Finding the quasipotential for nongradient SDEs

Abstract

The quasipotential with respect to a given attractor of a system, evolving according to a stochastic differential equation (SDE) with a small noise, is a function that allows us to find the expected exit time from its basin of attraction and to estimate the equilibrium probability density in its neighborhood. This work is devoted to the study of the quasipotential of nongradient SDEs with isotropic diffusion. The goals are: (i) to understand the theoretical behavior of the quasipotential, and (ii) to develop tools for its numerical study.This work is focused on 2D equations with asymptotically stable equilibria and limit cycles. In the theoretical part, we establish a number of properties of the quasipotential. For 2D linear equations, we derive an exact formula for the quasipotential. For 2D nonlinear equations, we establish how the quasipotential grows near the stable equilibria and the stable limit cycles. In the numerical part of this work we adjust the OUM for computing the quasipotential on a regular mesh. We show that the quasipotential satisfies a Hamilton–Jacobi–Bellman equation with an anisotropic and unbounded speed function. The unbounded speed function may lead to an additional consistency error in specific areas of the phase space. However, we conduct an error analysis and show that even if this error appears, it decays quadratically as we refine the mesh.We apply the proposed numerical algorithm to a number of examples: linear systems, the Maier–Stein model, a system with stochastic resonance, and a system with two stable limit cycles. Whenever it is appropriate, we compute the Minimum Action Paths by shooting characteristics. The Minimum Action Paths found in this way are necessarily the global minimizers of the Freidlin–Wentzell action.