  In this paper, we study the asymptotic behavior of randomly perturbed
path-dependent stochastic differential equations with small parameter
$\vartheta_{\varepsilon}$, when $\varepsilon \rightarrow 0$,
$\vartheta_\varepsilon$ goes to $0$. When $\varepsilon \rightarrow 0$, 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.
