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Ergodicity and fluctuations of a fluid particle driven by a diffusion process with jumps

Turbulence is one of the most important phenomena in nature and engineering. It is a flow regime characterized by the presence of irregular eddying motions, that is, motions with high level of vorticity. The key problem is to describe the chaotic motion of a turbulent fluid. In practice this is done by tracking the evolution of a specially marked physical entity which is immersed in the fluid, the so-called fluid particle. Clearly, such a particle must be light and small enough (noninertial) so that its presence does not affect the flow pattern. In this way, the motion of the fluid may be visualized through the evolution of this passively advected particle which follows the streamlines of the fluid. In this talk, I will discuss the long-time behavior of a fluid particle immersed in a turbulent fluid flow driven by a diffusion process with jumps, that is, Feller process associated with a non-local operator. I will present the law of large numbers and central limit theorem for the evolution process of the tracked particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic, or (iii) is a class of Levy processes. The presented results generalize the classical and well-known results for fluid flows driven by diffusions.

diff usion with jumps ; ergodicity ; Feller process ; Levy process ; semimartingale characteristics ; symbol

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