Escape rates over potential barriers: variational principles and the Hamilton–Jacobi equation

We describe a rigorous formalism to study some extrema statistics problems, like maximum probability events or escape rate processes, by taking into account that the Hamilton–Jacobi equation completes, in a natural way, the required set of boundary conditions of the Euler–Lagrange equation, for this kind of variational problem. We apply this approach to a one-dimensional stochastic process, driven by colored noise, for a double-parabola potential, where we have one stable and one unstable steady states.