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<p>A family of log-correlated Gaussian processes indexed by metric spaces is introduced, when the metric is conditionally negative definite. These processes arise as the limit of bi-fractional Brownian motions indexed by (<i>H</i>,&#xa0;<i>K</i>) scaled by <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1449_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(K^{-1/2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1449_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(K\downarrow 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1449_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\in (0,1/2]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> fixed. When the metric is in addition a measure definite kernel, stochastic-integral representations of the generalized processes when evaluated at a test function are provided. The introduced processes are also shown to be the scaling limits of certain aggregated models.</p>

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A Family of Log-Correlated Gaussian Processes

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  • 发表日期：

    2025年9月23日

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  • Yizao Wang

摘要

A family of log-correlated Gaussian processes indexed by metric spaces is introduced, when the metric is conditionally negative definite. These processes arise as the limit of bi-fractional Brownian motions indexed by (H, K) scaled by \(K^{-1/2}\) K - 1 / 2 as \(K\downarrow 0\) K ↓ 0 with \(H\in (0,1/2]\) H ∈ ( 0 , 1 / 2 ] fixed. When the metric is in addition a measure definite kernel, stochastic-integral representations of the generalized processes when evaluated at a test function are provided. The introduced processes are also shown to be the scaling limits of certain aggregated models.