Brownian Motion and Finite Approximations of Quantum Systems over Local Fields

Abstract

We give a stochastic proof of the finite approximability of a class of Schrödinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

References in corpus (1)

• Nonlocal Operators, Parabolic-type Equations, and Ultrametric Random Walks

Cited by in corpus (2)

• $p$-Adic Brownian Motion is a Scaling Limit
• On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion