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Preprint

2014-03-28

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.

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