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Stochasticity in Spatially Extended Deterministic Systems and via Homogenization of Deterministic Fast-Slow Systems

Final Report Summary - STOCHEXTHOMOG (Stochasticity in Spatially Extended Deterministic Systems and via Homogenization of Deterministic Fast-Slow Systems)

Ergodic theory is the analysis of probabilistic or statistical aspects of deterministic systems. Roughly speaking, deterministic systems are those that evolve without any randomness. Nevertheless, the probabilistic approach is appropriate since specific trajectories are unpredictable in “chaotic” systems. At the other extreme, stochastic systems evolve in a random manner by assumption.

One of the main topics of this proposal was to investigate how separation of time scales can cause a fast-slow deterministic system to converge to a stochastic differential equation (SDE). This is called homogenization; the fast variables are averaged out and the limiting SDE is generally of much lower dimension than the original system. Homogenization is reasonably well-understood when the underlying fast-slow system is itself stochastic. However there are very few results for deterministic
fast-slow systems. The aim is to make homogenization rigorous in a very general setting, and as a byproduct to determine how the stochastic integrals in the SDE are to be interpreted. Our method bypasses issues such as decay of correlations (which is very poorly understood for continuous time systems). In this respect, it is a radically new approach, completely contrary to the prevailing school of thought amongst researchers in homogenization of fast-slow systems. The project led to a definitive solution to the classical Wong-Zakai question concerning smooth approximation of stochastic integrals and made significant progress on the longer term aim to understand fully-coupled deterministic fast-slow systems.

A second main topic was to explore the idea that anomalous diffusion arises naturally in odd dimensions but not in even dimensions. The context is pattern formation in spatially extended systems with Euclidean symmetry, and this dichotomy can be seen as an extension of the classical Huygens principle that sound waves propagate in odd but not even dimensions. For anisotropic systems (where there are translation symmetries but not rotation symmetries), the situation is
simpler: chaotic dynamics leads to Brownian motion and weakly chaotic dynamics (of intermittent type) leads to a Lévy process. However in the isotropic case (rotations and translations), it is conjectured that anomalous diffusion is suppressed in even (but not odd) dimensions in favour of Brownian motion. This topic combines aspects of smooth ergodic theory, stochastic analysis and pattern formation. During this project, the conjecture in the even dimensional case, namely suppression of anomalous diffusion, was proved.

The classical Lorenz attractor, introduced by Lorenz in 1963, is the prototypical example of deterministic chaos, and is responsible for the term "butterfly effect" in everyday usage. An outstanding problem has been to establish strong randomness properties for this attractor, especially to prove exponential decay of correlations (exponential rate of loss of memory). Such a property is notoriously difficult to establish for continuous time dynamical systems. This question has now been
solved as a result of this project: the classical Lorenz attractor has exponential decay of correlations.