| The goal of this work is to initiate mathematical theory of tunneling at positive temperatures. The mathematical framework we develop here rests on the notion of resonance free energy, Fb (where b is inverse temperature). This quantity plays the role of the resonance energy at zero temperature. We prove that the probability of escape of the particle from a potential well due to tunneling can be expressed as P(t) = 1 - p(t) where roughly: e-Gt&plank;≤ pt≤e-G t&plank;+O tG&plank; 2 modulo OL2+O &plank;t infinity, where L is an exponentially small (in 1&plank; ) quantity specified in Section 8. Here &plank; is the Planck constant divided by 2pi and G=-2ImFb , is the "width" of the resonance free energy.;Furthermore, we give a semi-classical bound on G : 0 0, where Sb is the action of the instanton of period &plank;b . |
