Exponential Decay of Lifetimes and a Theorem of Kac on Total Occupation Times

Abstract

Let $(\tfrac{1}{2}D,H^1 (R^d ))$ be the Dirichlet integral and $(B_t ,P_z^W )$ the Brownian motion on R. Let μ be a finite positive measure in the Kato class and $A_t^\mu$ the additive functional associated with μ. We prove that for a regular domain D of R ^d

$$\begin{gathered} \mathop {\lim }\limits_{\beta \to \infty } \frac{1}{\beta }\log P_z^W (A_{\tau _D }^\mu > \beta )\;\; = \;\; - \inf \left\{ {\tfrac{1}{2}D(u,u):u \in C_0^\infty (D)\int_D {u^2 {\text{d}}} \mu = 1} \right\} \hfill \\ {\text{ for any }}x \in D, \hfill \\ \end{gathered} $$

where τ _D is the exit time from D. As an application, we consider the integrability of Wiener functional exp ($A_{\tau _D }^\mu$).