A direct proof of the exponential limit law for one dimensional small noise diffusion processes (Q1113529)

This paper is devoted to a study of the stochastic differential equation \[ dx_{\epsilon}(t)=\sqrt{\epsilon} dW(t)-G'(x_{\epsilon}(t))dt,\quad \epsilon >0, \] where G(x) is a double well potential (a precise definition of G appears in the paper). Let G(x) have a global maximum at M, a local maximum at S, and a local minimum at m. Further, m and M are stable equilibrium points. Thus for small \(\epsilon\) the process \(x_{\epsilon}(t)\) is expected to remain in a neighbourhood of m for a long time before escaping from the bottom of the well at (m,G(m)). Particular attention is paid to the time taken for the process, starting at \(x_ 0<S\), to reach a point \(z\in (S,M)\). The paper provides a proof of a limit theorem for the ``escape time'' \(T_{\epsilon}=\inf \{t:\) \(x_{\epsilon}(t)\geq z\}\).