No abstract

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 ≪ 1 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.

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.

We prove the existence and uniqueness of quasi-stationary and quasi-ergodic measures for a class of reaction-diffusion equations perturbed by additive cylindrical noise. To the author's knowledge, this is the first time that quasi-stationarity & quasi-ergodicity of a stochastic partial differential equation have been proven. Additionally, we prove a quasi-ergodic theorem, and in a general setting, demonstrate that the rate of convergence to the quasi-ergodic average in this theorem is, under certain conditions, exponential in time. These results allow us to qualitatively characterize the behaviour of solutions to these equations in neighbourhoods of an invariant manifold of the corresponding deterministic equations at some large time t > 0, conditioned on remaining in the neighbourhood at time t. In contrast to other approaches taken to the question of the long-term behaviour of stochastic dynamical systems in general, and stochastic reactiondiffusion equations in particular, the approach based on quasi-stationary measures works well outside of the small-noise regime.

In this note, we prove an Itô formula for the isochron map of a reaction-diffusion system. This follows the proof of a new result which states that the second derivative of the isochron map of a reaction-diffusion system is trace class. This result, in turn, is a corollary of Proposition 2.3, which guarantees that the first and second Fréchet derivatives of the flow of a reaction-diffusion system with respect to initial conditions are trace class.