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Continuous Time p-Adic Random Walks and Their Path Integrals

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  • Erik Bakken

    (The Norwegian University of Science and Technology)

  • David Weisbart

    (University of California Riverside)

Abstract

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

Suggested Citation

  • Erik Bakken & David Weisbart, 2019. "Continuous Time p-Adic Random Walks and Their Path Integrals," Journal of Theoretical Probability, Springer, vol. 32(2), pages 781-805, June.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:2:d:10.1007_s10959-018-0831-3
    DOI: 10.1007/s10959-018-0831-3
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    References listed on IDEAS

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    1. Albeverio, Sergio & Karwowski, Witold, 1994. "A random walk on p-adics--the generator and its spectrum," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 1-22, September.
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