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Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Mohamed Ben Alaya  and
Benjamin Jourdain
Mohamed Ben Alaya*
Affiliation:
Université Paris 13
Benjamin Jourdain*
Affiliation:
CERMICS
*
∗ Postal address: LAGA, Institut Galilée, Université Paris 13, 93430 Villetaneuse, France. Email address: mba@math.univ-paris13.fr
∗∗ Postal address: CERMICS, École des Ponts, ParisTech, 6–8 avenue Blaise Pascal, 77455 Marne la Vallée, France. Email address: jourdain@cermics.enpc.fr
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Abstract

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In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.

Keywords

Nonlinear martingale problempropagation of chaosstochastic particle method

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60F99: None of the above, but in this section 65C35: Stochastic particle methods
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 528 - 546
Copyright
Copyright © Applied Probability Trust 2007 

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