New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes
Year: 2026
Type: preprint
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 $\mathbb{R}^N\times(0,\infty)$, where $Δ_p$ is the $p$-Laplacian and $Δ_p^s$ 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 $p\ge4$. 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 $Δ_p+Δ_p^s$.
Source: arXiv (Cornell University)
Authors: Jeddaoui, Houssine El, Nabab, Dany
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213600329