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 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 Schrödinger operators with free part ${\Delta }_{\mathbb{A}}$ have path integral representations.

arXiv comment32 pages

References in corpus (6)

• p-Adic Mathematical Physics
• Application of p-adic analysis to models of spontaneous breaking of the replica symmetry
• p-Adic description of characteristic relaxation in complex systems
• Parabolic Type Equations and Markov Stochastic Processes on Adeles
• Path Integrals for Quadratic Lagrangians on p-Adic and Adelic Spaces
• Brownian Motion and Finite Approximations of Quantum Systems over Local Fields