Continuous Time p-Adic Random Walks and Their Path Integrals

Erik Bakken () and David Weisbart ()
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Erik Bakken: The Norwegian University of Science and Technology
David Weisbart: University of California Riverside

Journal of Theoretical Probability, 2019, vol. 32, issue 2, 781-805

Abstract: 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.

Keywords: p-adics; Brownian motion; Random walks; Feynman–Kac; 60B10; 60B11; 60G50; 60G51; 47D08; 47S10; 46S10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0831-3

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