# American Institute of Mathematical Sciences

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

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