A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise

2025/07/15 by Zelati, Michele Coti, Hairer, Martin, Villringer, David · 1 citation
#35Q35 #35R60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2507.11422

Abstract

We prove a stochastic version of the classical RAGE theorem that applies to the two-point motion generated by noisy transport equations. As a consequence, we identify a necessary and sufficient condition for the corresponding diffusive equation to be dissipation enhancing. This involves the identification of a non-trivial, finite dimensional subspace that is invariant for the family of self-adjoint operator characterizing the structure of the transport noise. We discuss several examples and prove a sharp enhanced dissipation rate for stochastic shear flows.

Citations

• Exponential mixing by random cellular flows
• A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows
• Enhanced Dissipation via the Malliavin Calculus
• A Harris theorem for enhanced dissipation, and an example of Pierrehumbert
• On anomalous diffusion in the Kraichnan model and correlated-in-time variants
• Exponential mixing for random dynamical systems and an example of Pierrehumbert
• Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows
• Enhanced dissipation and Hörmander's hypoellipticity
• Stabilization by transport noise and enhanced dissipation in the Kraichnan model
• Bounds on the rate of enhanced dissipation
• Almost-sure exponential mixing of passive scalars by the stochastic Navier-Stokes equations
• On the relation between enhanced dissipation time-scales and mixing\n rates
• Enhanced dissipation, hypoellipticity, and anomalous small noise\n inviscid limits in shear flows
• Sample path properties of the stochastic flows

Cited by

• Superexponential dissipation enhancement on \mathbbT^d

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