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A variety of physical phenomena involve the nonlinear transfer of energy fromweakly damped modes subjected to external forcing to other modes which are moreheavily damped. In this work we explore this in (finite-dimensional) stochasticdifferential equations in $\mathbb R^n$ with a quadratic, conservativenonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate inthe sense that $\mathrm{ker} A \neq \emptyset$. We investigate sufficientconditions to deduce the existence of a stationary measure for the associatedMarkov semigroups. Existence of such measures is straightforward if $A$ is fullrank, but otherwise, energy could potentially accumulate in $\mathrm{ker} A$and lead to almost-surely unbounded trajectories, making the existence ofstationary measures impossible. We give a relatively simple and generalsufficient condition based on time-averaged coercivity estimates alongtrajectories in neighborhoods of $\mathrm{ker} A$ and many examples where suchestimates can be made.

Stationary measures for stochastic differential equations with degenerate damping