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Negative regularity mixing for random volume preserving diffeomorphisms

Jacob Bedrossian, Patrick Flynn, Sam Punshon-Smith

Abstract

We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes $H^{-δ}$ observables exponentially fast in $H^{-δ}$ (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on $\mathbb T^2$. In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing "ideal" scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting.

Negative regularity mixing for random volume preserving diffeomorphisms

Abstract

We consider the negative regularity mixing properties of random volume preserving diffeomorphisms on a compact manifold without boundary. We give general criteria so that the associated random transfer operator mixes observables exponentially fast in (with a deterministic rate), a property that is false in the deterministic setting. The criteria apply to a wide variety of random diffeomorphisms, such as discrete-time iid random diffeomorphisms, the solution maps of suitable classes of stochastic differential equations, and to the case of advection-diffusion by solutions of the stochastic incompressible Navier-Stokes equations on . In the latter case, we show that the zero diffusivity passive scalar with a stochastic source possesses a unique stationary measure describing "ideal" scalar turbulence. The proof is based on techniques inspired by the use of pseudodifferential operators and anisotropic Sobolev spaces in the deterministic setting.

Paper Structure

This paper contains 26 sections, 14 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Abstract framework and statement of negative regularity mixing
  3. Lyapunov structure
  4. The linearization and projective process
  5. Two point geometric ergodicity
  6. Main result
  7. Comments on the proof of Theorem \ref{['thm:AM']}
  8. Applications
  9. Examples
  10. IID diffeomorphisms and stochastic flows
  11. Stochastic Navier-Stokes equations
  12. Checking the conditions for stochastic Navier-Stokes
  13. Checking Assumption \ref{['ass:Lyapunov-Flow-Assumption']}
  14. Construction of $\psi_p$ with degenerate noise
  15. Passive scalars
  16. ...and 11 more sections

Key Result

Theorem 1.8

Under Assumptions ass:Lyapunov-Flow-Assumption--ass:2pt, there exists constants $\mu >0$, $\delta \in (0,1)$ and $\beta_* \geq 1$, such for each Lyapunov function $V = V_{\beta_*,\eta}$, with $\eta >0$ sufficiently small there holds the following: for all distributions $f_0 \in H^{-\delta}$ with $\i As a corollary, there holds $\forall q\in (0,2)$

Theorems & Definitions (44)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 34 more