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PreprintCC BY-NC-ND 4.0arXiv

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

  • Houssine El Jeddaoui
  • Dany Nabab
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Abstract

We prove the existence of a McKean--Vlasov stochastic process with jumps associated to the nonlinear parabolic equation ∂t u = Δp u + Δps u in ℝN×(0,∞), where Δp is the p-Laplacian and Δps 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≥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 Δp+Δps.

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Cite

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BibTeX
@misc{jeddaoui2026link,
  title = {{New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes}},
  author = {Houssine El Jeddaoui and Dany Nabab},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.19492},
  eprint = {2609.19492},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19492},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes
AU  - Houssine El Jeddaoui
AU  - Dany Nabab
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.19492
UR  - https://arxiv.org/abs/2609.19492
AB  - We prove the existence of a McKean--Vlasov stochastic process with jumps associated to the nonlinear parabolic equation ∂_t u = Δ_p u + Δ_p^s u in ℝ^N×(0,∞), 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≥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 Δ_p+Δ_p^s.
ER  -