Spectral Equality for Novikov Integrability: Recursive Criticality and Unbounded Asymptotic Depth

Abstract

We determine the finiteness boundary of the Novikov exponential moment for a one-dimensional constant-volatility mean-reverting diffusion. A localised change of measure cancels the squared drift and leaves a Brownian Feynman-Kac functional with potential $q/2$, where $q=-{\lambda }^{\prime }$. If $q(\theta +y)\sim {\kappa }_{\infty }{y}^2$, the exact spectral boundary is ${\kappa }_{\infty }{\sigma }^2{T}^2={\pi }^2$; for cubic drift it becomes $c{\sigma }^2{T}^2={\pi }^2/3$, and equality is divergent. On the spectral equality surface, the first lower-order transition occurs at power $4/3$, where an explicit coefficient separates the two sides. For symmetric finite pure-power tails, exact tuning generates the recursion ${\beta }_n=1+3^{-(n+1)}$. We prove that this recursion gives a complete classification of the class. Each individual tail is decided after finitely many comparisons, but the required depth is unbounded: arbitrarily long common critical prefixes can lead to opposite outcomes. The drift-removing stochastic exponential nevertheless remains a true martingale; under the physical law it has no higher moments on the steep-drift class, while the reverse density is essentially bounded.

48 pages, 3 figures. Ancillary numerical Python script included