The Mather measure and a Large Deviation Principle for the Entropy Penalized Method

Abstract

We present a large deviation principle for the entropy penalized Mather problem when the Lagrangian L is generic (in this case the Mather measure $\mu$ is unique and the support of $\mu$ is the Aubry set). Consider, for each value of $\epsilon$ and h, the entropy penalized Mather problem $\min \{\int_{\tn\times\rn} L(x,v)dμ(x,v)+εS[μ]\},$ where the entropy S is given by $S[μ]=\int_{\tn\times\rn}μ(x,v)\ln\frac{μ(x,v)}{\int_{\rn}μ(x,w)dw}dxdv,$ and the minimization is performed over the space of probability densities $\mu (x,v)$ that satisfy the holonomy constraint It follows from D. Gomes and E. Valdinoci that there exists a minimizing measure ${\mu }_{\epsilon ,h}$ which converges to the Mather measure $\mu$. We show a LDP ${\lim }_{\epsilon ,h\to 0}\epsilon \ln {\mu }_{\epsilon ,h}(A),$ where $A\subset {\mathbb{T}}^N\times {\mathbb{R}}^N$. The deviation function I is given by $I(x,v)=L(x,v)+\nabla {\varphi }_0(x)(v)-{\bar{H}}_0,$ where ${\varphi }_0$ is the unique viscosity solution for L.

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