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On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

David Weisbart

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, $σ$, of the diffusion constants is finite. Finiteness of $σ$ implies that there is an adelic Vladimirov operator, $Δ_{\mathbb A}$, 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 Schrödinger operators with free part $Δ_{\mathbb A}$ have path integral representations.

On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

Abstract

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

Paper Structure

This paper contains 10 sections, 16 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Basic Facts about $\mathds Q_p$ and $\mathds A$
  4. Diffusion in the $\mathds Q_p$ Setting
  5. Products of $p$-Adic Path Spaces
  6. Exit Time Probabilities
  7. The Probability of the Adelic Path Subspace
  8. Moments of the ${{\mathds Z}_{\mathds A}}$ First Exit Number
  9. An Adelic Diffusion Equation
  10. Adelic Path Integrals and Propagators

Key Result

Proposition \oldthetheorem

A Hausdorff topological space that is second countable and locally compact is completely metrizable.

Theorems & Definitions (31)

  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 21 more