AI 中文总结
本文研究带公共噪声的McKean-Vlasov半群的微分结构，推导其Lions导数与后向表示公式，将其应用于Wasserstein空间线性方程的Feynman-Kac表示及无漂移情形的鞅表示问题，并讨论相关表示的扩展。
AI 中文摘要
我们引入了带有公共噪声的McKean-Vlasov半群并研究其一阶微分结构。给定终端泛函G，该半群通过初始律沿参考条件McKean-Vlasov流传播来定义。利用有限粒子近似以及切过程的对角/非对角分解，我们确定了传播泛函的Lions导数，并沿任意容许条件律流推导出后向表示公式。该公式仅包含测度变量的一阶导数，既给出了Wasserstein空间上线性方程的Feynman-Kac表示，又在无漂移情形下，得到了唯一解由McKean-Vlasov热半群及其Lions导数给出的鞅表示问题。我们还讨论了混合有限维/Wasserstein表示及离散路径依赖扩展。
英文摘要
We introduce a McKean--Vlasov semigroup in the presence of common noise and study its first-order differential structure. Given a terminal functional $G$, the semigroup is defined by propagating the initial law through a reference conditional McKean--Vlasov flow. Using a finite-particle approximation and a diagonal/off-diagonal decomposition of the tangent processes, we identify the Lions derivative of the propagated functional and derive a backward representation along arbitrary admissible conditional law flows. This formula involves only first-order derivatives in the measure variable and yields both a Feynman--Kac representation for linear equations on the Wasserstein space and, in the driftless case, a martingale representation problem whose unique solution is given by the McKean--Vlasov heat semigroup and its Lions derivative. We also discuss mixed finite-dimensional/Wasserstein representations and discrete path-dependent extensions.
