The differential equation for the Feynman-Kac formula with a Lebesgue- Stieltjes measure
type: Item
instance of: scholarly article
title: The differential equation for the Feynman-Kac formula with a Lebesgue- Stieltjes measure
zbMATH Open document ID: 0635.35081
DOI: 10.1007/BF00417458
published in: Letters in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 81Q30
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 4034263
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: Lebesgue-Stieltjes measure
zbMATH Keywords: time integration
zbMATH Keywords: Borel measure
zbMATH Keywords: discontinuity
zbMATH Keywords: Volterra-Stieltjes integral equation
zbMATH Keywords: explicit expression
zbMATH Keywords: product integral representation
Wikidata QID: Q115395009
author: Michel L. Lapidus
MaRDI profile type: Publication
cites work: Space-Time Approach to Non-Relativistic Quantum Mechanics
cites work: The Feynman-Kac Formula with a Lebesgue-Stieltjes Measure and Feynman's Operational Calculus
cites work: Q5516901
cites work: Perturbation theory and a dominated convergence theorem for Feynman integrals
cites work: An Operator Calculus Having Applications in Quantum Electrodynamics
description: scientific article; zbMATH DE number 4034263