On the stochastically-induced exponential stability of two nonlinear dynamics exhibiting energy conservation

Abstract

We present results concerning geometric ergodicity and exponential stability of the unique invariant measures for two nonlinear stochastic dynamics: Langevin dynamics in the presence of singular potential functions, and the degenerate stochastic Lorenz 96 model. In particular, these dynamics both have the property that, in the absence of their fluctuation dissipation (i.e., additive degenerate Ornstein-Uhlenbeck) forcings, they exhibit conservation laws on a suitable energy functional. We prove hypocoercivity in the singular Langevin dynamics by means of a modified, but equivalent, norm reliant on the existence of a Poincaré inequality in both the space and momentum marginals of the invariant measure. We prove geometric ergodicity of the 4-dimensional degenerate stochastic Lorenz 96 dynamics by means of a partitioning of the state space at high energies, and relevant pathwise computations of properly scaled dynamics in each partition.