July 2013 August 2013 September 2013 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 1 2 3 1 2 3 4 5 6 7 7 8 9 10 11 12 13 4 5 6 7 8 9 10 8 9 10 11 12 13 14 14 15 16 17 18 19 20 11 12 13 14 15 16 17 15 16 17 18 19 20 21 21 22 23 24 25 26 27 18 19 20 21 22 23 24 22 23 24 25 26 27 28 28 29 30 31 25 26 27 28 29 30 31 29 30 |

4:00 pm Wednesday, August 7, 2013 Analysis Seminar: Interpretation of stochastic integrals that arise as limits of deterministic systems by Ian Melbourne (University of Warwick) in HB 227- 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.
Submitted by dani@rice.edu |