Functional large deviations for Stroock's approximation to a class of Gaussian processes with application to small noise diffusions

Abstract

Letting~$N=\left\{N(t),t\ge 0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$\hat{I}{˜}_{\hat{I}}\mu (t)={\int }_0^t\hat{I}{¸}_{\hat{I}}\mu (r)dr,0\le t\le 1,$$ where $\hat{I}{¸}_{\hat{I}}\mu (r)=\frac{1}{\hat{I}}\mu (-1{)}^{N(\hat{I}{\mu }^{-2}r)}$, and proved that it weakly converges to a standard Brownian motion under the continuous function topology. We establish the functional large deviations principle (LDP) for the approximations of a class of Gaussian processes constructed by integrals over $\hat{I}{˜}_{\hat{I}}\mu (t)$, and find the explicit form for rate function. As an application, we consider the following (non-Markovian) stochastic differential equation \begin{equation*} \begin{aligned} X^ε(t) &=x_{0}+\int^{t}_{0}b(X^ε(s))ds+λ(ε)\int^{t}_{0}σ(X^ε(s))dΘ_ε(s), \end{aligned} \end{equation*} where $b$ and $\ddot{I}ƒ$ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as $\hat{I}\mu \to 0$. The rate function indicates a phase transition phenomenon as $\hat{I}»(\hat{I}\mu )$ moves from one region to the other.