Pseudodifferential operators and Markov processes on certain totally disconnected groups

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Abstract

This article describes a class of invariant Markov processes on certain totally disconnected groups. An invariant pseudodifferential operator on these groups, similar to the Vladimirov operator on the p-adic line, allows us to state an L2-abstract Cauchy problem for a homogeneous heat-type pseudodifferential equation. The fundamental solutions of these parabolic-type pseudodifferential equations give transition functions of time and space homogeneous Markov processes on these groups. Particularly interesting examples are polyadic rings, such as the ring of m-adic numbers, and the ring of finite adèles of the rational numbers.

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 p-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 m-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 G be a second countable, totally disconnected, locally compact, Abelian topological group with a collection {Hn}nZ of compact and open subgroups such that: they configure a filtration, i.e. {0}HnH0HnG,n1,and the following relations are satisfied: nZHn={0}, and nZHn=G.

Let μ be the Haar measure on G such that μ(H0)=1. The group G has a unique G-invariant ultrametric dG such that the collection of all balls centered at zero coincides with the filtration {Hn}nZ, and such that the radius of any ball is equal to its Haar measure.

Let Ĝ be the Pontryagin dual group of G and let KnĜ be the annihilator of Hn. The collection {Kn}nZ, of compact and open subgroups of Ĝ, is a filtration which satisfies the analogue of properties (1) in Ĝ. Let μˆ be the Haar measure on Ĝ such that μˆ(K0)=1. The group Ĝ has a unique Ĝ-invariant ultrametric dĜ such that the collection of all balls centered at zero coincide with the filtration {Kn}nZ, and such that the radius of any ball equals to its Haar measure. For our purposes denote |ξ|Ĝ=dĜ(0,ξ), for any ξĜ.

Let χ:Ĝ×G be a pairing function. The Fourier transform, F(f)(ξ)=Gf(x)χ(ξ,x)dμ(x), gives a topological and linear isomorphism between D(G) and D(Ĝ), the locally convex topological linear spaces given by the locally constant functions on G and Ĝ, respectively. Further, the extension of the Fourier transform F:L2(G)L2(Ĝ), is an isometry of Hilbert spaces.

For any α>0, define the pseudodifferential operator of fractional differentiation Dα:Dom(Dα)L2(G)L2(G) by the formula Dα(f)=Fξx1[|ξ|ĜαFxξ[f]],(fDom(Dα)).The operator Dα is a positive selfadjoint unbounded operator which is invariant under translations of G. The spectrum of Dα is essential and it consists of the countable number of eigenvalues which converge to zero, and zero.

The heat kernel, Zα(x,t)=Fξx1(exp(t|ξ|Ĝα)),is a well defined non-negative function and the distribution of a probability measure on G, for all t>0.

The following result encompasses our achievements.

Main Theorem

If f belongs to Dom(Dα)L2(G), the Cauchy problem for the heat-type pseudodifferential equation u(x,t)t+Dαu(x,t)=0,xG,t0,u(x,0)=f(x),has a classical solution u(x,t) which is determined by the convolution of f with the heat kernel Zα(x,t). In addition, Zα(x,t) is the transition density of a time and space homogeneous Markov process Wα(t) on G, 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 Dα defined on L2(G) 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 G

This section introduces a class of totally disconnected groups and its relevant algebraic and topological properties. An ultrametric on G and its Pontryagin dual group Ĝ, 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 G

This section introduces, for any α>0, a positive selfadjoint pseudodifferential unbounded operator Dα on L2(G) related to the ultrametric dG in G, and it presents the analysis of the abstract Cauchy problem for a homogeneous heat-type pseudodifferential equation on L2(G) related to Dα. The properties of general evolution equations on Banach spaces can be found in Engel and Nagel, 2000, Pazy, 1983.

A Markov process on G

In this section the fundamental solution of the heat equation Zα(x,t) will be interpreted as the transition density function of a Markov process on G. This Markov process is an analogue on G of the classical Brownian motion. The general theory of Markov processes can be found in the classical work (Dynkin, 2006).

Let B denote the Borel σ-algebra of G and for any BB write 1B for the characteristic function of B. Define pα(t,x,y)Zα(xy,t), (t>0,x,yG), and Pα(t,x,B)=Bpα(t,x,y)dμ(y),ift>0,xG,BB

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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