分数阶p-Laplacian算子与一类McKean-Vlasov飞行型过程之间的新联系
New link between the fractional p-Laplacian operators and a class of McKean-Vlasov flight type processes
AI总结：
本文证明了一类与非线性抛物方程（含p-Laplacian和分数阶p-Laplacian）相关联的带跳McKean-Vlasov过程的存在性，通过重写为FPKE并利用新的非线性叠加原理解决鞅问题，最终得到弱解。
AI中文摘要：
我们证明了与非线性抛物方程 $\partial_t u = \Delta_p u + \Delta_p^s u$（在 $\R^N\times(0,\infty)$ 中）相关联的带跳McKean--Vlasov随机过程的存在性，其中 $\Delta_p$ 是 $p$-Laplacian，$\Delta_p^s$ 是分数阶 $p$-Laplacian。所用算法如下：首先，在证明了前述PDE解的存在性之后，我们将其重写为非线性Fokker-Planck-Kolmogorov方程，当 $p\ge4$ 时其解测度得到保证。然后，我们通过一个新的非线性叠加原理解决了与我们的FPKE相关的鞅问题。最后，借助所获得的鞅解，我们推导出McKean-Vlasov型带跳SDE弱解的存在性，其无穷小生成元是算子 $\Delta_p+\Delta_p^s$ 的“混合版本”。
英文摘要：
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 $Δ_p+Δ_p^s$.
