Quantitative convergence to equilibrium for a class of hypoelliptic SDEs

Various physical systems (fluids, nonlinear waves, etc.) exhibit turbulent, generally chaotic behavior when subject to forcing and weak damping. A natural problem when studying such systems is to establish unique ergodicity and quantify the convergence of generic time averages to the unique stationary statistics. In this talk, we discuss recent work on the long-time behavior of a class of hypoelliptic SDEs that covers some prototypical chaotic/turbulent systems such as Lorenz-96 and Galerkin truncations of the stochastic Navier-Stokes equations. Our main result is an optimal quantitative estimate on the exponential convergence to equilibrium in the limit of vanishing, balanced noise and dissipation. Exponential convergence for the model under consideration has been known for some time, but our quantitative estimates are new. Our proof uses a scheme that combines a weak Poincaré inequality argument with a quantitative hypoelliptic regularization estimate for the associated time-dependent Kolmogorov equation. As a necessary step in carrying out our approach, we obtain quantitative pointwise estimates (uniform in the small noise/dissipation parameter) on the stationary density. This is accomplished with hypoelliptic De Giorgi and Moser type iterations.

This is joint work with Jacob Bedrossian (University of Maryland).