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

Abstract

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.

Survey of the Theories for Analogue of Wiener Measure Space

“…The following theorem is the counterpart for the measure-valued measure V ' of the integral equation for the Feynman-Kac formula with Lebesgue-Stieljes measure, obtained by Lapidus in [20][21][22] and for the Feynman-Kac formula with an operator-valued measure, obtained by Kluvanek in [18].…”

“…Johnson and Lapidus established the existence theorem of the operator-valued function space integral as an operator from L 2 ðR N Þ to itself for certain functionals involving some Borel measures [16]. And in 1987, Lapidus proved that the integral satisfies the Schrödinger wave equation [20]. In 1992, Chang and the first author established the existence theorem of the operatorvalued function space integral as an operator from L p to L p 0 ð1 < p < 2Þ for certain functionals involving some Borel measures [7].…”

A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula

“…From the multinomial expansion, In this section, we prove that the equality (5.16) in Theorem 5.5, satisfies a suitable Volterra integral equation (see [10], [12], [13], [14]). …”

“…The following theorem is the counterpart for the measure-valued measure V ϕ of the integral equation for the Feynman-Kac formula with Lebesgue-Stieltjes measure, obtained by Lapidus in [12], [13], [14] and for the Feynman-Kac formula with an operator-valued measure, obtained by Kluvanek in [10]. Step (1) follows from (2.17).…”

“…In 1986, Johnson and Lapidus obtained a perturbation expansion for the operatorvalued function space integral under quite general conditions. Further, in 1987, Lapidus showed that for an exponential functional, the result in their paper satisfies a Volterra integral equation [9], [12], [13], [14], thereby establishing in this context the Feynman-Kac formula with a (complex-valued) measure. In 1983, Kluvanek introduced a measure µ ϕ on the space C 0 [0, t] of all continuous functions on a closed interval by a method similar to the Wiener case and introduced an operator-valued measure M t on C 0 [0, t] associated with a measure µ ϕ [10].…”