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 $L(x, v):Tt^ntimesRr^nto Rr$ 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 $phi:Tt^nto Rr$. 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.