Stationary measures for stochastic differential equations with degenerate damping

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in ${\mathbb{R}}^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\mathrm{k}\mathrm{e}\mathrm{r}A\mathrm{\ne }\mathrm{\varnothing }$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if $A$ is full rank, but otherwise, energy could potentially accumulate in $\mathrm{k}\mathrm{e}\mathrm{r}A$ and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of $\mathrm{k}\mathrm{e}\mathrm{r}A$ and many examples where such estimates can be made.