Feynman and Feynman-Kac formulas for evolution equations with Vladimirov operator

Abstract

A Feynman formula is a representation of a solution to the Cauchy problem for an evolution differential or pseudodifferential equation in terms of a limit of integrals over the Cartesian degrees of some space E . A Feynman‐Kac formula is a representation of a solution to the same problem in terms of a path integral. We assume that, on the path space, a countably additive measure or a pseudomeasure (of the type of the Feynman measure; see [2, 3]) is defined, and the multiple integrals in the Feynman formulas coincide with integrals of finite multiplicity approximating integrals with respect to this measure or pseudomeasure. In this paper, we obtain Feynman and Feynman‐ Kac formulas for solutions to the Cauchy problems for the heat equation with respect to complex-valued functions on the product of the real half-line and the p -adic line � � ; the role of the Laplace operator in these equations is played by the Vladimirov operator. Similar formulas can be obtained for Schrodinger-type equations and for the case of a multidimensional space over � � . Such equations may be useful in constructing mathematical models of processes on scales characterized by Planck length and time and phenomenological models in chemistry, continuum mechanics, and psychology (see [1, 4‐6] and the references therein).