Spectral Selection under Exponential Tilting: Covariance Dynamics and Edge Asymptotics
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| Sprache: | Englisch |
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2026
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| _version_ | 1866902337761574912 |
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| author | Morissette, Louis |
| author_facet | Morissette, Louis |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19074037 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Selection under Exponential Tilting: Covariance Dynamics and Edge Asymptotics Morissette, Louis Laplace mixtures replicator equation hazard rate dynamics Riccati equation variance dissipation spectral dynamics covariance dynamics completely monotone functions replicator dynamics Price equation moment cascade <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> |
| title | Spectral Selection under Exponential Tilting: Covariance Dynamics and Edge Asymptotics |
| topic | Laplace mixtures replicator equation hazard rate dynamics Riccati equation variance dissipation spectral dynamics covariance dynamics completely monotone functions replicator dynamics Price equation moment cascade |
| url | https://doi.org/10.5281/zenodo.19074037 |