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

2025/05/28 by Jacob Bedrossian, Bedrossian, Jacob, Alex Blumenthal +3
Engineering · Mathematics · Physics and Astronomy · #35R60 #37A50 (Secondary) #37H15 #60H10 (Primary) 37D25 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2505.22903

openalex publication_date 2025/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Non-equilibrium steady state for a three-mode energy cascade model
• Stationary measures for stochastic differential equations with degenerate damping
• Exponential mixing for random dynamical systems and an example of Pierrehumbert
• Chaos in stochastic 2d Galerkin-Navier-Stokes
• A noise-induced transition in the Lorenz system
• Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations
• Stochastic Persistence
• Yet another look at Harris' ergodic theorem for Markov chains
• Sample path properties of the stochastic flows

Related

• Stationary flows and uniqueness of invariant measures
• Non uniqueness of stationary measures for self-stabilizing processes
• Invariant manifold reduction for stochastic dynamical systems
• Uniqueness of the Φ⁴₃ measures on closed Riemannian 3-manifolds
• Stochastic stability of physical measures in conservative systems