Abstract
In this paper, we explain how to associate a nonlinear martingale problem with some nonlinear parabolic evolution equations starting at bounded signed measures. Our approach generalizes the classical link made when the initial condition is a probability measure. It consists in giving to each sample-path a signed weight which depends on the initial position. After dealing with the classical McKean-Vlasov equation as an introductory example, we are interested in a viscous scalar conservation law. We prove uniqueness for the corresponding nonlinear martingale problem and then obtain existence thanks to a propagation of chaos result for a system of weakly interacting diffusion processes. Last, we study the behavior of the associated fluctuations and present numerical results which confirm the theoretical rate of convergence.
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Jourdain, B. Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures. Methodology and Computing in Applied Probability 2, 69–91 (2000). https://doi.org/10.1023/A:1010059302049
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DOI: https://doi.org/10.1023/A:1010059302049