# Long time behaviour of periodic stochastic flows

Kaloshin V, DOLGOPYAT D, KORALOV L. 2006. Long time behaviour of periodic stochastic flows. XIVth International Congress on Mathematical Physics. International Congress on Mathematical Physics, 290–295.

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*Conference Paper*|

*Published*|

*English*

Author

Kaloshin, Vadim

^{ISTA}^{}; DOLGOPYAT, D.; KORALOV, L.Abstract

We consider the evolution of a set carried by a space periodic incompressible stochastic flow in a Euclidean space. We
report on three main results obtained in [8, 9, 10] concerning long time behaviour for a typical realization of the stochastic flow. First, at time t most of the particles are at a distance of order √t away from the origin. Moreover, we prove a Central Limit Theorem for the evolution of a measure carried by the flow, which holds for almost every realization of the flow. Second, we show the existence of a zero measure full Hausdorff dimension set of points, which
escape to infinity at a linear rate. Third, in the 2-dimensional case, we study the set of points visited by the original set by time t. Such a set, when scaled down by the factor of t, has a limiting non random shape.

Publishing Year

Date Published

2006-03-01

Proceedings Title

XIVth International Congress on Mathematical Physics

Page

290-295

Conference

International Congress on Mathematical Physics

Conference Location

Lisbon, Portugal

Conference Date

2003-07-28 – 2003-08-02

ISBN

IST-REx-ID

### Cite this

Kaloshin V, DOLGOPYAT D, KORALOV L. Long time behaviour of periodic stochastic flows. In:

*XIVth International Congress on Mathematical Physics*. World Scientific; 2006:290-295. doi:10.1142/9789812704016_0026Kaloshin, V., DOLGOPYAT, D., & KORALOV, L. (2006). Long time behaviour of periodic stochastic flows. In

*XIVth International Congress on Mathematical Physics*(pp. 290–295). Lisbon, Portugal: World Scientific. https://doi.org/10.1142/9789812704016_0026Kaloshin, Vadim, D. DOLGOPYAT, and L. KORALOV. “Long Time Behaviour of Periodic Stochastic Flows.” In

*XIVth International Congress on Mathematical Physics*, 290–95. World Scientific, 2006. https://doi.org/10.1142/9789812704016_0026.V. Kaloshin, D. DOLGOPYAT, and L. KORALOV, “Long time behaviour of periodic stochastic flows,” in

*XIVth International Congress on Mathematical Physics*, Lisbon, Portugal, 2006, pp. 290–295.Kaloshin, Vadim, et al. “Long Time Behaviour of Periodic Stochastic Flows.”

*XIVth International Congress on Mathematical Physics*, World Scientific, 2006, pp. 290–95, doi:10.1142/9789812704016_0026.