Sobolev space and Dirichlet space associated with symmetric Markov process on a local field
type: Item
instance of: scholarly article
title: Sobolev space and Dirichlet space associated with symmetric Markov process on a local field
zbMATH Open document ID: 1089.31005
published in: Potential Analysis
publication date: 2006-03-21T00:00:00Z
review text: The author is focused on a counterpart of the Sobolev space based on Albeverio, Karwowski and Yasuda's additive process on a finite extension \(K\) of \(Q_p\) characterized by a nonnegative nonincreasing sequence \(A=\{a(m)\}\). The counterpart \({\mathcal F}_{r,p}^{(A)}\) of the Sobolev space \(W^{r,p}(\mathbb{R}^n)\) is obtained as the image of the map \(V_p^{(A)}:L^p(K;\mu)\to L^p(K;\mu)\), where \(V_p^{(A)}\) denotes an operator associated with Albeverio, Karwowski and Yasuda's additive process determined by \(A=\{a(m)\}\) and \(\mu\) denotes the Haar measure on \(K\). In Section 2, a sufficient condition is given for the refinement \({\mathcal F}_{r,2}^{(A)}\) of the Dirichlet space to be isometric to the domain \({\mathcal F}^{(B)}\) of the Dirichlet form associated with another sequence \(B=\{b(m)\}\). This isometry enables to find a purely probabilistic characterization of \((r, 2)\)-polar set in the case with non-integer \(r\). In Section 3, a characterization of the kernel \(V_r^{(A)}\) by the method of Fourier transform, is given. As a consequence, there are many cases where the composite of two kernels \(V_r^{(A)}\) and \(V_r^{(B)}\) provides us with another kernel \(V_r^{(C)}\) corresponding to some sequence \(C=\{c(m)\}\).
reviewed by: Sophia L. Kalpazidou
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 5014108
zbMATH Keywords: nonlinear capacity
zbMATH Keywords: local field
zbMATH Keywords: Dirichlet form
zbMATH Keywords: Dirichlet space
author: Hiroshi Kaneko
MaRDI profile type: Publication
full work available at URL: s11118-005-8570-3
OpenAlex ID: W1963756764
cites work: Q4764531
cites work: A random walk on \(p\)-adics -- the generator and its spectrum
cites work: Real time random walks on \(p\)-adic numbers
cites work: Trace formula for \(p\)-adics
cites work: Asymptotics and spectral results for random walks on \(p\)-adics
cites work: Q4512961
cites work: Measure-valued branching processes associated with random walks on \(p\)-adics.
cites work: A decomposition theorem for Lévy processes on local fields
cites work: Dirichlet forms on totally disconnected spaces and bipartite Markov chains
cites work: Stable laws arising from hitting distributions of processes on homogeneous trees and the hyperbolic half-plane.
cites work: Q3726220
cites work: Local properties of Lévy processes on a totally disconnected group
cites work: Q3692580
cites work: Dirichlet forms and symmetric Markov processes
cites work: Markov properties of multiparameter processes and capacities
cites work: Pseudo differential operators and Markov processes. In 3 vol. Vol. 1: Fourier analysis and semigroups
cites work: Q3149351
cites work: On (r,p)-capacities for Markov processes
cites work: A class of spatially inhomogeneous Dirichlet spaces on the \(p\)-adic number field.
cites work: Recurrence and transience criteria for symmetric Hunt processes
cites work: Time-inhomogeneous stochastic processes on the \(p\)-adic number field
cites work: Capacities associated with Dirichlet space on an infinite extension of a local field
cites work: Stochastic processes on ℚp induced by maps and recurrence criteria
cites work: Hierarchical structures and asymmetric stochastic processes on p-adics and adeles
cites work: Q4841577
cites work: Q4138813
cites work: Stochastic integrals and stochastic differential equations over the field of \(p\)-adic numbers
cites work: Q3887567
cites work: Random walk on \(p\)-adics with non-zero killing part
cites work: Generalized functions over the field ofp-adic numbers
cites work: Q4003475
cites work: Q4337519
cites work: On infiniteley divisible distributions on locally compact Abelian groups
cites work: Extension of measures to infinite dimensional spaces over \(p\)-adic field.
DOI: 10.1007/S11118-005-8570-3
description: scientific article; zbMATH DE number 5014108