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| 第一著者: | |
|---|---|
| フォーマット: | Recurso digital |
| 言語: | 英語 |
| 出版事項: |
Zenodo
2026
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| 主題: | |
| オンライン･アクセス: | https://doi.org/10.5281/zenodo.19021390 |
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目次:
- <p>We study the dynamical structure induced by Laplace mixtures of exponential modes,</p> <p><span><span><span>F(t)=∫e−λt dμ(λ),</span><span><span><span>F</span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span>∫</span><span><span>e</span><span><span><span><span><span><span>−<span>λ</span><span>t</span></span></span></span></span></span></span></span><span>d</span><span>μ</span><span>(</span><span>λ</span><span>)</span><span>,</span></span></span></span></span></p> <p>where <span><span>μ</span><span><span><span>μ</span></span></span></span> is a positive measure with finite first moment. Introducing the normalized tilted spectral measure</p> <p><span><span><span>νt(dλ)=e−λt dμ(λ)F(t),</span><span><span><span><span>ν</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>(</span><span>d</span><span>λ</span><span>)</span><span>=</span></span><span><span><span><span><span><span><span>F</span><span>(</span><span>t</span><span>)</span><span>e</span><span><span><span><span>−<span>λ</span><span>t</span></span></span></span></span><span>d</span><span>μ</span><span>(</span><span>λ</span><span>)</span></span><span></span></span></span></span></span><span>,</span></span></span></span></span></p> <p>we show that observables of the spectral variable satisfy the covariance law</p> <p><span><span><span>ddt Et[g(λ)]=−Covt(λ, g(λ)).</span><span><span><span><span><span><span><span><span>d</span><span>t</span><span>d</span></span><span></span></span></span></span></span><span><span>E</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>[</span><span>g</span><span>(</span><span>λ</span><span>)]</span><span>=</span></span><span><span>−</span><span><span>Cov</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>(</span><span>λ</span><span>,</span><span>g</span><span>(</span><span>λ</span><span>))</span><span>.</span></span></span></span></span></p> <p>In particular, the effective rate <span><span>r(t)=−F′(t)/F(t)</span><span><span><span>r</span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span>−</span><span><span>F</span><span><span><span><span><span><span>′</span></span></span></span></span></span></span><span>(</span><span>t</span><span>)</span><span>/</span><span>F</span><span>(</span><span>t</span><span>)</span></span></span></span> obeys the variance flow identity</p> <p><span><span><span>r′(t)=−Vart(λ)≤0,</span><span><span><span><span>r</span><span><span><span><span><span><span>′</span></span></span></span></span></span></span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span>−</span><span><span>Var</span><span><span><span><span><span><span>t</span></span></span><span></span></span></span></span></span><span>(</span><span>λ</span><span>)</span><span>≤</span></span><span><span>0</span><span>,</span></span></span></span></span></p> <p>revealing a dissipative dynamics on the spectral distribution. The special case of two exponential modes yields an exact autonomous Riccati equation <span><span>r′(t)=−(r−p)(q−r)</span><span><span><span><span>r</span><span><span><span><span><span><span>′</span></span></span></span></span></span></span><span>(</span><span>t</span><span>)</span><span>=</span></span><span><span>−</span><span>(</span><span>r</span><span>−</span></span><span><span>p</span><span>)</span><span>(</span><span>q</span><span>−</span></span><span><span>r</span><span>)</span></span></span></span>, which is the unique quadratic closure of the moment hierarchy. These results show that Laplace mixtures naturally carry a covariance-driven spectral dynamics, linking Laplace transform theory with dissipative flows on probability measures.</p>