{"title":"Resolvent stochastic processes","authors":"I. Ibragimov, N. Smorodina, M. Faddeev","doi":"10.1090/spmj/1797","DOIUrl":null,"url":null,"abstract":"<p>A family <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r Subscript lamda\"> <mml:semantics> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">r_\\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda element-of double-struck upper C\"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lambda \\in \\mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of complex stochastic processes is introduced, which makes it possible to construct a probabilistic representation for the resolvent of the operator <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"minus one half StartFraction d squared Over d x squared EndFraction\"> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mfrac> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">-\\frac {1}{2}\\frac {d^2}{dx^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lambda =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the process <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r Subscript lamda\"> <mml:semantics> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">r_\\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the Brownian local time process.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"45 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1797","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A family rλr_\lambda, λ∈C\lambda \in \mathbb {C}, of complex stochastic processes is introduced, which makes it possible to construct a probabilistic representation for the resolvent of the operator −12d2dx2-\frac {1}{2}\frac {d^2}{dx^2}. For λ=0\lambda =0, the process rλr_\lambda coincides with the Brownian local time process.
期刊介绍：
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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