A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise

TL;DR

This paper introduces a stochastic version of the RAGE theorem applicable to noisy transport equations, providing conditions for enhanced dissipation and demonstrating sharp decay rates in stochastic shear flows.

Contribution

It develops a stochastic RAGE theorem and characterizes conditions for dissipation enhancement in noisy transport equations, including explicit examples and decay rates.

Findings

Identifies a necessary and sufficient condition for dissipation enhancement.

Proves a sharp decay rate for stochastic shear flows.

Establishes a stochastic RAGE theorem for transport noise.

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.