The differential equation for the Feynman-Kac formula with a Lebesgue- Stieltjes measure
We investigate the Feynman-Kac formula with a Lebesgue-Stieltjes measure, in which the time integration is performed with respect to a Borel measure and not just an ordinary Lebesgue measure as in the classical case. We derive the corresponding differential equation, which reduces to the heat or the Schrödinger equation in the standard cases. Actually, we observe two distinct phenomena: first, the differential equation is governed by the continuous part of the measure; secondly, the solution undergoes a discontinuity at every point in the support of the discrete part of the measure. Alternately, we obtain a Volterra-Stieltjes integral equation. We conclude by finding an explicit expression for the solution and a certain product integral representation. These results and the methods of proof suggest possible physical interpretations and various mathematical developments.
scientific article; zbMATH DE number 4034263