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

We prove the existence of a McKean–Vlasov stochastic process with jumps associated to the nonlinear parabolic equation ${\mathrm{\partial }}_{t}u={\mathrm{\Delta }}_{p}u+{\mathrm{\Delta }}_p^{s}u$ in ${\mathbb{R}}^N\times (0,\mathrm{\infty })$, where ${\mathrm{\Delta }}_p$ is the $p$-Laplacian and ${\mathrm{\Delta }}_p^s$ 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 $p\ge 4$. 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 ${\mathrm{\Delta }}_p+{\mathrm{\Delta }}_p^s$.