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

摘要

Let (1/2D, H~1 (R~d)) be the Dirichlet integral and (B_t, P_x~W) the Brownian motion on R~d. Let #mu# be a finite positive measure in the Kato class and A_t~(#mu#) the additive functional associated with #mu#. We prove that for a regular domain D of R~d lim_(#beta# -> infinity) 1/#beta# log P_x~W (A_(#tau#D)~(#mu#) > #beta#) = -inf {1/2D(u,u): u implied by C_0~(#infinity#) (D), integral_D u~2 d#mu# = 1} for any x implied by D, where #tau#_D is the exit time from D. As an application, we consider the integrability of Wiener functional exp (A_(#tau#_D)~(#mu#)).