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| Hlavní autor: | |
|---|---|
| Médium: | Recurso digital |
| Jazyk: | angličtina |
| Vydáno: |
Zenodo
2026
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| Témata: | |
| On-line přístup: | https://doi.org/10.5281/zenodo.19074037 |
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- <p>We study the dynamical structure induced by exponential tilting of a positive measure μ on [0,∞) with finite second moment. The tilted family ν_t(dλ) ∝ e^{−λt} dμ(λ) satisfies the exact covariance identity<br><br><span>d/dt E_t[g] = −Cov_t(λ,g),</span><br><br>from which the variance dissipation law<br><br><span>r′(t) = −Var_t(λ) ≤ 0</span><br><br>follows, where r(t) = E_t[λ].<br><br>The family (ν_t) is a one-parameter exponential family whose Fisher information satisfies<br><br><span>I(t) = Var_t(λ) = −r′(t).</span><br><br>This identity provides a closed scalar law governing spectral selection dynamics.<br><br>We prove convergence r(t) → λ* = inf supp(μ) and classify rates: exponential under a spectral gap, and algebraic r(t) − λ* ~ β/t under regular variation μ([λ*,λ*+x]) ~ x^β L(x). We establish the global dissipation identity<br><br><span>∫₀^∞ Var_t(λ) dt = r(0) − λ*,</span><br><br>under a finite first-moment condition. For bi-atomic spectra, the dynamics closes exactly as a Riccati equation r′(t) = −(r−p)(q−r), saturating the variance bound.<br><br>To our knowledge, the explicit formulation r′(t) = −I(t) as a standalone identity governing spectral selection does not appear explicitly in the literature.<br><br>This framework provides a minimal and exact backbone for spectral flow, linking covariance dynamics, Fisher information, and asymptotic edge selection.</p>