Exponential decay of correlation for the Stochastic Process associated to the Entropy Penalized Method

In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):\Tt^n\times\Rr^n\to \Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 - U(x) + < P, v>. For each value of $\epsilon $ and $h$, consider the operator \Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N} e ^{-\frac{hL(x,v)+\phi(x+hv)}{\epsilon h}}dv], as well as the reversed operator \bar \Gg[\phi](x):= -\epsilon h {ln}[\int_{\re^N} e^{-\frac{hL(x+hv,-v)+\phi(x+hv)}{\epsilon h}}dv], both acting on continuous functions $\phi:\Tt^n\to \Rr$. Denote by $\phi_{\epsilon,h} $ the solution of $\Gg[\phi_{\epsilon,h}]=\phi_{\epsilon,h}+\lambda_{\epsilon,h}$, and by $\bar \phi_{\epsilon,h} $ the solution of $\bar \Gg[\phi_{\epsilon,h}]=\bar \phi_{\epsilon,h}+\lambda_{\epsilon,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {\cal L} (\phi) (x) = \int e^{- \frac{h L (x,v)}{\epsilon}} \phi(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.