Stochastic Persistence in Infinite Dimensions
Juraj Foldes, Declan Stacy
TL;DR
This work extends stochastic persistence theory from finite to infinite dimensions by introducing an average Lyapunov-exponent framework that applies to SPDEs and stochastic functional differential equations. It develops a robust methodology that blends mild (convolution) and variational (Lyapunov) techniques, along with a projective-coordinate reduction for nonnegative SPDEs, to prove persistence via tightness of occupation measures and convergence of averaged drifts. The authors provide general theorems showing persistence whenever the average Lyapunov exponent is negative, derive a perturbation result for small noise, and instantiate the theory in stochastic Kolmogorov equations and SPDEs, including Lotka–Volterra, SIR, and turbulence models. The approach yields almost-sure persistence results, explicit Lyapunov-function constructions, and a unified framework that accommodates delays, diffusion, and regime-switching, with broad implications for ecology, epidemiology, and physics.
Abstract
Motivated by infinite-dimensional ecological and biological models such as reaction-diffusion SPDEs and stochastic functional differential equations, we develop a general criteria for stochastic persistence (coexistence) in terms of an average lyapunov function, which was previously known only in finite dimensions. To apply our results to SPDEs we analyze the projective process, and we employ a combination of mild (stochastic convolution) and variational (lyapunov function) techniques. Our analysis also requires some nontrivial well-posedness and nonnegativity results for reaction-diffusion SPDEs, which we state and prove in great generality, extending the known results in the literature. Finally, we present several examples including ecological models (Lotka-Volterra), an epidemic model (SIR), and a model for turbulence. Notably we show that, as in the SDE case, coexistence in the Lotka-Volterra model is determined by the invasion rates.