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 $\partial_t u = Δ_p u + Δ_p^s u$ in $\R^N\times(0,\infty)$, where $Δ_p$ is the $p$-Laplacian and $Δ_p^s$ is the fractional $p$-Laplacian. The algorithm used 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 solution-measure is guaranted when $p\ge4$. Then we solve the martingale problem associated to our FPKE via a new nonlinear supersition principle. 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 $Δ_p+Δ_p^s$.