Unpredictability of an exit time

We consider a process uε(x,t), for x in a bounded interval, t ⩾ 0 and ε a small parameter, given as the solution of a nonlinear heat equation perturbed by a space-time white noise multiplied by ε. The nonlinear part is the derivative of a one-well polynomial, with a nondegenerate minimum at 0. We study, in the limit as ε goes to zero, the time required by uε to escape from the unitary ball (in the sup norm), when it is close to the null function at time zero. We prove that, when conveniently normalized, this time has an exponential limit distribution. The proof is based on a coupling constructed by Mueller (1993), and answers a question posed by Martinelli et al. in (1989).