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

Kotelenez, Peter

doi:10.1090/S0033-569X-08-01102-6

Author: Peter M. Kotelenez
Journal: Quart. Appl. Math. 66 (2008), 539-564
MSC (2000): Primary 60H10, 60H05, 60H30, 60F17
DOI: https://doi.org/10.1090/S0033-569X-08-01102-6
Published electronically: July 2, 2008
MathSciNet review: 2445528
Full-text PDF Free Access

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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.

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