On recurrence and transience of some Lévy-type processes in ℝ

IF 0.4 Q4 STATISTICS & PROBABILITY Theory of Probability and Mathematical Statistics Pub Date : 2023-05-02 DOI:10.1090/tpms/1187
V. Knopova
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For the proof the Foster–Lyapunov approach is used.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1187","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0

Abstract

In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in R \mathbb {R} , whose generator defined on the test functions is of the form L f ( x ) = R ( f ( x + u ) f ( x ) f ( x ) u 1 | u | 1 ) ν ( x , d u ) , f C 2 ( R ) . \begin{equation*} Lf(x) =\int _{\mathbb {R}} \left ( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\leq 1} \right ) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{equation*} Here ν ( x , d u ) \nu (x,du) is a Lévy-type kernel, whose tails are either extended regularly varying or decaying fast enough. For the proof the Foster–Lyapunov approach is used.

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若干l型过程的递归性和暂态性
在本文中,我们证明了R \mathbb R{中l型过程的暂态和递归的几个充分条件。其生成函数定义在测试函数上的形式为L f (x) =∫R (f (x + u) - f (x) -∇f (x)·u 1 | u |≤1)ν (x, du), f∈C∞2 (R)。}\begin{equation*} Lf(x) =\int _{\mathbb {R}} \left ( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb {1}_{|u|\leq 1} \right ) \nu (x,du), \quad f\in C_\infty ^2(\mathbb {R}). \end{equation*}这里的ν (x,du) \nu (x,du)是一个l型核,它的尾部要么有规律地扩展变化,要么衰减得足够快。为了证明,使用了Foster-Lyapunov方法。
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CiteScore
1.30
自引率
0.00%
发文量
22
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