Non-uniqueness of stationary measures for stochastic systems with almost surely invariant manifolds

Abstract

We develop a general framework for establishing non-uniqueness of stationary measures for stochastically forced dynamical systems possessing an almost surely invariant submanifold. Our main abstract result provides sufficient conditions for the existence of multiple stationary measures on compact manifolds, though the underlying methodology extends to non-compact settings. The key insight is to construct additional stationary measures by exploiting the linear instability of the invariant submanifold, as quantified by a positive transverse Lyapunov exponent. To demonstrate the practical applicability of our framework, we apply it to the Lorenz 96 model with degenerate stochastic forcing, which serves as an example of both non-compact and high-dimensional dynamics. We prove that as the damping parameter becomes sufficiently small, the unique stationary measure bifurcates, giving rise to exactly two distinct stationary measures. The proof combines our general theory with computer-assisted verification of certain Lie algebra generation properties that ensure the required hypoellipticity and irreducibility conditions.