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Brownian motion and finite approximations of quantum systems over local fields

Erik Makino Bakken

Department of Mathematical Sciences, The Norwegian University of Science and Techmology (NTNU), 7491 Trondheim, Norway

E-mail Address: Erik-Makino.Bakken@ffi.no

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Trond Digernes

Department of Mathematical Sciences, The Norwegian University of Science and Techmology (NTNU), 7491 Trondheim, Norway

E-mail Address: digernes@math.ntnu.no

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David Weisbart

Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA

E-mail Address: weisbart@math.ucr.edu

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https://doi.org/10.1142/S0129055X17500167Cited by:7 (Source: Crossref)

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.

Keywords:

AMSC: 81Q65, 60B10, 47G30, 41A99

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