Ergodicity and fluctuations of a fluid particle driven by diffusions with jumps

TL;DR

This paper investigates the long-term behavior of a fluid particle influenced by jump-driven diffusions, establishing laws of large numbers and central limit theorems for various classes of driving processes, extending classical fluid flow results.

Contribution

It introduces a general framework for analyzing fluid particles driven by non-local jump diffusions, broadening the scope of classical ergodic and fluctuation results.

Findings

Law of large numbers for jump-driven fluid particles

Central limit theorem established for ergodic and Lévy process drivers

Generalization of classical diffusion-driven fluid flow results

Abstract

In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic or (iii) is a class of L\'evy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.