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Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures

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  • B. Jourdain

    (ENPC-CERMICS)

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:2:y:2000:i:1:d:10.1023_a:1010059302049
    DOI: 10.1023/A:1010059302049
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    Cited by:

    1. Jourdain, B., 2000. "Probabilistic approximation for a porous medium equation," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 81-99, September.
    2. Andrey Sarantsev, 2019. "Comparison Techniques for Competing Brownian Particles," Journal of Theoretical Probability, Springer, vol. 32(2), pages 545-585, June.
    3. Shkolnikov, Mykhaylo, 2013. "Large volatility-stabilized markets," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 212-228.
    4. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    5. Rémillard, Bruno & Vaillancourt, Jean, 2014. "On signed measure valued solutions of stochastic evolution equations," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 101-122.
    6. BOSSY Mireille & JOURDAIN Benjamin, 2001. "A Stochastic Particle Method For The Solution Of A 1d Viscous Scalar Conservation Law In A Bounded Interval," Monte Carlo Methods and Applications, De Gruyter, vol. 7(1-2), pages 45-54, December.
    7. Tran, Viet Chi, 2008. "A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 284-318, February.
    8. Kolli, Praveen & Sarantsev, Andrey, 2019. "Large rank-based models with common noise," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 29-35.

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