McKean--Vlasov 随机微分方程在 Krylov 稳定系数空间中的一般强适定性（含跳）
Generic Strong Well-Posedness for McKean--Vlasov SDEs with Jumps in a Krylov-Stable Coefficient Space
- Jiangxi University of Finance and Economics（江西财经大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明带跳的 McKean--Vlasov 随机微分方程在 Krylov 稳定系数空间中强适定性是普遍的，通过集值稳定性定理和 Baire 范畴论证，得到稠密剩余系数集上的唯一强解及连续依赖。
AI中文摘要：
我们证明了在允许真正不连续状态依赖的完备系数空间中，带跳的 McKean--Vlasov 随机微分方程的强适定性是普遍的。对每个 $T>0$，存在一个与初始状态无关的稠密剩余子集 $\mathfrak R_T$，使得每个系数 $a\in\mathfrak R_T$ 对每个确定性初始条件 $x\in\mathbb R^d$ 都生成唯一强解。此外，解映射 $x\mapsto X^{a,x}$ 在 $\mathcal S_T^2$ 中是连续的，并且每个由空间 Lipschitz 系数逼近的序列在 $x$ 上局部一致收敛到相同的典范解族。系数是一致有界的，在扩散分量上一致非退化，并且在 $2$-Wasserstein 距离下关于分布是 Lipschitz 连续的，而其状态依赖可能仅仅是可测的。环境空间是通过在基于局部 $L^{2(d+1)}$ 系数误差的 Krylov 稳定度量下完备化空间 Lipschitz 核心而获得的。一个停止的条件 Krylov--Melnikov 占据估计将该度量中的收敛转化为沿非退化 Itô--Lévy 轨迹的占据收敛。关键的随机成分是一个集值稳定性定理：在核心系数附近，每个邻近可解方程的每个强解都保持接近唯一的核心解，而不需要对邻近方程作唯一性假设。结合强解关系的闭性，这允许在完备系数空间上进行 Baire 范畴论证。我们还给出了一个内在的空间平移准则，表明该完备化包含具有本质空间不连续性的自然类。一个射影极限论证产生了相应的全局时间一般适定性结果。
英文摘要：
We prove that strong well-posedness is generic for McKean--Vlasov stochastic differential equations with jumps in a complete coefficient space that allows genuinely discontinuous state dependence. For every $T>0$, there exists a dense residual subset $\mathfrak R_T$, independent of the initial state, such that every coefficient $a\in\mathfrak R_T$ generates a unique strong solution for every deterministic initial condition $x\in\mathbb R^d$. Moreover, the solution map $x\mapsto X^{a,x}$ is continuous in $\mathcal S_T^2$, and every approximation by spatially Lipschitz coefficients converges locally uniformly in $x$ to the same canonical solution family. The coefficients are uniformly bounded, uniformly non-degenerate in the diffusion component, and Lipschitz continuous in the law with respect to the $2$-Wasserstein distance, while their state dependence may be merely measurable. The ambient space is obtained by completing a spatially Lipschitz core under a Krylov-stable metric based on local $L^{2(d+1)}$ coefficient errors. A stopped conditional Krylov--Melnikov occupation estimate converts convergence in this metric into occupation convergence along non-degenerate Itô--Lévy trajectories. The key probabilistic ingredient is a set-valued stability theorem: near a core coefficient, every strong solution of every nearby solvable equation remains close to the unique core solution, without any uniqueness assumption on the nearby equation. Together with closedness of the strong-solution relation, this permits a Baire-category argument on the completed coefficient space. We also give an intrinsic spatial-translation criterion showing that the completion contains natural classes with essential spatial discontinuities. A projective-limit argument yields the corresponding global-time generic well-posedness result.
