Stochastic probability9/26/2023 Nonequilibrium statistical mechanics (Dr Tobias Kuna, Professor Valerio Lucarini) Our research focuses on studying the properties of the extremes of observables of chaotic dynamical systems. Clearly, understanding the properties of the tail of the probability distribution of a stochastic variable attracts a lot of interest in many sectors of science and technology because extremes sometimes relate to situations of high stress or serious hazard, so that it is crucial to be able to predict their return times in order to cushion and gauge risks. While extreme events in a given physical system obey the same laws as typical events, extreme events are rather special both from a mathematical point of view and in terms of their impacts. The study of extreme events has long been a very relevant field of investigation at the intersection of different fields, most notably mathematics, geosciences, engineering, finance. Lyons' study of stochastic differential equation, the theory has inspired Martin Hairer's (Fields Medal 2014) work on stochastic partial differential equations and subsequently the theory of regularity structures.Įxtreme events (Dr Tobias Kuna, Professor Valerio Lucarini) Rough path theory also gives a canonical way to define differential equations driven by non-semi-martingales. It simplifies and strengthens classical results in Stochastic Analysis such as large deviation principle and stochastic flow. Rough path is a deterministic theory of calculus purposed built for such paths. Many standard stochastic processes, such as Brownian motion, have no-where differentiable sample paths. Research in this area includes the investigation of mixing properties. The mathematics of random as well as deterministic dynamical systems is central in the description of processes appearing in many areas of science. Random dynamical systems and time series analysis (Dr Tobias Kuna, Dr Jochen Broecker, Professor Valerio Lucarini) In particular, we do research in derivation of scaling limits, effective description for large systems in terms of few low-dimensional equations. Of enormous practical relevance in applications are memoryless time developments, known as Markov processes, it is a well studied area of probability theory having many deep connections to real and complex analysis, in particular ordinary differential equations, spectral theory, partial differential equations. Markov processes (Dr Tobias Kuna, Dr Jochen Broecker, Professor Valerio Lucarini) Main areas of research are the moment problem for point processes, geometry of configuration spaces and analyticity properties. The aim is a rigorous derivation of cooperative effects. A classical area of application for such systems is the description of solid and soft matter from microscopic principles. Systems consisting out of a large number of point like subsystems arise in many areas, for example if the system is based on a large number of interacting individual agents as in economics, biology or sociology. Our research group in Reading is focused on the following areas: Point processes and interacting particle systems Probability theory is key to providing robust foundations for statistical mechanics, an interdisciplinary research area between mathematics and theoretical physics focussing on the study of the emerging properties of systems with many degrees of freedom. Located between pure and applied mathematics, this field overlaps with many different branches of mathematics and provides a background, as well as tools, to properly formulate and solve problems from a range of other sciences.įrom its inception as an analysis of "chance", modern probability theory is indispensable in mathematical fields as different as combinatorics, real and complex analysis, and group theory. Probability theory naturally appears in the description and analysis of all these different areas. Is there a connection between the distribution of prime numbers, the structure of Mandelbrot sets, compression algorithms for data sets, the description of polymers and the analysis of heat flow in media?
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