The Berlin stochastics community encompasses groups at HU, TU, and WIAS, as well as the jointly organized Berlin Probability Colloquium and the Research Seminar in Stochastic Analysis and Stochastics of Financial Markets. There is a long tradition of joint activity in graduate education, as realized in the Int'l RTG Stochastic Models of Complex Processes and their Applications with ETH and the University of Zurich.

Randomness is ubiquitous in real-life phenomena. Modern stochastic models start with descriptions on different spatial and temporal scales, using advanced mathematical approaches and techniques: microscopic modeling through interacting systems of individual agents or particles, random walks, or random media, and macro-/mesoscopic modeling using stochastic (partial) differential equations, stochastic delay equations, and stochastic dynamics. Typically, the same system can be modeled on different scales, and it is of crucial importance to study the interplay of different levels of modeling through limit theorems. This involves advanced methods from the statistics of stochastic systems and time series, such as parameter identification, calibration, regression or dimension reduction techniques, and reaches out into the numerical-statistical treatment of complex systems (Spokoiny).

Recent research in these core fields of stochastics is highlighted by the strong impact methods of statistical physics and dynamics or biostatistics have found in explaining biological systems in random environment, or by the boom in demand for tools of risk management in banking and insurance, starting with the paradigm of Merton–Scholes in financial economics.

Stochastic analysis (Becherer, Föllmer, Imkeller, Küchler, Scheutzow) describes real world phenomena on a macro-/mesoscopic level. Randomness enters by th important paradigm of processes of independent increments. They originate in microscopic models, and lead to the main objects of stochastic analysis: finite or infinite dimensional stochastic differential equations, possibly with memory effects. Dynamical properties of systems described by such equations have been attracting much attention worldwide in recent years. This was triggered by new demands for mathematical concepts for instance in biology, climate dynamics and environmental economics. Mathematical techniques related to large deviations for randomly perturbed dynamical systems, including a systematic use of spectral and elliptic regularity theory are of central importance. Here we see strong links to dynamical systems and applied and functional analysis.

Stochastic and statistical analysis has been the realm of models of financial markets with mesoscopic price dynamics. Recent developments in this area have seen a movement towards a new generation of market models, in which a multitude of highly interdependent risk factors become integrated, including climate risks, volatility clusters, and memory effects. In these models, the implementation of more sophisticated measures for the downside risk becomes vital for hedging and valuation purposes fitting the demands of practitioners. This creates challenging mathematical problems on all levels, from stochastic and time series analysis, through control and optimization, to PDEs and stochastic numerical simulation.

The main paradigm of microscopic stochastic models (Blath, Deuschel, Gärtner) is the notion of systems of particles evolving subject to some stochastic dynamics. This embraces both mutual interactions and interactions with a random environment. Beyond statistical physics, recent applications face evolving populations in biology and agent-based modeling in mathematical finance, with a strong appeal to techniques from biostatistics and parameter identification for time series. A key phenomenon in large systems is the occurrence of phase transitions. In contrast to classical statistical mechanics, the modern dynamical perspective views nonuniqueness of invariant measures in the scope of the dynamics of phase transitions. A first-order phase transition is reflected through the phenomenon of metastability. In the macro-/mesoscopic limit, this basic feature is encountered in the context of diffusion processes in stochastic analysis. The study of the connection between the two levels of modeling has been of primary importance. Links to dynamical systems and mathematical physics are apparent.

The analysis of equilibrium and dynamics goes hand in hand. This is particularly pronounced in systems with quenched disorder, where both topics abound of highly challenging problems of modern probability theory. For equilibrium theory, this amounts to the investigation of the geometric properties of correlated random fields on high-dimensional spaces, requiring considerable refinements of the theory of extreme values and processes on Banach spaces. These methods are relevant for extreme portfolio risks in finance and for combinatorial optimization.

A recurrent theme for dynamical processes in random environments are random walks in random environments. This topic has been at the heart of modern probability for many years, but key questions remain wide open. They will also require methods from mathematical physics and applied and functional analysis.