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<p>Suppose that we have a canonical Gibbs measure <i>μ</i> defined on a marked configuration space Ω that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on Ω<i>.</i> Then <i>μ</i> is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to <i>μ</i>-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version <i>β</i> of this cocycle. Further, we characterize all the measures on Ω that possess <i>β</i> as their cocycle. If <i>μ</i> is obtained (e.g.) from a particular two-body potential <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11253_2025_2479_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\widehat{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> (satisfying some mild regularity assumptions), then <i>β</i> takes a certain explicit form, and the class of canonical Gibbs measures characterized by <i>β</i> contains exactly the measures associated with the potential <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11253_2025_2479_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\widehat{V}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>V</mi> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation><i>.</i> Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general <i>G</i>-spaces for local infinite-dimensional groups.</p>

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Characterizing Measures According to Their Radon–Nikodym Cocycles: Canonical Marked Gibbs Measures Under the Action of the Diffeomorphism Group

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Tobias Kuna,
  • Gerald A. Goldin,
  • Yuri G. Kondratiev,
  • José L. Silva

摘要

Suppose that we have a canonical Gibbs measure μ defined on a marked configuration space Ω that describes a system of infinitely many indistinguishable particles with internal degrees of freedom together with a diffeomorphism group action on Ω. Then μ is quasiinvariant under the group action, and we obtain a class of associated cocycles from its Radon–Nikodym derivatives. The cocycles are defined up to μ-measure zero sets. We show that it is possible to choose a suitable pointwise-defined version β of this cocycle. Further, we characterize all the measures on Ω that possess β as their cocycle. If μ is obtained (e.g.) from a particular two-body potential \(\widehat{V}\) V ^ (satisfying some mild regularity assumptions), then β takes a certain explicit form, and the class of canonical Gibbs measures characterized by β contains exactly the measures associated with the potential \(\widehat{V}\) V ^ . Our result is based on the inheritance properties for the characterization by cocycles of Radon–Nikodym derivatives, which are proved for general G-spaces for local infinite-dimensional groups.