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Convergence of a stochastic particle approximation for fractional scalar conservation laws

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  • Jourdain, Benjamin
  • Roux, Raphaël

Abstract

We are interested in a probabilistic approximation of the solution to scalar conservation laws with fractional diffusion and nonlinear drift. The probabilistic interpretation of this equation is based on a stochastic differential equation driven by an [alpha]-stable Lévy process and involving a nonlinear drift. The approximation is constructed using a system of particles following a time-discretized version of this stochastic differential equation, with nonlinearity replaced by interaction. We prove convergence of the particle approximation to the solution of the conservation law as the number of particles tends to infinity whereas the discretization step tends to 0 in some precise asymptotics.

Suggested Citation

  • Jourdain, Benjamin & Roux, Raphaël, 2011. "Convergence of a stochastic particle approximation for fractional scalar conservation laws," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 957-988, May.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:5:p:957-988
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    References listed on IDEAS

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    1. BOSSY Mireille & FEZOUI Loula & PIPERNO Serge, 1997. "Comparison Of A Stochastic Particle Method And A Finite Volume Deterministic Method Applied To Burgers Equation," Monte Carlo Methods and Applications, De Gruyter, vol. 3(2), pages 113-140, December.
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    Cited by:

    1. Li, Can, 2017. "Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 240-250.
    2. Vassili Kolokoltsov & Marianna Troeva & Wei Yang, 2014. "On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players," Dynamic Games and Applications, Springer, vol. 4(2), pages 208-230, June.

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