Bounds for the L 2 Spectrum of Reversible Markov Chains with Dirichlet Boundary

Let (X(t); t 2 IR) be a reversible Markov chain on a nite set , with invariant measure. For A , let TA := infft 0; X(t) 2 Ag. We consider the tail distribution of TA for a start with a probability measure 0 such that 0(A) = 0. We prove that P 0 (TA > t) jj0jj2;; A c exp(?At), t > 0, where Ac is restricted to A c , jj0jj 2 2;; A c := P x2A c (0(x) 2 ==Ac(x)), and A > 0 is the spectral gap associated with TA.