Georgia Institute of Technology College of Sciences

We study nonlinear energy transfer and the existence of stationary measures in a class of degenerately forced SDEs on 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 dim(kerA) ≫ 1. Existence of a stationary measure is straightforward if kerA = {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 dim(kerA) < 2d/3 and the stochastic forcing acts directly on at least two degrees of freedom. We also show that the restriction dim(kerA) < 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.