Ergodic and Mixing Properties of the Boussinesq Equations with a Degenerate Random Forcing

2013/11/14 by Juraj Földes, Nathan Glatt-Holtz, Földes, Juraj +5 · 1 citation
Economics, Econometrics and Finance · Mathematics · #37L55 #60H07 #60H15 #76D06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1311.3620

openalex publication_date 2013/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish the existence, uniqueness and attraction properties of an ergodic invariant measure for the Boussinesq Equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales. The central challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. Towards this aim we encounter a Lie bracket structure in the associated vector fields with a complicated dependence on solutions. This leads us to develop a novel Hörmander-type condition for infinite-dimensional systems. Demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the Malliavin covariance matrix.

Citations

• Computation of the Energy Spectrum in Homogeneous Two-Dimensional Turbulence

Cited by

• Notes on Statistically Invariant States in Stochastically Driven Fluid Flows

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