Convergence of moderately interacting particle systems to a diffusion-convection equation
type: Item
instance of: scholarly article
title: Convergence of moderately interacting particle systems to a diffusion-convection equation
zbMATH Open document ID: 0942.60095
DOI: 10.1016/S0304-4149(97)00111-7
author: Benjamin Jourdain
published in: Stochastic Processes and their Applications
publication date: 2000-03-01T00:00:00Z
review text: The diffusion-convection equation \[ \frac {\partial u}{\partial t}+|u|^{q-1}\frac {\partial u}{\partial x}=\frac 12\frac {\partial ^2u}{\partial x^2}\tag{1} \] in the domain \((t,x)\in(0,\infty)\times R \) with initial condition \(\delta _0\), where \(q\geq 2\), admits a unique positive solution \(v_q\) in \(C((0,\infty),L^1(R))\cap C^{\infty }((0,\infty)\times R)\). By interpreting the equation (1) as a Fokker-Planck equation a probabilistic interpretation to the solution \(v_q\) is given. To this end it is shown that an appropriate martingale problem has a unique solution. Consequently, the solution is obtained as the propagation of chaos limit of a sequence of moderately interacting particle systems.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1409173
zbMATH Keywords: interacting particle systems
zbMATH Keywords: diffusion-convection equation
zbMATH Keywords: nonlinear martingale problem
MaRDI profile type: Publication
cites work: Q4692970
cites work: Q4748596
cites work: Asymptotic behaviour and source-type solutions for a diffusion-convection equation
cites work: Large time behavior for convection-diffusion equations in \(\mathbb{R}{} ^ N\)
cites work: A propagation of chaos result for a system of particles with moderate interaction
cites work: Q5512463
cites work: A law of large numbers for moderately interacting diffusion processes
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q3342966
cites work: Instability of certain nonlinear stochastic differential equations
cites work: Ecole d'été de probabilités de Saint-Flour XIX, France, du 16 août au 2 septembre 1989
Recommended article: Convergence of Stochastic Particle Systems Undergoing Advection and Coagulation
Recommended article: Convergence of a branching and interacting particle system to the solution of a nonlinear stochastic PDE
Recommended article: Convergence of Diffusion-Drift Many Particle Systems in Probability Under a Sobolev Norm
Recommended article: Convergence to equilibrium for many particle systems
Recommended article: Convergence of stochastic interacting particle systems in probability under a Sobolev norm
Recommended article: Rate of convergence of a stochastic particle system for the Smoluchowski coagulation equation
Recommended article: Convergence of a non-failable mean-field particle system
Recommended article: Q3142700
description: scientific article; zbMATH DE number 1409173