New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes

Abstract

We prove the existence of a McKean--Vlasov stochastic process with jumps associated to the nonlinear parabolic equation ${\partial }_{t}u={\Delta }_{p}u+{\Delta }_p^{s}u$ in ${\mathbb{R}}^N\times (0,\infty )$, where ${\Delta }_p$ is the $p$-Laplacian and ${\Delta }_p^s$ is the fractional $p$-Laplacian. Our approach is the following: first, after proving the existence of a solution for the PDE presented earlier, we rewrite it as a nonlinear Fokker-Planck-Kolmogorov equation whose measure-valued solution is guaranteed when $p\ge 4$. Then we solve the martingale problem associated to our FPKE via a new superposition principle for non-local Fokker-Planck-Kolmogorov equations, which only requires an integrability condition on the local coefficients. Finally, thanks to the martingale solution obtained, we derive the existence of a weak solution for the McKean-Vlasov's type SDE with jumps whose infinitesimal generator is a << hybrid version >> of the operator ${\Delta }_p+{\Delta }_p^s$.