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Probabilistic approximation for a porous medium equation

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  • Jourdain, B.

Abstract

In this paper, we are interested in the one-dimensional porous medium equation when the initial condition is the distribution function of a probability measure. We associate a nonlinear martingale problem with it. After proving uniqueness for the martingale problem, we show existence owing to a propagation of chaos result for a system of weakly interacting diffusion processes. The particle system obtained by increasing reordering from these diffusions is proved to solve a stochastic differential equation with normal reflection. Last, we obtain propagation of chaos for the reordered particles to a probability measure which does not solve the martingale problem but is also linked to the porous medium equation.

Suggested Citation

  • Jourdain, B., 2000. "Probabilistic approximation for a porous medium equation," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 81-99, September.
    • Handle: RePEc:eee:spapps:v:89:y:2000:i:1:p:81-99
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    References listed on IDEAS

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    1. B. Jourdain, 2000. "Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures," Methodology and Computing in Applied Probability, Springer, vol. 2(1), pages 69-91, April.
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    Cited by:

    1. Philipowski, Robert, 2007. "Interacting diffusions approximating the porous medium equation and propagation of chaos," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 526-538, April.
    2. Mykhaylo Shkolnikov & Lane Chun Yeung, 2024. "From rank-based models with common noise to pathwise entropy solutions of SPDEs," Papers 2406.07286, arXiv.org.
    3. Belaribi, Nadia & Cuvelier, François & Russo, Francesco, 2011. "A probabilistic algorithm approximating solutions of a singular PDE of porous media type," Monte Carlo Methods and Applications, De Gruyter, vol. 17(4), pages 317-369, December.

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