Weak synchronization for isotropic flows

TL;DR

This paper investigates weak synchronization phenomena in isotropic Brownian flows on manifolds, establishing conditions under which trajectories converge to a singleton equilibrium, especially in cases with negative or zero top Lyapunov exponents.

Contribution

It provides a sufficient boundary condition on the distance process to ensure singleton equilibrium and weak point attractors in isotropic flows, extending understanding beyond negative Lyapunov exponents.

Findings

Singleton equilibrium under certain boundary conditions

Weak point attractor established for specific flows

Conditions fulfilled in isotropic Brownian and Ornstein-Uhlenbeck flows

Abstract

We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide a sufficient condition on the boundary behavior of $r$ at $0$ which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on $S^{d-1}$ as well as isotropic Ornstein-Uhlenbeck flows on $\mathbb{R}^d$.