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

Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$Θ_ε(t)=\int_0^tθ_ε(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where $θ_ε(r)=\frac{1}ε(-1)^{N(ε^{-2}r)}$, 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 $Θ_ε(t)$, 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 $σ$ are both Lipschitz functions, and establish its Freidlin-Wentzell type LDP as $ε\rightarrow 0$. The rate function indicates a phase transition phenomenon as $λ(ε)$ moves from one region to the other.