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

Gomes, Diogo A.; Lopes, Artur O.

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

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Description

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 $ε$ and $h$, consider the operator \Gg[ϕ](x):= -εh {ln}[\int_{\re^N} e ^{-\frac{hL(x,v)+ϕ(x+hv)}{εh}}dv], as well as the reversed operator \bar \Gg[ϕ](x):= -εh {ln}[\int_{\re^N} e^{-\frac{hL(x+hv,-v)+ϕ(x+hv)}{εh}}dv], both acting on continuous functions $ϕ:\Tt^n\to \Rr$. Denote by $ϕ_{ε,h} $ the solution of $\Gg[ϕ_{ε,h}]=ϕ_{ε,h}+λ_{ε,h}$, and by $\bar ϕ_{ε,h} $ the solution of $\bar \Gg[ϕ_{ε,h}]=\bar ϕ_{ε,h}+λ_{ε,h}$. In order to analyze the decay of correlation for this process we show that the operator $ {\cal L} (ϕ) (x) = \int e^{- \frac{h L (x,v)}ε} ϕ(x+h v) d v,$ has a maximal eigenvalue isolated from the rest of the spectrum.