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{"title":"若干l<s:1>型过程的递归性和暂态性","authors":"V. Knopova","doi":"10.1090/tpms/1187","DOIUrl":null,"url":null,"abstract":"<p>In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, whose generator defined on the test functions is of the form <disp-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Endscripts left-parenthesis f left-parenthesis x plus u right-parenthesis minus f left-parenthesis x right-parenthesis minus nabla f left-parenthesis x right-parenthesis dot u double-struck 1 Subscript StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 Baseline right-parenthesis nu left-parenthesis x comma d u right-parenthesis comma f element-of upper C Subscript normal infinity Superscript 2 Baseline left-parenthesis double-struck upper R right-parenthesis period\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:msub>\n <mml:mo>∫<!-- ∫ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>+</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi mathvariant=\"normal\">∇<!-- ∇ --></mml:mi>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\n <mml:mi>u</mml:mi>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn mathvariant=\"double-struck\">1</mml:mn>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mi>u</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mspace width=\"1em\" />\n <mml:mi>f</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:msubsup>\n <mml:mi>C</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>.</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\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*}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</disp-formula>\n Here <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu left-parenthesis x comma d u right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ν<!-- ν --></mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>d</mml:mi>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\nu (x,du)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> 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.</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":"{\"title\":\"On recurrence and transience of some Lévy-type processes in ℝ\",\"authors\":\"V. Knopova\",\"doi\":\"10.1090/tpms/1187\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note we prove some sufficient conditions for transience and recurrence of a Lévy-type process in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, whose generator defined on the test functions is of the form <disp-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L f left-parenthesis x right-parenthesis equals integral Underscript double-struck upper R Endscripts left-parenthesis f left-parenthesis x plus u right-parenthesis minus f left-parenthesis x right-parenthesis minus nabla f left-parenthesis x right-parenthesis dot u double-struck 1 Subscript StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to 1 Baseline right-parenthesis nu left-parenthesis x comma d u right-parenthesis comma f element-of upper C Subscript normal infinity Superscript 2 Baseline left-parenthesis double-struck upper R right-parenthesis period\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>L</mml:mi>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:msub>\\n <mml:mo>∫<!-- ∫ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mrow>\\n <mml:mo>(</mml:mo>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>+</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∇<!-- ∇ --></mml:mi>\\n <mml:mi>f</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mn mathvariant=\\\"double-struck\\\">1</mml:mn>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mi>u</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mo>≤<!-- ≤ --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:mo>)</mml:mo>\\n </mml:mrow>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>,</mml:mo>\\n <mml:mspace width=\\\"1em\\\" />\\n <mml:mi>f</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:msubsup>\\n <mml:mi>C</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>.</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\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*}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</disp-formula>\\n Here <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"nu left-parenthesis x comma d u right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ν<!-- ν --></mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>d</mml:mi>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\nu (x,du)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> 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.</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}","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}
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