On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

TL;DR

This paper explores the construction of adelic diffusion processes using Vladimirov operators across all primes, establishing conditions for their measure and path integral representations of associated Schrödinger operators.

Contribution

It introduces an adelic Vladimirov operator and demonstrates the existence of path integral representations for adelic Schrödinger operators with a broad class of potentials.

Findings

Finiteness of the sum of diffusion constants ensures full measure of adelic paths.

Existence of an adelic Vladimirov operator and associated diffusion equation.

Path integral representations for adelic Schrödinger operators with various potentials.

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 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, $\sigma$, of the diffusion constants is finite. Finiteness of $\sigma$ implies that there is an adelic Vladimirov operator, $\Delta_{\mathbb A}$, 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 to adelic Schr\"{o}dinger operators with free part $\Delta_{\mathbb A}$ have path integral representations.