Continuous time \(p\)-adic random walks and their path integrals. (English) Zbl 1447.60007
Summary: 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.
MSC:
60B10 | Convergence of probability measures |
60B11 | Probability theory on linear topological spaces |
60G50 | Sums of independent random variables; random walks |
60G51 | Processes with independent increments; Lévy processes |
47D08 | Schrödinger and Feynman-Kac semigroups |
47S10 | Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory |
46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |
References:
[1] | Albeverio, S., Karwowski, W.: A random walk on \[p\] p-adics—the generator and its spectrum. Stoch. Process. Appl. 53, l-22 (1994) · Zbl 0810.60065 · doi:10.1016/0304-4149(94)90054-X |
[2] | Avetisov, V.A., Bikulov, A.K., Al Osipov, \[V.: p\] p-adic description of characteristic relaxation in complex systems. J. Phys. A 36(15), 4239-4246 (2003) · Zbl 1049.82051 · doi:10.1088/0305-4470/36/15/301 |
[3] | Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, Hoboken (1999) · Zbl 0944.60003 · doi:10.1002/9780470316962 |
[4] | Chacón-Cortes, L.F., Zúñiga-Galindo, W.A.: Nonlocal operators, parabolic-type equations, and ultrametric random walks. J. Math. Phys. 54, 113503 (2013) · Zbl 1288.82027 · doi:10.1063/1.4828857 |
[5] | Digernes, T., Varadarajan, V.S., Varadhan, S.R.S.: Finite approximations to quantum systems. Rev. Math. Phys. 6(4), 621-648 (1994) · Zbl 0855.47046 · doi:10.1142/S0129055X94000213 |
[6] | Digernes, T., Varadarajan, V.S., Weisbart, D.: Schrödinger operators on local fields: self-adjointness and path integral representations for propagators. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 11(4), 495-512 (2008) · Zbl 1162.47038 · doi:10.1142/S0219025708003294 |
[7] | Varadarajan, V.S.: Path integrals for a class of \[p\] p-adic Schrödinger equations. Lett. Math. Phys. 39(2), 97-106 (1997) · Zbl 0868.47047 · doi:10.1023/A:1007364631796 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.