A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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 L 1∩L p. 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 L 1 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.
Mathematics Subject Classifications (2000)
- Aldous, D.: ‘Stopping times and tightness’, Ann. Probab. 6 (1978), 335–340.
- Bichteler, K., Gravereaux, J.B. and Jacod, J.: Malliavin Calculus for Processes with Jumps, Stochastics Monographs 2, Gordon and Breach Science Publishers, 1987.
- Biler, P. and Nadzieja, T.: ‘Existence and nonexistence of solutions for a model of gravitational interaction of particles’, Colloquium Math. 66 (1994), 319–334.
- Biler, P. and Nadzieja, T.: ‘A singular problem in electrolytes theory’, Math. Methods Appl. Sci. 20 (1997), 767–782.
- Biler, P. and Woyczynski, W.A.: ‘Global and exploding solutions for nonlocal quadratic evolution problems’, SIAM J. Appl. Math. 59 (1998), 845–869.
- Biler, P., Funaki, T. and Woyczynski, W.A.: ‘Interacting particle approximations for nonlocal quadratic evolution problems’, Probab. Math. Statist. 19(2) (1999), 267–286.
- Biler, P., Karch, G. and Woyczynski, W.A.: ‘Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws’, Ann. Inst. Henri Poincaré Nonlinear Anal. 18 (2001), 613–637.
- Cannone, M.: Ondelettes, Paraproduits et Navier–Stokes, Diderot Editeur, Paris, 1995.
- Dobrushin, R.L.: ‘Prescribing a system of random variables by conditional expectations’, Theory Probab. Appl. 15(3) (1970), 450.
- Ethier, S.N. and Kurtz, T.G.: Markov Processes, Characterization and Convergence, Wiley, 1986.
- Fontbona, J.: ‘Nonlinear martinagle problems involving singular integrals’, J. Funct. Anal. 200(1) (2003), 198–236.
- Jourdain, B. and Méléard, S.: ‘Propagation of chaos and fluctuations for a moderate model with smooth initial data’, Ann. Inst. Henri Poincaré Probab. 34(6) (1998), 727–766.
- Jourdain, B.: ‘Diffusion processes associated with nonlinear evolution equations for signed measures’, Methodology and Computing in Applied Probability 2(1) (2000), 69–91.
- Mann, J.A. and Woyczynski, W.A.: ‘Growing fractal interfaces in the presence of self-similar hopping surface diffusion’, Physica A 291 (2001), 159–183.
- Liggett, T.: Interacting Particle Systems, Springer, 1985.
- Méléard, S.: ‘Asymptotic behaviour of some interacting particle systems: McKean–Vlasov and Boltzmann models’, in CIME 1995, Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math. 1627, Springer, 1996, pp. 42–95.
- Méléard, S.: ‘A trajectorial proof of the vortex method for the two-dimensional Navier–Stokes equation’, Ann. Appl. Probab. 10(4) (2000), 1197–1211.
- Méléard, S.: ‘Monte-Carlo approximations for 2d Navier–Stokes equations with measure initial data’, Probab. Theory Related Fields 121 (2001), 367–388.
- Metzler, R. and Klafter, J.: ‘The random walk’s guide to anomalous diffusion: A fractional dynamics approach’, Phys. Rep. 339 (2000), 1–77. CrossRef
- Saichev, A.I. and Woyczynski, W.A.: Distributions in the Physical and Engineering Sciences. Volume 2: Linear, Nonlinear, Random and Fractal Dynamics in Continuous Media, Birkhäuser, Boston, 2004.
- Sato, K.: Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Math. 68, Cambridge University Press, 1999.
- Stroock, D.W. and Varadhan, S.R.S.: Multidimensional Diffusion Processes, Springer-Verlag, 1979.
- Sznitman, A.S.: Topics in Propagation of Chaos, Ecole d’été de probabilités de Saint-Flour XIX – 1989, Lecture Notes in Math. 1464, Springer-Verlag, 1991.
- Zheng, W.: ‘Conditional propagation of chaos and a class of quasilinear PDE’s’, Ann. Probab. 23 (1995), 1389–1413.
- A Probabilistic Approach for Nonlinear Equations Involving the Fractional Laplacian and a Singular Operator
Volume 23, Issue 1 , pp 55-81
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- propagation of chaos
- nonlinear stochastic differential equations driven by Lévy processes
- partial differential equation with fractional Laplacian
- nonlinear singular operator
- Author Affiliations
- 001. ENPC-CERMICS, 6-8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455, Marne la Vallée Cedex 2, France
- 002. Université Paris 10, MODALX, 200 av. de la République, 92000, Nanterre, France
- 003. Department of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH, 44106, U.S.A.