Existence of stationary measures for partially damped SDEs with generic, Euler-type nonlinearities

TL;DR

This paper establishes conditions under which stationary measures exist for a class of partially damped stochastic differential equations with nonlinearities similar to fluid models, highlighting the role of energy transfer and damping structure.

Contribution

It provides new dynamical criteria ensuring stationary measures for partially damped SDEs, including generic conditions and modifications allowing fewer damped modes.

Findings

Stationary measures exist when energy transfer from undamped to damped modes is sufficient.

Existence of stationary measures is guaranteed under generic nonlinearities if the kernel dimension of damping is less than 2d/3.

Perturbing the nonlinearity at discrete times allows stationary measures with only a single damped mode.

Abstract

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on $\mathbb R^d$ with a quadratic, conservative nonlinearity $B(x,x)$ constrained to possess various properties common to finite-dimensional fluid models and a linear damping term $-Ax$ that acts only on a proper subset of phase space in the sense that $\mathrm{dim}(\mathrm{ker}A) \gg 1$. Existence of a stationary measure is straightforward if $\mathrm{ker}A = \{0\}$, but when the kernel of $A$ is nontrivial a stationary measure can exist only if the nonlinearity transfers enough energy from the undamped modes to the damped modes. We develop a set of sufficient dynamical conditions on $B$ that guarantees the existence of a stationary measure and prove that they hold ``generically'' within our constraint class of nonlinearities provided that $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction $\mathrm{dim}(\mathrm{ker}A) < 2d/3$ can be removed if one allows the nonlinearity to change by a small amount at discrete times. In particular, for a Markov chain obtained by evolving our SDE on approximately unit random time intervals and slightly perturbing the nonlinearity within our constraint class at each timestep, we prove that there exists a stationary measure whenever just a single mode is damped.