2022/10/19 by Zachary P. Adams, Adams, Zachary P., James MacLaurin +1 · 2 citations
Economics, Econometrics and Finance · Physics and Astronomy · #35R60 #60H15 #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2210.10681
openalex publication_date 2022/10/19 · openalex created_date 2022/10/22 · openalex updated_date 2026/07/28
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.