  In this paper, we consider the convergence rate with respect to Wasserstein
distance in the invariance principle for deterministic nonuniformly hyperbolic
systems, where both discrete time systems and flows are included. Our results
apply to uniformly hyperbolic systems and large classes of nonuniformly
hyperbolic systems including intermittent maps, Viana maps, finite horizon
planar periodic Lorentz gases and others. Furthermore, as a nontrivial
application to homogenization problem, we investigate the
$\mathcal{W}_2$-convergence rate of a fast-slow discrete deterministic system
to a stochastic differential equation.
