Regularity of semigroups generated by Lévy type operators via coupling
type: Item
instance of: scholarly article
title: Regularity of semigroups generated by Lévy type operators via coupling
zbMATH Open document ID: 1204.60071
author: Jian Wang
published in: Stochastic Processes and their Applications
publication date: 2010-08-18T00:00:00Z
review text: Let \(X_t\) be the \(d\)-dimensional Markov process generated by the Lévy type operator \(Lf(x)=\frac{1}{2}\sum_{i,j}a_{ij}(x)\partial_{ij}f(x)+\sum_ib_i(x)\partial_if(x)+\int(f(x+z)-f(x)-\nabla f(x)\cdot z1_{\{|z|\leq 1\}})\nu(x,dz),\) where \((a,b,\nu)\) are Lévy characteristics. Let \(P_t\) denote the semigroup of operators generated by \(L.\) The process \(X_t\) has \(C_b\)-Feller continuous property if for all \(t>0,\) \(P_t:C_b(\mathbb R^d)\to C_b(\mathbb R^d);\) \(X_t\) has the strong Feller continuous property if \(P_t:C_b(\mathbb R^d)\to C_b(\mathbb R^d);\) finally, \(X_t\) has Lipschitz continuous property if for all \(t>0,\) \(P_t:\mathrm{Lip}_b(\mathbb R^d)\to\mathrm{Lip}_b(\mathbb R^d),\) where \(\mathrm{Lip}_b\) stands for Lipschitz bounded. The main results of the paper give the verifiable sufficient conditions which guarantee that \(X_t\) has the above mention properties. As a corollary the criterion for the equi-continuity of \(P_t\) on \(\mathrm{Lip}_b(\mathbb R^d)\) (the so-called \(e\)-property) is also given. It is shown that the genuine Lévy processes as well as the Ornstein-Uhlenbeck type processes have \(e\)-property.
reviewed by: Roman Urban
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5772501
zbMATH Keywords: Lévy type operators
zbMATH Keywords: \(c_b(\mathbb R^d)\)-continuity
zbMATH Keywords: Lipschitz continuity
zbMATH Keywords: strong Feller continuity
zbMATH Keywords: coupling
zbMATH Keywords: Feller process
zbMATH Keywords: \(e\)-property
MaRDI profile type: Publication
full work available at URL: j.spa.2010.04.007
OpenAlex ID: W2072484172
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DOI: 10.1016/J.SPA.2010.04.007
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description: scientific article; zbMATH DE number 5772501