  Consider an It\^{o} process $X$ satisfying the stochastic differential
equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a
multidimensional Brownian motion. Suppose that $W_n$ has smooth sample paths
and that $W_n$ converges weakly to $W$. A central question in stochastic
analysis is to understand the limiting behavior of solutions $X_n$ to the
ordinary differential equation $dX_n=a(X_n)\,dt+b(X_n)\,dW_n$. The classical
Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges
weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given
the Stratonovich interpretation. The sufficient conditions are automatic in one
dimension, but in higher dimensions the correct interpretation of $\int
b(X)\,dW$ depends sensitively on how the smooth approximation $W_n$ is chosen.
In applications, a natural class of smooth approximations arise by setting
$W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$ where $\phi_t$ is a flow
(generated, e.g., by an ordinary differential equation) and $v$ is a mean zero
observable. Under mild conditions on $\phi_t$, we give a definitive answer to
the interpretation question for the stochastic integral $\int b(X)\,dW$. Our
theory applies to Anosov or Axiom A flows $\phi_t$, as well as to a large class
of nonuniformly hyperbolic flows (including the one defined by the well-known
Lorenz equations) and our main results do not require any mixing assumptions on
$\phi_t$. The methods used in this paper are a combination of rough path theory
and smooth ergodic theory.
