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<p>This paper is devoted to the Dirichlet problem associated to operators defined on the nuclear algebra of entire functions. Firstly, we give a probabilistic representation of the solution of the Dirichlet problem associated to the extended <i>K</i>-Gross Laplacian in terms of <i>K</i>-Wiener process. Secondly, we prove that the Dirichlet problem associated to the operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_565_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {F}_{t(-K),I}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>I</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-transform has an explicit solution. Finally, an application to the large deviation principle is given.</p>

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The Dirichlet Problem Associated to Operators Defined on the Nuclear Algebra of Entire Functions

  • 期刊论文

  • 发表日期：

    2025年4月28日

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  • Sonia Chaari,
  • Afef Ben Farah

摘要

This paper is devoted to the Dirichlet problem associated to operators defined on the nuclear algebra of entire functions. Firstly, we give a probabilistic representation of the solution of the Dirichlet problem associated to the extended K-Gross Laplacian in terms of K-Wiener process. Secondly, we prove that the Dirichlet problem associated to the operator \(\mathcal {F}_{t(-K),I}\) F t ( - K ) , I -transform has an explicit solution. Finally, an application to the large deviation principle is given.