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

摘要

We present a large deviation principle for the entropy penalized Matherproblem 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 valueof $\epsilon $ and h, the entropy penalized Mather problem $\min\{\int_{\tn\times\rn} L(x,v)d\mu(x,v)+\epsilon S[\mu]\},$ where the entropy S is given by$S[\mu]=\int_{\tn\times\rn}\mu(x,v)\ln\frac{\mu(x,v)}{\int_{\rn}\mu(x,w)dw}dxdv,$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 $\mu_{\epsilon, h}$ whichconverges to the Mather measure $\mu$. We show a LDP $\lim_{\epsilon,h\to0}\epsilon \ln \mu_{\epsilon,h}(A),$ where $A\subset\mathbb{T}^N\times\mathbb{R}^N$. The deviation function I is given by $I(x,v)=L(x,v)+\nabla\phi_0(x)(v)-\bar{H}_{0},$ where $\phi_0$ is the unique viscositysolution for L.