## Probability and Stochastic Analysis

**Probability and Stochastic Analysis | Research | People | Publications **

### Research

Our research group in Reading is focused on the following areas:

#### Analysis of Many-Particle Systems (Tobias Kuna)

Systems with many particles arise in many different areas, for example if the description of a system is based on a large number of interacting individual agents, such as molecules which form a solid or soft state of matter, animals, plants, stars. They play an important role in several simulation algorithms. More specifically, our research is about moment problems in statistical mechanics and the description of rare events.

#### Asymptotics of Markov Processes (Tobias Kuna)

The analysis of the large time behaviour is of enormous practical relevance for most applications. For memoryless stochastic processes, known as Markov processes, it is a well studied-area of probability theory having many deep connections to real and functional analysis, in particular spectral analysis. Our investigation of these spectral properties uses probabilistic as well as analytic tools. We also use this analysis as the base for the derivation of scaling limits, that is the derivation of effective description for large systems in terms of few low-dimensional equations.

#### Classical Probability Theory

Random walks and Levy processes as their continuous analogue are classical objects being right at the centre of probability theory. Our research in this area includes the analysis of the fine structure of Levy processes and random walks, such as laws of iterated logarithms for Levy processes as well as the analysis of areas under random walk excursion.

#### (Random) Dynamical Systems (Tobias Kuna/Jochen Broecker/Valerio Lucarini)

The mathematics of random as well as deterministic dynamical systems is central in the description of processes appearing in many areas of science. Research in this area includes the development of extreme value theory for dynamical systems, application of response theory and the investigation of mixing properties.

#### Applications

Our research group is open for interdisciplinary research. We are interested in applications mainly in the following areas:

**- Meteorology**

Atmospheric dynamics is the interplay of processes ranging over enormous temporal and spatial scales. Despite ever-increasing computer power, we are still not able to simulate all these processes faithfully; we might never be able to, and we might in fact not want to. Nonetheless, fast and small-scale processes might still be taken into account in some sense by modelling them as stochastic processes.

**- Filtering problems**

Important in various areas such as engineering, systems biology, and meteorology, filtering aims at reconstruction trajectories of dynamical systems (such as Markov processes) from a history of noise-corrupted and incomplete observations.

**- Demography and Biology**

Demography has always been one of the important fields of applications of probabilistic and statistical methods. The dynamics of populations must often be considered as inherently stochastic and therefore it is often appropriate to describe the dynamics via stochastic processes. The development of probabilistic and statistical tools for the investigation of populations undergoing ageing and mortality is of particular interest.

**- Theory of classical fluids**

Liquids and gases are formed by the cooperative behaviour of a myriad of molecules. In time spans perceivable by human beings, liquids and gases are very well-distinguished states of matter. Conversely, in the world of molecules, the striking difference between these two states of matter disappears. Only probabilistic tools can tell apart a gas from a liquid. All properties of liquids perceivable by human beings have to be in principle derivable from properties of the molecules alone. Most existing physical theories contain *ad hoc* steps which can be justified only afterwards by apparent similarity with observed properties of the liquid. Probability theory is already able to fill some of these gaps.