Representation of solutions to evolution equations with Vladimirov operator in terms of Feynman path integrals

Abstract

It is well known that solutions of initial‐boundary value problems for classical evolution equations with a pseudodifferential operator on the right-hand side can be represented by integrals over the trajectory space in the configuration, momentum, or phase space. In the first case, integration is performed with respect to the Wiener measure or Feynman pseudomeasure (and their generalizations); in the second case, with respect to the Maslov‐Chebotarev(‐Poisson) complex measure; and, in third case, with respect to the Hamiltonian Feynman pseudomeasure, its analogues, or an analogue of the Maslov‐Chebotarev measure. In this paper, such representations are derived for evolution equations with the Vladimirov pseudodifferential operator, which is a p -adic analogue of a positive power of the Laplacian acting on the space of complexvalued functions of � -adic argument.