Stochastic Analysis

The meeting took place on May 30-June 3, 2011, with over 55 people in attendance. Each day had 6 to 7 talks of varying length (some talks were 30 minutes long), except for Thursday: the traditional hike was moved to Thursday due to the weather (and weather on thursday was indeed fine). The talks reviewed directions in which progress in the general field of stochastic analysis occurred since the last meeting of this theme in Oberwolfach three years ago. Several themes were covered in some depth, in addition to a broad overview of recent developments. Among these themes a prominent role was played by random matrices, random surfaces/planar maps and their scaling limits, the KPZ universality class, and the interplay between SLE (Schramm-Loewner equation) and the GFF (Gaussian free field)

Laws of rare events for deterministic and random dynamical systems

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against $L^1$ observables will play a central role in our investigations

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic

A study of the sensitivity of topological dynamical systems and the Fourier spectrum of chaotic interval maps

We study some topological properties of dynamical systems. In particular the rela- tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems establishing that for minimal dynamical systems those properties are equivalent. In the same direction we show that being a Li-Yorke sensitive dynamical system implies that the system is also Li-Yorke chaotic. On the other hand we survey the possibility of lifting some topological properties from a given dynamical system (Y, S) to an- other (X, T). After studying some basic facts about topological dynamical systems, we move to the particular case of interval maps. We know that through the knowl- edge of interval maps, f : I → I, precious information about the chaotic behavior of general nonlinear dynamical systems can be obtained. It is also well known that the analysis of the spectrum of time series encloses important material related to the signal itself. In this work we look for possible connections between chaotic dynamical systems and the behavior of its Fourier coefficients. We have found that a natural bridge between these two concepts is given by the total variation of a function and its connection with the topological entropy associated to the n-th iteration, fn(x), of the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how the Fourier coefficients are related to the chaoticity of interval maps

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Large Scale Stochastic Dynamics

Equilibrium statistical mechanics studies random fields distributed according to a Gibbs probability measure. Such random fields can be equipped with a stochastic dynamics given by a Markov process with the correspondingly high-dimensional state space. One particular case are stochastic partial differential equations suitably regularized. Another common version is to consider the evolution of random fields taking only values 0 or 1. The workshop was concerned with an understanding of qualitative properties of such high-dimensional Markov processes. Of particular interest are nonreversible dynamics for which the stationary measures are determined only through the dynamics and not given a priori (as would be the case for reversible dynamics). As a general observation, properties on a large scale do not depend on the precise details of the local updating rules. Such kind of universality was a guiding theme of our workshop