Stability and invariant measure asymptotics in a model for heavy particles in rough turbulent flows

Abstract

We study a system of Skorokhod stochastic differential equations (SDEs) modeling the pairwise dispersion (in spatial dimension $d=2$) of heavy particles transported by a rough self-similar, turbulent flow with Hölder exponent $h\in (0,1)$. Under the assumption that $h>0$ is sufficiently small, we use Lyapunov methods and control theory to show that the Markovian system is nonexplosive and has a unique, exponentially attractive invariant probability measure. Furthermore, our Lyapunov construction is radially sharp and gives partial confirmation on a predicted asymptotic behavior with respect to the Hölder exponent $h$ of the invariant probability measure. A physical interpretation of the asymptotics is that intermittent clustering is weakened when the carrier flow is sufficiently rough.