Potential Analysis

, Volume 23, Issue 1, pp 55–81

A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator

  • Benjamin Jourdain
  • Sylvie Méléard
  • Wojbor A. Woyczynski
Article

DOI: 10.1007/s11118-004-3264-9

Cite this article as:
Jourdain, B., Méléard, S. & Woyczynski, W.A. Potential Anal (2005) 23: 55. doi:10.1007/s11118-004-3264-9

Abstract

We consider a class of nonlinear integro-differential equations involving a fractional power of the Laplacian and a nonlocal quadratic nonlinearity represented by a singular integral operator. Initially, we introduce cut-off versions of this equation, replacing the singular operator by its Lipschitz continuous regularizations. In both cases we show the local existence and global uniqueness in L1Lp. Then we associate with each regularized equation a stable-process-driven nonlinear diffusion; the law of this nonlinear diffusion has a density which is a global solution in L1 of the cut-off equation. In the next step we remove the cut-off and show that the above densities converge in a certain space to a solution of the singular equation. In the general case, the result is local, but under a more stringent balance condition relating the dimension, the power of the fractional Laplacian and the degree of the singularity, it is global and gives global existence for the original singular equation. Finally, we associate with the singular equation a nonlinear singular diffusion and prove propagation of chaos to the law of this diffusion for the related cut-off interacting particle systems. Depending on the nature of the singularity in the drift term, we obtain either a strong pathwise result or a weak convergence result.

Keywords

propagation of chaosnonlinear stochastic differential equations driven by Lévy processespartial differential equation with fractional Laplaciannonlinear singular operator

Copyright information

© Springer 2005

Authors and Affiliations

  • Benjamin Jourdain
    • 1
  • Sylvie Méléard
    • 2
  • Wojbor A. Woyczynski
    • 3
  1. 1.ENPC-CERMICSMarne la Vallée Cedex 2France
  2. 2.Université Paris 10, MODALXNanterreFrance
  3. 3.Department of Statistics and Center for Stochastic and Chaotic Processes in Science and TechnologyCase Western Reserve UniversityClevelandU.S.A.