Scaling limits of interacting diffusions in domains

doi:10.1007/s11464-014-0399-x

Front. Math. China

2014, Vol. 9

Issue (4) : 717-736 https://doi.org/10.1007/s11464-014-0399-x

SURVEY ARTICLE

Scaling limits of interacting diffusions in domains

Zhen-Qing CHEN(

),Wai-Tong(Louis) FAN

Department of Mathematics, University of Washington, Seattle, WA 98195, USA

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Abstract

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the population dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.

Keywords Hydrodynamic limit fluctuation interacting diffusion reflected diffusion Dirichlet form non-linear boundary condition coupled partial differential equation martingales stochastic partial differential equation Guassian process

Corresponding Author(s): Zhen-Qing CHEN

Issue Date: 26 August 2014

Cite this article:

Zhen-Qing CHEN,Wai-Tong(Louis) FAN. Scaling limits of interacting diffusions in domains[J]. Front. Math. China, 2014, 9(4): 717-736.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0399-x
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I4/717

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