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

Let (1/2D, H¹(R^d)) be the Dirichlet integral and (B_t, P_{cursive Greek chi}^W) the Brownian motion on R^d. Let μ be a finite positive measure in the Kato class and A^μ_t the additive functional associated with μ. We prove that for a regular domain D of R^d lim_β→∞ 1/β log P_{cursive Greek chi}^W (A^μ_τD > β) = - inf {1/2D(u, u) : u ∈ C₀^∞(D), ∫_D u² dμ = 1} for any cursive Greek chi ∈ D, where τ_D is the exit time from D. As an application, we consider the integrability of Wiener functional exp (A^μ_τD).