Passive tracer in non-Markovian, Gaussian velocity field

TL;DR

This paper extends the understanding of passive tracer trajectories in Gaussian velocity fields by analyzing non-Markovian cases with covariance structures defined by Bernstein functions, beyond the classical Markovian framework.

Contribution

It introduces a novel analysis of tracer behavior in non-Markovian Gaussian fields with Bernstein function covariance, expanding previous Markovian results.

Findings

Established convergence in law of tracer trajectories in non-Markovian fields

Generalized spectral gap conditions to non-Markovian Gaussian fields

Demonstrated normal distribution limit for tracer displacement

Abstract

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see [K-F;C-X;K-L-O], that $\bbX(t)/\sqrt{t}$ converges in law, as $t\to+\infty$, to a normal, zero mean vector, provided that the field $V(t,x)$ is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.