Objective 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 is 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. The focus is mainly on situations where the SDE limit is driven by Brownian motion, but SDEs driven by stable Lévy processes are also of interest. 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.A second main topic is to explore the idea that anomalous diffusion in the form of a superdiffusive Lévy process 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 only), 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), anomalous diffusion is suppressed in even dimensions in favour of Brownian motion. Fields of science natural sciencesmathematicspure mathematicsmathematical analysisdifferential equations Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Call for proposal ERC-2012-ADG_20120216 See other projects for this call Funding Scheme ERC-AG - ERC Advanced Grant Host institution THE UNIVERSITY OF WARWICK EU contribution € 1 577 880,00 Address Kirby Corner Road - University House CV4 8UW Coventry United Kingdom See on map Region West Midlands (England) West Midlands Coventry Activity type Higher or Secondary Education Establishments Principal investigator Ian Melbourne (Prof.) Administrative Contact Catherine Cochrane (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all THE UNIVERSITY OF WARWICK United Kingdom EU contribution € 1 577 880,00 Address Kirby Corner Road - University House CV4 8UW Coventry See on map Region West Midlands (England) West Midlands Coventry Activity type Higher or Secondary Education Establishments Principal investigator Ian Melbourne (Prof.) Administrative Contact Catherine Cochrane (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data