Pseudodifferential operators and Markov processes on certain totally disconnected groups
Introduction
The theory of pseudodifferential operators and Markov stochastic processes on second countable, totally disconnected, locally compact Abelian topological groups has been a classical subject in the general theory of probability (see, e.g., Applebaum, 2006, Evans, 1989, Parthasarathy, 1967, Saloff-Coste, 1986, Urban, 2012, Yasuda, 2013 ). In recent decades this research has been marvelously motivated by -adic mathematical physics and more general ultrametric modeling of hierarchical structures (see Aguilar-Arteaga and Estala-Arias, 2019, Aguilar-Arteaga et al., 2020, Albeverio et al., 2010, Cruz-López and Estala-Arias, 2016, Del Muto and Figà-Talamanca, 2004, Dolgopolov and Zubarev, 2011, Dragovich et al., 2017, Karwowski and Vilela Mendes, 1994, Khrennikov and Radyno, 2003, Kochubei, 2001, Vladimirov et al., 1994, Zúñiga–Galindo, 2016 and the references therein).
This essay deals with an illustration of time and space homogeneous Markov processes on certain second countable, totally disconnected, locally compact Abelian topological groups. Noteworthy examples of the locally compact Abelian topological groups considered here are polyadic rings like the ring of -adic numbers and the ring of finite adèles of the rational numbers (Dolgopolov and Zubarev, 2011, Aguilar-Arteaga and Estala-Arias, 2019). This last example has been the motivation of this work and it will be portrayed in Section 4.1.
Let be a second countable, totally disconnected, locally compact, Abelian topological group with a collection of compact and open subgroups such that: they configure a filtration, i.e. and the following relations are satisfied:
Let be the Haar measure on such that . The group has a unique -invariant ultrametric such that the collection of all balls centered at zero coincides with the filtration , and such that the radius of any ball is equal to its Haar measure.
Let be the Pontryagin dual group of and let be the annihilator of . The collection , of compact and open subgroups of , is a filtration which satisfies the analogue of properties (1) in . Let be the Haar measure on such that . The group has a unique -invariant ultrametric such that the collection of all balls centered at zero coincide with the filtration , and such that the radius of any ball equals to its Haar measure. For our purposes denote , for any .
Let be a pairing function. The Fourier transform, , gives a topological and linear isomorphism between and , the locally convex topological linear spaces given by the locally constant functions on and , respectively. Further, the extension of the Fourier transform , is an isometry of Hilbert spaces.
For any , define the pseudodifferential operator of fractional differentiation by the formula The operator is a positive selfadjoint unbounded operator which is invariant under translations of . The spectrum of is essential and it consists of the countable number of eigenvalues which converge to zero, and zero.
The heat kernel, is a well defined non-negative function and the distribution of a probability measure on , for all .
The following result encompasses our achievements.
Main Theorem If belongs to , the Cauchy problem for the heat-type pseudodifferential equation has a classical solution which is determined by the convolution of with the heat kernel . In addition, is the transition density of a time and space homogeneous Markov process on , which is bounded, right-continuous and has no discontinuities other than jumps.
This exposition is organized as follows. Section 2 introduces the class of groups relevant to this work and presents preliminary results of non-Archimedean and Harmonic analysis on these groups. In Section 3 the pseudodifferential operator defined on is introduced, and also the corresponding Cauchy problem related to the homogeneous heat-type pseudodifferential equation is treated. In Section 4 the Markov process associated to the fundamental solution of the parabolic-type pseudodifferential equation is presented. We also include some examples of the totally disconnected groups considered here.
Section snippets
Ultrametric and harmonic analysis of
This section introduces a class of totally disconnected groups and its relevant algebraic and topological properties. An ultrametric on and its Pontryagin dual group , invariant under translations, are introduced. For complete information on the theory of locally compact Abelian topological groups the reader can consult the classical work (Hewitt and Ross, 1970).
A parabolic-type equation on
This section introduces, for any , a positive selfadjoint pseudodifferential unbounded operator on related to the ultrametric in , and it presents the analysis of the abstract Cauchy problem for a homogeneous heat-type pseudodifferential equation on related to . The properties of general evolution equations on Banach spaces can be found in Engel and Nagel, 2000, Pazy, 1983.
A Markov process on
In this section the fundamental solution of the heat equation will be interpreted as the transition density function of a Markov process on . This Markov process is an analogue on of the classical Brownian motion. The general theory of Markov processes can be found in the classical work (Dynkin, 2006).
Let denote the Borel -algebra of and for any write for the characteristic function of . Define , (), and
CRediT authorship contribution statement
Samuel Estala-Arias: Conceptualization, Investigation, Writing.
Acknowledgments
The author would like to thank Profs. Manuel Cruz-López and Victor A. Aguilar-Arteaga for their kind support and very useful discussions.
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