A class of stochastic evolutions that scale to the porous medium equation
type: Item
instance of: scholarly article
title: A class of stochastic evolutions that scale to the porous medium equation
zbMATH Open document ID: 0947.60090
DOI: 10.1007/BF02174218
author: Shui Feng
author: Ian Iscoe
author: Timo Seppäläinen
published in: Journal of Statistical Physics
publication date: 1999-06-17T00:00:00Z
review text: A class of reversible Markov jump processes on a periodic lattice is described. The process can be viewed as a randomly interacting configuration of sticks that evolves through exchanges of stick pieces between nearest neighbours. Namely, after an exponentially distributed time the stick \(x_k\) at a site \(k\in Z_N^d\), where \(Z_N^d\) is a finite-dimensional cube lattice, breaks and transfers a randomly long piece \(u\) to a stick at a randomly chosen neighbouring site \(\ell\) (the distribution depends on a parameter \(\alpha>1\)). Let \(\mu_t^N(d\theta)\) be a random measure induced by the above described random evolution. Then the main result of the paper states that \(\mu_{N^2t}^N(d\theta)\) converges in probability to a weak solution \(u(t,\theta)\) of the porous medium equation which is the nonlinear heat equation \(\partial_t u = \kappa \Delta(u^\alpha)\) on the \(d\)-dimensional torus.
reviewed by: Martin Janžura
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 82C20
zbMATH DE Number: 1279723
zbMATH Keywords: porous medium equation
zbMATH Keywords: hydrodynamic scaling limit
MaRDI profile type: Publication
cites work: Q4896352
cites work: Nonlinear diffusion limit for a system with nearest neighbor interactions
cites work: Hydrodynamic limit for a spin system on a multidimensional lattice
full work available at URL: https://doi.org/10.1007/bf02174218
OpenAlex ID: W2086287295
description: scientific article; zbMATH DE number 1279723