Annals of Probability 44 (2016) 479-520.
David Kelly and Ian Melbourne
Abstract Consider an Itô 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 Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behaviour of solutions Xn to the ordinary differential equation dXn=a(Xn)dt+b(Xn)dWn.
The classical Wong-Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic integral ∫ b(X)dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of
∫ b(X)dW depends sensitively on how the smooth approximation Wn is chosen.
In applications, a natural class of smooth approximations arise by setting
$Wn(t)=n-1/2∫0ntv o φs ds$ where φt is a flow (generated for instance by an ordinary differential equation) and v is a mean zero observable.
Under mild conditions on φt we give a definitive answer to the interpretation question for the stochastic integral ∫ b(X)dW.
Our theory applies to Anosov or Axiom A flows φt, 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 φt.
The methods used in this paper are a combination of rough path theory and smooth ergodic theory.