  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.
