Master Structural Theorem for Exponential Persistence (SOEP)

A structural formulation of exponential persistence in metastable stochastic dynamical systems is developed. For a broad admissible class of dissipative Markov processes with light-tailed noise, geometric ergodicity and minorization yield a positive spectral gap and a simple principal Dirichlet eigenvalue governing survival asymptotics. Small-noise scaling is derived via quasipotential large deviation theory, with sharp asymptotics in gradient systems and extensions to non-gradient settings. A renormalization framework for survival kernels is introduced, and structural openness and genericity of the exponential sector in generator space are analyzed. The results provide a unified structural perspective on persistence in finite- and infinite-dimensional stochastic systems. Related resourcesAdditional preprints, theoretical frameworks, and ongoing work by the author are available at:https://murad-ahmadov.github.io/