Action minimizing stochastic invariant measures for a class of Lagrangian systems

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September 2008, 7(5): 1211-1223. doi: 10.3934/cpaa.2008.7.1211

Action minimizing stochastic invariant measures for a class of Lagrangian systems

1.	College of Mathematics, Jilin University, Changchun 130012, China

Received October 2007 Revised April 2008 Published June 2008

Abstract
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In this paper we discuss a variational method of constructing an action minimizing stochastic invariant measure for positive definite Lagrangian systems. Then we study some main properties of the stochastic minimal measures. Finally we give the definitions of stochastic Mather's functions with respect to the stochastic differential equation d$x=v(t)$d$t+\sigma(x)$d$w$ and prove their differentiability.

Keywords: Lagrangian systems, Action minimizing stochastic invariant measures, variational method., stochastic Mather's functions.

Mathematics Subject Classification: Primary: 37H99; Secondary: 37J5.

Citation: Kaizhi Wang. Action minimizing stochastic invariant measures for a class of Lagrangian systems. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1211-1223. doi: 10.3934/cpaa.2008.7.1211

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