Deterministic homogenization for fast-slow systems with chaotic noise

Abstract

Consider a fast-slow system of ordinary differential equations of the form $\dot{x}=a(x,y)+{\epsilon }^{-1}b(x,y)$, $\dot{y}={\epsilon }^{-2}g(y)$, where it is assumed that $b$ averages to zero under the fast flow generated by $g$. We give conditions under which solutions $x$ to the slow equations converge weakly to an Itô diffusion $X$ as $\epsilon \to 0$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.

31 pages

References in corpus (2)

• Smooth approximation of stochastic differential equations
• Rough path recursions and diffusion approximations

Cited by in corpus (5)

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