On infinitesimal generators and Feynman–Kac integrals of adelic diffusion

Abstract

For each prime p, a Vladimirov operator with a positive exponent specifies a p-adic diffusion equation and a measure on the Skorokhod space of p-adic paths. The product, P, of these measures with a fixed exponent is a probability measure on the product of the p-adic path spaces. The adelic paths have full measure if and only if the sum, σ, of the diffusion constants is finite. Finiteness of σ implies that there is an adelic Vladimirov operator, ΔA, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by P on an adelic Skorokhod space. For a wide class of potentials, the dynamical semigroups associated with adelic Schrödinger operators with free part ΔA have path integral representations.