  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 $\mathrm{ker} A \neq \emptyset$. 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 $\mathrm{ker} A$
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 $\mathrm{ker} A$ and many examples where such
estimates can be made.
