The Feynman–Kac Formula: A Measure-Theoretic, Analytic, and Probabilistic Synthesis

The Feynman–Kac formula stands among the rare mathematical results that elegantly bridge distinct conceptual worlds — analysis, probability, and physics — by asserting that the evolution of analytic structures such as parabolic partial differential equations can be represented through the expectations of random paths. It is a synthesis that makes rigorous Feynman’s intuitive vision of quantum propagation via path integrals and Kac’s probabilistic representation of parabolic equations. The purpose of this monograph is to present the Feynman–Kac formula and its far-reaching generalizations within a rigorous measure-theoretic and functional-analytic framework. While many excellent expositions introduce the formula as a computational tool or as a heuristic bridge between stochastic processes and partial differential equations, few texts aim to unify its analytical, probabilistic, and physical interpretations within a single, fully self-contained narrative.