The Feynman‐Kac Formula with a Lebesgue‐Stieltjes Measure and Feynman's Operational Calculus

We investigate what happens if in the Feynman‐Kac functional, we perform the time integration with respect to a Borel measure η rather than ordinary Lebesgue measure l . Let u ( t ) be the operator associated with this functional through path integration. We show that u ( t ), considered as a function of time t , satisfies a certain Volterra‐Stieltjes integral equation. This result establishes a “FeynmanKac formula with Lebesgue‐Stieltjes measure η .” One recovers the classical Feynman‐Kac formula by letting η = l . We deduce from the integral equation that u ( t ) satisfies a differential equation associated with the continuous part μ of η when η = μ = l , this differential equation reduces to the heat or the Schrödinger equation in the probabilistic or quantum‐mechanical case, respectively. Moreover, we observe a new phenomenon, due to the discrete part v of η : the function u ( t ) undergoes a discontinuity at every point in the support of v , assumed here to be finite. Further, one obtains an explicit expression for u ( t ) in terms of operators alternatively associated with μ and v . Our results are new even in the probabilistic or “imaginary time” case and allow us to unify various concepts. The derivation of our integral equation has an interesting combinatorial structure and makes essential use of the “generalized Dyson series”— recently introduced by G. W. Johnson and the author—that “disentangle” the operator u ( t ). We provide natural physical interpretations of our results in both the diffusion and quantum‐mechanical cases. We also suggest further connections with FeynmanȈs operational calculus for noncommuting operators.