SPADE2 · Deterministic and stochastic dynamics, fractals, turbulence

Final Activity Report Summary - SPADE2 (Deterministic and stochastic dynamics, fractals, turbulence)

The transfer of knowledge (ToK) programme titled 'Deterministic and stochastic dynamics, fractals, turbulence' (SPADE2), hosted by the Institute of Mathematics of the Polish Academy of Sciences (IMPAN) and was organised in four related and interweaving tasks, namely: 1. dynamical systems 2. partial differential equations (PDEs), turbulence 3. stochastic processes, scaling limits in physical processes 4. function spaces. It embraced physical processes with analytical background, as well as applications. During its four year duration IMPAN had 66 incoming and 46 outgoing person-months visitors, including 21 incoming and 15 outgoing researchers. The partner institutions were the University of Warwick, Universite Paris 6, Scuola Normale Superiore di Pisa, INRIA Rocquencourt and Christian-Albrechts-Universitaet (CAU) Kiel, in accordance with the proposal. SPADE2 cooperated with the Marie Curie research training network (RTN) 'Conformal structures and dynamics' (CODY), having a node at IMPAN. Several regular seminars accompanied the programme, with some conference schools and workshops, supported by the Polish Ministry of Sciences and Higher Education complementary funds. A SPADE2 summarising workshop was organised at IMPAN in 2009, where the main research achievements were presented by SPADE2 fellows and guests. In the first task thermodynamical formalism was used to study geometric pressure and fractal dimensions and geometry of limit sets for iteration of holomorphic, i.e. rational, entire and meromorphic, maps by F. Przytycki, J. Kotus, A. Zdunik, M. Rams et al. in cooperation with M. Urbanski, J. Rivera-Letelier, G. Levin, K. Gelfert and W. Bergweiler. In several cases conformal and related invariant measures with their statistical properties were fairly completely understood. Moreover, iterated function systems (IFS) and their fractal limit sets, including deterministic and random self-intersecting Cantor sets were studied. A highlight was a study, by M. Rams with J. Levy-Vehel from INRIA, of a model of internet traffic, its multifractal queuing behaviour. In population dynamics R. Rudnicki presented a model of blood stem cells production exhibiting 'chaos', with M.B. Adioui, along with models of genome evolution, random walks on neural networks and models for animal aggregations. In the second task most results concerned Navier-Stokes equations (NSE), in two-dimensional and special three-dimensional domains, existence and regularity of solutions, in connection with the millennium problem, and attractors, both deterministic and random. The researchers that were involved in the second task were W. Zajaczkowski, J. Renclawowicz, P. Mucha, G. Lukaszewicz and W. Sadowski, in cooperation with K. Pileckas, G. Seregin, M. Arnold, B. da Veiga from Pisa and J.C Robinson from Warwick. Gevrey classes of solutions of PDE's were studied by G. Lysik and S. Michalik in collaboration with T. Gramchev. The researcher T. Reginska used numerical methods for the reconstruction of acoustic or electromagnetic fields from inexact data. A highlight of the third task was a cycle of papers by J. Zabczyk, S. Peszat at al. in cooperation with Z. Brzezniak, P. Imkeller and E. Priola on analytic and probabilistic properties of solutions of stochastic differential equations (SDEs), perturbed by a discontinuous noise, i.e. Levy processes. Conditions on strong Feller property, existence of stationary measures, regularity and integrability of trajectories were obtained. They implied lack of explosion in related models of bond markets. On his stay at IMPAN Imkeller gave a course titled 'Malliavin calculus and applications to stochastic control and finance', published in the new book series 'Banach Center Lecture Notes' at IMPAN. Remarkable were the results by T. Komorowski with S. Olla on heat transport and radiation. In scaling limits M. Zinsmeister gave a course at IMPAN on stochastic Loewner evolution (SLE), which was also planned to appear in Banach Center Lecture Notes. Finally, with respect to the fourth task, remarkable were the results by P. Wojtaszczyk regarding the 'Stability of l_1 minimisation in compressed sensing' and the 'Stability and instance optimality for Gaussian measurements in compressed sensing', Foundations of Computational Mathematics. The papers provided the first 'numerically feasible decoder'. Other results concerned analysis on metric spaces, approximation theory and geometry of Banach spaces, e.g. Sobolev spaces, Hardy-type inequalities, Franklin bases and wavelets, by A. Kalamajska, A. Kamont, S. Gogyan, T. Szankowski, A. Pelczynski, M. Wojciechowski, M. Roginskaya, A. Koldobskiy et al.

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