  In this paper we develop via Girsanov's transformation a perturbation
argument to investigate weak convergence of Euler-Maruyama (EM) scheme for
path-dependent SDEs with H\"older continuous drifts. This approach is available
to other scenarios, e.g., truncated EM schemes for non-degenerate SDEs with
finite memory or infinite memory. Also, such trick can be applied to study weak
convergence of truncated EM scheme for a range of stochastic Hamiltonian
systems with irregular coefficients and with memory, which are typical
degenerate dynamical systems. Moreover, the weak convergence of path-dependent
SDEs under integrability condition is investigated by establishing, via the
dimension-free Harnack inequality, exponential integrability of irregular
drifts w.r.t. the invariant probability measure constructed explicitly in
advance.
