Communications in Mathematical Sciences

Volume 14 (2016)

Number 2

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

Pages: 327 – 362

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n2.a2

Authors

Guodong Pang (Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, Penn., U.S.A.)

Nikola Sandrić (Institut für Mathematische Stochastik, Fachrichtung Mathematik, Technische Universität Dresden, Germany; and Department of Mathematics, Faculty of Civil Engineering, University of Zagreb, Croatia)

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 Lévy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.

Keywords

diffusion with jumps, ergodicity, Feller process, Lévy process, semi-martingale characteristics, symbol

2010 Mathematics Subject Classification

60F17, 60G17, 60J75

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Published 14 December 2015