Invariant measure selection bynoise. An example

Jonathan C. Mattingly; Etienne Pardoux

doi:10.3934/dcds.2014.34.4223

Article Contents

2014, Volume 34, Issue 10: 4223-4257. Doi: 10.3934/dcds.2014.34.4223

This issue Previous Article Invariant measures for non-autonomous dissipative dynamical systems Next Article The Penrose-Fife phase-field model with coupled dynamic boundary conditions

Invariant measure selection by noise. An example

Jonathan C. Mattingly^1, and
Etienne Pardoux^2,

1.
Mathematics Department and Department of Statisical Science, Statisical Science Duke University, Box 90320, Durham, NC 27708-0320
2.
Laboratoire d'Analyse, Topologie, Probabilités, Université de Provence 39, rue F. Joliot-Curie, F-13453 Marseille cedex 13

Received: February 28, 2013

Revised: September 30, 2013

Published: September 2014

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Abstract

We consider a deterministic system with two conserved quantities and infinity many invariant measures. However the systems possess a unique invariant measure when enough stochastic forcing and balancing dissipation are added. We then show that as the forcing and dissipation are removed a unique limit of the deterministic system is selected. The exact structure of the limiting measure depends on the specifics of the stochastic forcing.

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Mathematics Subject Classification: Primary: 60H10, 37L40; Secondary: 37A60, 34C29.

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