Itô and Stratonovich stochastic partial differential equations: Transition from microscopic to macroscopic equations

Abstract

We review the derivation of stochastic ordinary and quasi-linear stochastic partial differential equations (SODE’s and SPDE’s) from systems of microscopic deterministic equations in space dimension $d\geq 2$ as well as the macroscopic limits of the SPDE’s. The macroscopic limits are quasi-linear (deterministic) PDE’s. Both noncoercive and coercive SPDE’s, driven by Itô differentials with respect to correlated Brownian motions, are considered. For the solutions of semi-linear noncoercive SPDE’s with smooth and homogeneous diffusion kernels we show that these solutions can be obtained as solutions of first-order SPDE’s, driven by Stratonovich differentials and their macroscopic limit, and are solutions of a class of semi-linear second-order parabolic PDE’s. Further, the space-time covariance structure of correlated Brownian motions is described and for space dimension $d\geq 2$ the long-time behavior of the separation of two uncorrelated Brownian motions is shown to be similar to the independent case.