Continuous Time p-Adic Random Walks and Their Path Integrals

The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in R to the groups p - n Z p / p n Z p give rise to probability measures on the space of Skorokhod paths on these finite groups. These measures induce probability...

Full description

Saved in:
Bibliographic Details
Published in:Journal of theoretical probability 2019-06, Vol.32 (2), p.781-805
Main Authors: Bakken, Erik, Weisbart, David
Format: Article
Language:English
Subjects:
Brownian motion
Differential equations
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Multiplication
Operators (mathematics)
Probability Theory and Stochastic Processes
Random walk
Random walk theory
Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in R to the groups p - n Z p / p n Z p give rise to probability measures on the space of Skorokhod paths on these finite groups. These measures induce probability measures on the Skorokhod space of Q p -valued paths that almost surely take values on finite grids. We study the convergence of these induced measures to their continuum limit, a p -adic Brownian motion. We additionally prove a Feynman–Kac formula for the matrix-valued propagator associated to a Schrödinger type operator acting on complex vector-valued functions on p - n Z p / p n Z p where the potential is a Hermitian matrix-valued multiplication operator.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-018-0831-3