期刊
JOURNAL OF APPLIED PROBABILITY
卷 44, 期 2, 页码 528-546出版社
CAMBRIDGE UNIV PRESS
DOI: 10.1239/jap/1183667419
关键词
nonlinear martingale problem; propagation of chaos; stochastic particle method
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 rho(t, x) with respect to the Lebesgue measure, where the function rho 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 infinity. 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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