Abstract <p>The paper investigates a model of the motion of a Brownian particle on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11972_2025_8752_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}^{2}\)</EquationSource>  </InlineEquation>. A six-dimensional random process is considered, whose states are described by the coordinates of the particle and the complex analog of the area swept out by the radius vector of the point as it moves along the trajectories of a four-dimensional Brownian bridge. Using the methods of integration over the conditional Wiener measure, an expression for the transition probability density of a random process from an arbitrary state to a subsequent one is obtained. A connection is revealed between the trajectories of the process under consideration and the Heisenberg group constructed over the field of complex numbers <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11972_2025_8752_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{3}(\mathbb{C})\)</EquationSource>  </InlineEquation>. The continuity in time and Hölder property with order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11972_2025_8752_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha<1/2\)</EquationSource>  </InlineEquation> of the oriented area function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11972_2025_8752_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(S(t)\)</EquationSource>  </InlineEquation> are proven, as well as the Heisenberg Markov property of the process. The heat equation describing the system’s evolution and the corresponding sub-Laplacian are derived. The solution to the equation is obtained in the form of a functional integral. Performing a Wick rotation allows for drawing an analogy between the considered process and the motion of an electron in a magnetic field.</p>

Markov Random Process on the Heisenberg Group

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发表日期：

2025年2月1日

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I. A. Bogatyrev,
M. B. Vavilov,
I. A. Sukharev,
E. T. Shavgulidze

摘要

Abstract

The paper investigates a model of the motion of a Brownian particle on \(\mathbb{C}^{2}\) . A six-dimensional random process is considered, whose states are described by the coordinates of the particle and the complex analog of the area swept out by the radius vector of the point as it moves along the trajectories of a four-dimensional Brownian bridge. Using the methods of integration over the conditional Wiener measure, an expression for the transition probability density of a random process from an arbitrary state to a subsequent one is obtained. A connection is revealed between the trajectories of the process under consideration and the Heisenberg group constructed over the field of complex numbers \(H_{3}(\mathbb{C})\) . The continuity in time and Hölder property with order \(\alpha<1/2\) of the oriented area function \(S(t)\) are proven, as well as the Heisenberg Markov property of the process. The heat equation describing the system’s evolution and the corresponding sub-Laplacian are derived. The solution to the equation is obtained in the form of a functional integral. Performing a Wick rotation allows for drawing an analogy between the considered process and the motion of an electron in a magnetic field.