Conditional Brownian Motion, Whitney Squares and the Conditional Gauge Theorem

Abstract

Let (X, P ^x) be Brownian motion killed at τ _D = inf {t > 0: X _t ∉ D}, D a domain in ℝ² and (X, P ^x _z ) this motion conditioned on X _τD = z. For Kato class potentials q we show $E_x^x\left[ {\exp \left\{ { - \int\limits_0^{{\tau D}} {q\left( {{X_s}} \right)ds} } \right\}} \right]$is bounded from zero and infinity with little or no assumption on the smoothness of the boundary.