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

Abstract: 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.