Melnikov's criterion for nondifferentiable weak-noise potentials

The stationary probability density of Fokker-Planck models with weak noise has an asymptotic form containing a pseudopotential theta. If theta is smooth, it satisfies a Hamilton-Jacobi equation at zero energy and can be interpreted as the action of an associated Hamiltonian system. Under this assumption, theta has the properties of a Lyapunov function, and can be used, for example, as a thermodynamic potential in nonequilibrium steady states. The author considers systems having several attractors and shows, by applying Melnikov's method to the associated Hamiltonian, that in general theta is not differentiable. A small perturbation of a model with differentiable theta leads to a nondifferentiable theta. The method is illustrated on a model used in the treatment of the unstable mode in a laser