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

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.

Multivariate Hawkes processes on inhomogeneous random graphs

“…We need a way to define a proper phase reduction of U N along U. We have two ways to do so that we use in our results that are well explained in the recent work [1], which takes the NFE as a good class of examples and motivation. The first one is via the variational phase, defined in the following Proposition 2.7: Proposition 2.7 (Variational phase).…”

“…In 2a, we start in a vicinity of U as we take for initialization ρ(x) = A(κ) cos(x) + cos(2x), where A(κ) solving (2.2) for f = fκ,ϱ is found by a numerical root finding method, with a final time Tmax = 500 (of the same order that the size of the population). In 2b, we initialize the system with ρ(x) = 1 4 A(κ) cos(x). It is too far from the manifold U and we can see that the dynamics is attracted to U A where A is the smallest solution of (2.2) (in Figure 1 it corresponds to the far left intersection of the black and blue lines) which is approximately 0, hence we only run the simulation with a final time Tmax = 5.…”