Large deviation for small noise path-dependent stochastic differential equations

In this paper, we study the asymptotic behavior of randomly perturbed path-dependent stochastic differential equations with small parameter ${\vartheta }_{\epsilon }$, when $\epsilon \to 0$, ${\vartheta }_{\epsilon }$ goes to $0$. When $\epsilon \to 0$, we establish large deviation principle. The proof of the results relies on the weak convergence approach. As an application, we establish the large deviation for functionals of path-dependent SDEs in small time intervals.