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

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 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 cutoff 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. (2000): 60K35, 35S10. Mathematics Subject Classifications

Far field asymptotics of solutions to convection equation with anomalous diffusion

“…The probabilistic interpretation of nonlinear evolution problems with an anomalous diffusion, obtained recently by Jourdain, Méléard, and Woyczyński [9], motivated us to study (1.1)- (1.2). The authors of [9] considered 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. They associated with the equation a nonlinear singular diffusion and proved propagation of chaos to the law of this diffusion for the related interacting particle systems.…”

Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications

“…Indeed, (4.1) is obvious if u belongs to the Schwartz class of rapidly decreasing functions, since, in this case, one can represent (−∆) s via a Fourier transform (see, for instance, [54,63,64,69]) and check (4.1).…”

“…In the first equality in (4.2), we have used one of the classical representations of the fractional Laplacian, see, e.g., [54,63,64,69] On the other hand,…”

“…We refer to [54,63,64] for the basics of fractional Laplacian theory. We would like to recall that the fractional Laplacian is a very important operator, since it naturally surfaces in many different areas, such as: the thin obstacle problem [13], optimization [33], finance [27], phase transitions [1,2,24,67], stratified materials [66], anomalous diffusion [55], crystal dislocation [57,68], soft thin films [53], some models of semipermeable membranes and flame propagation [21], conservation laws [9], the ultrarelativistic limit of quantum mechanics [40], quasigeostrophic flows [20,56], multiple scattering [18,31,47], minimal surfaces [22,28], materials science [3], probability [5,7,51,52,69], and water waves [17,19,25,26,29,34,35,46,50,59,60,65,71,72].…”

Derivation of Fokker–Planck equations for stochastic systems under excitation of multiplicative non-Gaussian white noise

“…The connection between the Lévy driven SDE and the related partial integrodifferential equations has been extensively studied since [21] and has found wide applications in many areas including finance [4]. This connection has been extended into nonlinear cases recently, see [8] among others. For SDEs driven by Lévy processes, there are two popular definitions, i.e., they are defined in sense of Itô or in sense of Marcus [13,14,11,1].…”