Journal of Theoretical Probability

, Volume 32, Issue 2, pp 781–805 | Cite as

Continuous Time p-Adic Random Walks and Their Path Integrals

  • Erik Bakken
  • David WeisbartEmail author


The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in \(\mathbb {R}\) to the groups \(p^{-n}\mathbb {Z}_p/p^{n} \mathbb {Z}_p\) 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 \(\mathbb {Q}_p\)-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 \(p^{-n}\mathbb {Z}_p/p^{n} \mathbb {Z}_p\) where the potential is a Hermitian matrix-valued multiplication operator.


p-adics Brownian motion Random walks Feynman–Kac 

Mathematics Subject Classification

60B10 60B11 60G50 60G51 47D08 47S10 46S10 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesThe Norwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of MathematicsUniversity of California RiversideRiversideUSA

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