seminar: Effective Macroscopic Dynamics of Stochastic Partial Differential
speaker: Jinqiao Duan (Laboratory for Stochastics and Dynamics, Chicago)abstract: The need to take stochastic effects into account for modeling complex systems has now become widely recognized. Stochastic partial differential equations arise naturally as mathematical models for multiscale systems under random influences. We consider macroscopic dynamics of microscopic systems described by stochastic partial differential equations. The microscopic systems are characterized by small scale heterogeneities (spatial domain with small holes or oscillating coefficients), fast scale boundary impact (random dynamic boundary condition), and, of course, random fluctuations.
An effective macroscopic model for such a stochastic microscopic system is derived. The homogenized effective model is still a stochastic partial differential equation, but defined on a unified spatial domain and the random impact is represented by an extra term in the effective model. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of convergence in probability distribution, and in the sense of convergence in energy are also proved.
timetable:
Thu 27 Jul, 9:50 - 10:30, Aula Dini
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