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Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology
Published online by Cambridge University Press: 14 July 2016
Abstract
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.
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- Copyright © Applied Probability Trust 2007
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