发表机构
École Polytechnique Paris, Centre de Mathématiques Appliquées（巴黎综合理工学院，应用数学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带共同噪声的Wasserstein空间倒向随机微分方程（W-BSDE），通过特征方程建立适定性，构造解并应用于McKean-Vlasov控制与平均场博弈，证明其作为有限维BSDE的平均场极限。
AI 中文摘要
我们研究了Wasserstein空间上的一类倒向随机微分方程，称为W-BSDE。这项工作将第一作者近期的结果推广到共同噪声的设定。该方程以马尔可夫形式表述，沿着由带共同噪声的McKean-Vlasov动力学生成的条件律的容许流。解是一对场$(Y,Z)$，其中$Y$是概率测度空间上的泛函，$Z$是相应的内禀Wasserstein梯度。我们的主要结果表明，W-BSDE的适定性可以归结为关于场$p$的特征方程。只要该方程存在合适的解，我们就构造出W-BSDE的解，并将$Z$同时识别为特征场$p$和$Y$的内禀梯度。这给出了W-BSDE的存在性结果，同时也给出了Wasserstein空间上泛函及其内禀梯度的概率构造。随后我们发展了该理论的若干推论。我们获得了McKean-Vlasov控制问题和带共同噪声的平均场博弈的验证原理，无需依赖凸性或可分性假设。我们还证明了特征场通过McKean-Vlasov正倒向SDE具有动态表示。第一个分量$Y$在粘性意义下求解概率测度空间上相关的半线性PDE，包括由共同噪声引起的二阶项。在额外光滑性条件下，$Y$是经典解。最后，我们证明W-BSDE作为经典有限维BSDE在经验测度上的内禀平均场极限出现，并且值分量和重标度鞅被积项均收敛。
英文摘要
We study a notion of backward stochastic differential equation on the Wasserstein space, called a W-BSDE. This work extends to the common noise setting recent results of the first author. The equation is formulated in Markovian form along admissible flows of conditional laws generated by McKean-Vlasov dynamics with common noise. A solution is a pair of fields $(Y,Z)$, where $Y$ is a functional on the space of probability measures and $Z$ is the corresponding intrinsic Wasserstein gradient. Our main result shows that the well-posedness of the W-BSDE can be reduced to a characteristic equation for a field $p$. Whenever this equation admits a suitable solution, we construct a W-BSDE solution and identify $Z$ with both the characteristic field $p$ and the intrinsic gradient of $Y$. This gives an existence result for the W-BSDE and, at the same time, a probabilistic construction of a functional on the Wasserstein space together with its intrinsic gradient. We then develop several consequences of the theory. We obtain verification principles for McKean-Vlasov control problems and for mean field games with common noise, without relying on convexity or separability assumptions. We also show that the characteristic field admits a dynamic representation through a McKean-Vlasov forward-backward SDE. The first component $Y$ solves the associated semilinear PDE on the space of probability measures in the viscosity sense, including the second-order term induced by common noise. Under additional smoothness, $Y$ is a classical solution. Finally, we prove that W-BSDEs arise as intrinsic mean-field limits of classical finite-dimensional BSDEs written on empirical measures, with convergence of both the value component and the rescaled martingale integrands.
