60J65 Brownian motion
60J45 Probabilistic potential theory

Citations:

Zbl 0659.00010