Structural Origins of Exponential Persistence (SOEP) is a multi-part theoretical program deriving exponential persistence laws from minimal structural assumptions governing open metastable systems.
Rather than treating exponential survival as an empirical regularity, SOEP establishes it as a structural consequence of dissipativity, spectral reduction, constraint geometry, and large-deviation scaling. The program develops a layered derivation connecting generator structure, quasi-compact semigroup theory, principal Dirichlet eigenvalues, and renormalization stability.
Establishes minimal admissible dynamical conditions under which exponential persistence emerges. Introduces constraint-class closure arguments, large-deviation barrier amplification, and structural separation of persistence regimes.
Develops the killed semigroup and Dirichlet generator framework. Proves quasi-compactness and existence of a finite slow spectral cluster, yielding dimensional reduction of long-time dynamics.
Derives resolvent representations of survival distributions. Establishes pole-dominance via contour deformation and spectral gap arguments, yielding sharp exponential asymptotics.
Introduces a renormalization operator on persistence kernels. Proves local contraction toward exponential fixed points and classifies structural universality sectors.
Consolidates the program into a unified structural theorem. Within a broad admissible class of dissipative Markov systems with light-tailed noise, the following hold:
The SOEP program applies to:
Heavy-tailed Lévy systems, long-memory non-Markov systems, and conservative Hamiltonian dynamics are treated as distinct persistence universality classes.
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Active multi-stage theoretical research program. Further formal refinements and structural extensions in progress.