标题： 费曼-卡茨理论中的非线性漂移：保留早期的概率论洞见 显示英文标题

标题： Nonlinear Drift in Feynman-Kac Theory: Preserving Early Probabilistic Insights

摘要： 1905年，爱因斯坦的布朗运动理论支持了扩散方程的分子基础，并引入了两种互补的观点：一种是确定性的场描述，另一种是以随机粒子系综为基础的概率表述。其后果在现代物理学关键概念的发展中具有深远影响，例如量子力学中的波粒二象性。 20世纪40年代，费曼和卡茨通过将路径积分纳入测度论框架内，定义解为数学期望，并将该方法扩展到一大类微分算子，从而推进了这一理论体系。 尽管这一理论影响深远，但将其确定性-概率对应关系应用于受限几何内的流动问题却一直难以实现：如何从由随机速度输运的粒子中恢复确定性的流线？这些随机速度从未与真实的流动场相匹配？ 优雅的粒子系统模型已被设计用于碰撞等离子体、半导体晶体、球状星团以及生物微泳动者，但它们偏离了用单一过程传播源来表示解的初衷。 在这里，我们证明费曼-卡茨理论可以严格扩展到带有漂移的非线性动力学中，同时保持其概率论的起源。这产生了新的传播子表达方式，并在解决复杂几何问题的应用数学、计算机图形学以及工程学界形成了思想的融合。 显示英文摘要

摘要： In 1905, Einstein's theory of Brownian motion supported the molecular basis of the diffusion equation and introduced two complementary viewpoints: a deterministic field description and a probabilistic formulation based on stochastic particle ensembles. The consequences were far-reaching in the development of key concepts of modern physics such as wave-particle duality in quantum mechanics. In the 1940s, Feynman and Kac advanced this framework by casting path integrals within measure theory, defining solutions as mathematical expectations and extending the method to a broad class of differential operators. Despite its influence, applying this deterministic-probabilistic correspondence to flows within confined geometries has remained elusive: how can one recover deterministic streamlines from particles advected by a random velocity that never matches the true flow field? Elegant particle-system models have been devised for collisional plasmas, semiconducting crystals, globular clusters, and biological microswimmers, yet they depart from the original intent of representing the solution as an expectation of sources propagated by a single process. Here, we show that Feynman-Kac's theory can be rigorously extended to nonlinear dynamics with drift, staying true to its probabilistic origin. This yields novel propagator representations and forges a convergence of ideas across applied mathematics, computer graphics, and engineering communities tackling complex geometries.