Stochastic Persistence in Infinite Dimensions

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.

💡 Research Summary

This paper develops a unified theory of stochastic persistence (coexistence) for infinite‑dimensional stochastic dynamical systems, with a particular focus on reaction‑diffusion stochastic partial differential equations (SPDEs) and stochastic functional differential equations (SFDEs) that incorporate spatial dependence and delays. The authors extend the finite‑dimensional “average Lyapunov function” method—previously limited to systems where a Lyapunov function W and a stronger function W′ satisfy L W ≤ K − W′ and where W′ has compact sub‑level sets—to Banach‑space settings where such compactness does not exist.

Key contributions include:

1. Projective (polar) coordinates – By separating the state into a radial component r = ‖u‖ (distance to the extinction set M₀) and an angular component v = u/‖u‖, the authors “blow up’’ the extinction set into a sphere, allowing the definition of a continuous generator‑applied function H even at r = 0. Restricting to non‑negative solutions (biologically natural) ensures that the invariant measures of the angular process are supported on a single non‑negative eigenfunction, yielding compactness of the invariant‑measure set.

2. Combination of mild and variational techniques – For SPDEs, mild solutions are expressed via stochastic convolutions, while variational Lyapunov functions (e.g., W(u)=‖u‖²_{L²}, W′(u)=‖u‖²_{H¹}) provide the dissipativity estimate L W ≤ K − W′. This dual approach overcomes the failure of traditional compact‑sublevel arguments in infinite dimensions.

3. Tightness of empirical occupation measures – Using Kolmogorov’s continuity criterion, the authors show that W(Xₜ) remains uniformly bounded over sufficiently long time windows, which implies almost‑sure tightness of the empirical measures μₜ = (1/t)∫₀ᵗ δ_{Xₛ} ds. They further require that the time‑averaged magnitude of H is negligible compared with that of W′, guaranteeing convergence of μₜ to an invariant measure μ.

4. Average Lyapunov exponent – For the extinction set M₀, the average Lyapunov exponent λ̄(M₀) = sup_{μ∈Inv(M₀)} ∫ H dμ is defined. The main theorem states that λ̄(M₀) < 0 implies stochastic persistence: trajectories starting near M₀ stay away with probability one. When λ̄(M₀) is hard to compute, a perturbation result shows that if the deterministic counterpart has a negative exponent, then weak enough stochastic forcing preserves persistence.

5. Well‑posedness and non‑negativity – The paper proves existence, uniqueness, and a weak maximum principle for a broad class of reaction‑diffusion SPDEs, even under noise that is not Hilbert‑Schmidt (e.g., space‑time white noise). New Itô formulas for mild solutions are derived, extending earlier works.

6. Applications – Four concrete models illustrate the theory:

   • Logistic growth with harvesting on a spatial domain, yielding explicit criteria for survival of all species.
   • A turbulence model on the torus driven by multiplicative space‑time white noise; small noise guarantees multiple invariant measures.
   • Spatial SIR epidemic model with multiplicative noise; the authors obtain almost‑sure endemicity conditions, improving on prior expectation‑based bounds.
   • Competitive Lotka‑Volterra systems with diffusion; invasion rates (average Lyapunov exponents) determine coexistence, extending finite‑dimensional results to SPDEs.

Overall, the manuscript bridges a gap in the literature by providing a robust, average‑Lyapunov‑function framework that works in infinite dimensions, accommodates non‑negative constraints, and handles both mild and variational aspects of SPDEs. It opens pathways for further research on relaxing the non‑negativity assumption, treating more general noise structures, and exploring the geometry of invariant‑measure sets in high‑dimensional stochastic dynamics.