arXiv subjects

Zachary P. Adams

Publications and source records attributed to Zachary P. Adams.

5 recordsLinked to original sources

arXiv2025

Separation of time scales in weakly interacting diffusions

We study metastable behaviour in systems of weakly interacting Brownian particles with localised, attractive potentials which are smooth and globally bounded. In this particular setting, numerical evidence suggests that the particles converge on a short time scale to a "droplet state" which is $metastable$, i.e. persists on a much longer time scale than the time scale of convergence, before eventually diffusing to $0$. In this article, we provide rigorous evidence and a quantitative characterisation of this separation of time scales. Working at the level of the empirical measure, we show that (after quotienting out the motion of the centre of mass) the rate of convergence to the quasi-stationary distribution, which corresponds with the droplet state, is $O(1)$ as the inverse temperature $β\to \infty$. Meanwhile the rate of leakage away from its centre of mass is $O(e^{-β})$. Futhermore, the quasi-stationary distribution is localised on a length scale of order $O(β^{-\frac12})$. We thus provide a partial answer to a question posed by Carrillo, Craig, and Yao (Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits. In: Active Particles, Volume 2, 2019) in the microscopic setting.

math.PR↗

arXiv2024

Quasi-Ergodicity of Transient Patterns in Stochastic Reaction-Diffusion Equations

We study transient patterns appearing in a class of SPDE using the framework of quasi-stationary and quasi-ergodic measures. In particular, we prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion systems perturbed by additive cylindrical noise. We obtain convergence results in $L^2$ and almost surely, and demonstrate an exponential rate of convergence to the quasi-stationary measure in an $L^2$ norm. These results allow us to qualitatively characterize the behaviour of these systems in neighbourhoods of an invariant manifold of the corresponding deterministic systems at some large time $t>0$, conditioned on remaining in the neighbourhood up to time $t$. The approach we take here is based on spectral gap conditions, and is not restricted to the small noise regime.

math.PR↗

arXiv2024

Existence, Regularity, and a Strong Itô Formula for the Isochronal Phase of SPDE

We prove the existence and regularity of the isochron map for stable invariant manifolds of a large class of evolution equations. Our results apply in particular to the isochron map of reaction-diffusion equations and neural field equations. Using the regularity properties proven here, we are able to obtain a strong Itô formula for the isochronal phase of stochastically perturbed travelling waves, spiral waves, and other patterns appearing in SPDEs driven by white noise, even for SPDEs that only admit mild solutions.

math.PR↗

arXiv2023

The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties

We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighbourbhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T.~Winfree and J.~Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than $O(σ^{-2})$, but less than $O(\exp(Cσ^{-2}))$, where $σ\ll1$ is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.

math.PR↗

arXiv2022

The asymptotic frequency of stochastic oscillators

We study stochastic perturbations of ODE with stable limit cycles -- referred to as stochastic oscillators -- and investigate the response of the asymptotic (in time) frequency of oscillations to changing noise amplitude. Unlike previous studies, we do not restrict our attention to the small noise limit, and account for the fact that large deviation events may push the system out of its oscillatory regime. To do so, we consider stochastic oscillators conditioned on their remaining in an oscillatory regime for all time. This leads us to use the theory of quasi-ergodic measures, and to define quasi-asymptotic frequencies as conditional, long-time average frequencies. We show that quasi-asymptotic frequencies always exist, though they may or may not be observable in practice. Our discussion recovers previous results on stochastic oscillators in the literature. In particular, existing results imply that the asymptotic frequency of a stochastic oscillator depends quadratically on the noise amplitude. We describe scenarios where this prediction holds, though we also show that it is not true in general -- even for small noise.

math.PR↗