Exponential mixing for random dynamical systems and an example of Pierrehumbert

Abstract

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions. CONTENTS1. Introduction 2. Markov chains and random dynamical systems 3. Random dynamical systems with absolutely continuous noise 4. Almost-sure mixing for incompressible RDS with absolutely continuous noise 5. Application to Pierrehumbert Appendix A. Proof of Theorem 2.3 Appendix B. Proofs of Proposition 2.9 and Corollary 2.10 Appendix C. The Weyl Equidistribution Theorem References

Exponential mixing by shear flows

“…Finally, we show that even if T > 0 is arbitrarily large, it is possible that b τ mixes at a subexponential rate if τ n = T for all n ∈ N, which answers a question of Blumenthal-Coti Zelati-Gvalani (see Remark 1.2 of [2]) in the negative.…”

“…where the supremum is over all balls B ⊆ R 2 /(2πZ) 2 and u(t, x) is the solution to the transport equation (1.1) with initial data u 0 = 21 x1≤π − 21 x1>π .…”

“…2 ; R 2 )) be a divergence-free shear at every time t ∈ [0, 1], that is, we assume that b(t, x) is parallel to b(t, y) for all t ≥ 0 and x, y ∈ R 2 /(2πZ) 2 . Then there is a constant…”

“…Our second result is an example of a particularly simple shear flow which mixes at the optimal rate. The argument follows the path of the recent work of Blumenthal-Coti Zelati-Gvalani [2]. To explain, we define the horizontal sine field b horiz (x) := (sin(x 2 ), 0) and the vertical sine field b vert (x) := (0, sin(x 1 )).…”

“…To explain, we define the horizontal sine field b horiz (x) := (sin(x 2 ), 0) and the vertical sine field b vert (x) := (0, sin(x 1 )). In [2], Blumenthal-Coti Zelati-Gvalani establish a general framework to show that the vector field…”