Anomalous behavior of the Kramers rate at bifurcations in classical field theories

We consider a Ginzburg–Landau partial differential equation in a bounded interval, perturbed by weak spatio–temporal noise. As the interval length increases, a transition between activation regimes occurs, in which the classical Kramers rate diverges [MS01]. We determine a corrected Kramers formula at the transition point, yielding a finite, though noise-dependent rate prefactor, confirming a conjecture by Maier and Stein [MS03]. For both periodic and Neumann boundary conditions, we obtain explicit expressions for the prefactor in terms of Bessel and error functions.