arXiv subjects

Declan Stacy

Publications and source records attributed to Declan Stacy.

3 recordsLinked to original sources

arXiv2026

Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing

In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or viscosity) varies, the number of invariant measures of the system switches from one to several. We are also able to obtain precise asymptotics for the top Lyapunov exponent associated to the unstable invariant measure. The crux of our proof is using a Wiener chaos expansion to show that mass quickly transfers from stable modes to unstable ones.

math.PR↗

arXiv2026

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.

math.PR↗

arXiv2024

Stochastic Extinction, An Average Lyapunov Function Approach

We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. In many examples we improve existing results by removing unnecessary assumptions or providing sharper criteria for the extinction.

math.PR↗