Spectral Mean Flow of Laplace Mixtures: Variance Dissipation, Moment Cascade, and Riccati Closure
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2026
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| author | Morissette, Louis |
| author_facet | Morissette, Louis |
| contents | <p> </p> <p>We study the dynamical structure induced by Laplace mixtures of exponential modes, F(t) = ∫ e^{−λt} dμ(λ), where μ is a positive measure with finite first moment. Introducing the normalized tilted spectral measure ν_t(dλ) = e^{−λt} dμ(λ) / F(t), we show that observables of the spectral variable satisfy the covariance law d/dt E_t[g(λ)] = −Cov_t(λ, g(λ)). In particular, the effective rate r(t) = −F′(t)/F(t) obeys the variance flow identity r′(t) = −Var_t(λ) ≤ 0, revealing a dissipative dynamics on the spectral distribution. This implies monotone decrease of the effective rate and asymptotic dominance of the slowest spectral mode p = inf supp(μ).</p> <p>The special case of two exponential modes, F(t) = Ae^{−pt} + Be^{−qt}, yields an exact autonomous Riccati equation r′(t) = −(r−p)(q−r), the quadratic closure associated with a bi-atomic spectral measure. We prove that this closure characterizes bi-atomic spectral measures: quadratic variance closure forces μ = Aδ_p + Bδ_q.</p> <p>The covariance law generates an infinite hierarchy of centered moment dynamics, m′<em>k(t) = −m</em>{k+1}(t), in which each level drives the next. This moment cascade does not close unless the spectral measure has finite support; in the bi-atomic case it closes at order two, recovering the Riccati equation.</p> <p>These results show that Laplace mixtures naturally carry a covariance-driven spectral dynamics, linking Laplace transform theory with dissipative flows on probability measures.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19022141 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Mean Flow of Laplace Mixtures: Variance Dissipation, Moment Cascade, and Riccati Closure Morissette, Louis Laplace mixtures replicator equation hazard rate dynamics Riccati equation variance dissipation spectral dynamics covariance dynamics completely monotone functions <p> </p> <p>We study the dynamical structure induced by Laplace mixtures of exponential modes, F(t) = ∫ e^{−λt} dμ(λ), where μ is a positive measure with finite first moment. Introducing the normalized tilted spectral measure ν_t(dλ) = e^{−λt} dμ(λ) / F(t), we show that observables of the spectral variable satisfy the covariance law d/dt E_t[g(λ)] = −Cov_t(λ, g(λ)). In particular, the effective rate r(t) = −F′(t)/F(t) obeys the variance flow identity r′(t) = −Var_t(λ) ≤ 0, revealing a dissipative dynamics on the spectral distribution. This implies monotone decrease of the effective rate and asymptotic dominance of the slowest spectral mode p = inf supp(μ).</p> <p>The special case of two exponential modes, F(t) = Ae^{−pt} + Be^{−qt}, yields an exact autonomous Riccati equation r′(t) = −(r−p)(q−r), the quadratic closure associated with a bi-atomic spectral measure. We prove that this closure characterizes bi-atomic spectral measures: quadratic variance closure forces μ = Aδ_p + Bδ_q.</p> <p>The covariance law generates an infinite hierarchy of centered moment dynamics, m′<em>k(t) = −m</em>{k+1}(t), in which each level drives the next. This moment cascade does not close unless the spectral measure has finite support; in the bi-atomic case it closes at order two, recovering the Riccati equation.</p> <p>These results show that Laplace mixtures naturally carry a covariance-driven spectral dynamics, linking Laplace transform theory with dissipative flows on probability measures.</p> |
| title | Spectral Mean Flow of Laplace Mixtures: Variance Dissipation, Moment Cascade, and Riccati Closure |
| topic | Laplace mixtures replicator equation hazard rate dynamics Riccati equation variance dissipation spectral dynamics covariance dynamics completely monotone functions |
| url | https://doi.org/10.5281/zenodo.19022141 |