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<p>In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in Park and Xiao (Math. Nachr. <b>296</b>(9), 4192–4205, 2023), and the present work serves as a natural extension of Park and Xiao (Math. Nachr. <b>296</b>(9), 4192–4205, 2023) for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha =d-\mathfrak {b}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>d</mi> <mo>-</mo> <mi mathvariant="fraktur">b</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak {b}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">b</mi> </math></EquationSource> </InlineEquation> is the interior Minkowski dimension of the boundary of the underlying open set.</p>

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Small-time Heat Decay for Stable Processes on Fractal Drums

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Hyunchul Park,
  • Yimin Xiao

摘要

In this paper, we study the spectral heat content for isotropic stable processes on fractal drums (namely, open sets with fractal boundaries). The spectral heat content for subordinate killed Brownian motions by stable subordinators was investigated in Park and Xiao (Math. Nachr. 296(9), 4192–4205, 2023), and the present work serves as a natural extension of Park and Xiao (Math. Nachr. 296(9), 4192–4205, 2023) for the spectral heat content for stable processes. Under suitable geometric conditions on the underlying domains, we show that the decay rate of the spectral heat content for stable processes differs substantially from that for subordinate killed Brownian motions when \(\alpha =d-\mathfrak {b}\) α = d - b , where \(\mathfrak {b}\) b is the interior Minkowski dimension of the boundary of the underlying open set.