Dynamical Systems Seminar

Homogenization is a mechanism whereby multiscale deterministic systems converge to stochastic differential equations. In achieving a rigorous theory, the key problem is the interpretation (Stratonovich, Ito, other) of the stochastic integrals present in the limit. This boils down to the following question in ergodic theory. For a discrete time dynamical system f : X to X, given a vector valued observable v : X to \R^d with mean zero, consider the normalised sum \(W_n(t) = n^{-1/2}\sum_{j=1}^{[nt]}v\circ f^j\). Under certain conditions (eg Axiom A or nonuniformly hyperbolic), it is possible to prove that W_n converges weakly to d-dimensional Brownian motion. (This is known as the functional central limit theorem or weak invariance principle.) Now suppose that \(1\le b,c \le d\). What is the weak limit of \(\int_0^t W_n^b dW_n^c\)? In this talk, we present the solution to this problem for both discrete and continuous dynamical systems. Our solution sheds light on the general question of how to correctly interpret stochastic integrals arising as limits of deterministic systems. This is joint work with David Kelly.