Research Programme
Stochastic dynamics builds on probability theory and Itô’s stochastic analysis to study the evolution of systems under the influence of randomness, with a profound impact on many fields, including statistical physics, mathematical finance, uncertainty quantification, quantum field theory, mathematical biology, economics. Rough analysis, on the other hand, stands for recent breakthroughs in mathematics, rooted in Lyons’ rough path theory. With the original motivation of introducing robustness in noise/signal, rough analysis offers a nonlinear extension of distribution theory that is crucial for understanding singular stochastic dynamics and their possible renormalisations, and to capture nonlinear effects of signals. Transcending its origins, rough analysis recently saw the emergence of deep mathematical structures with significant geometric and algebraic components. Together, they form the fertile grounds for this TRR Rough Analysis, Stochastic Dynamics and Related Fields.
With an intense interplay of analysis, algebra/geometry and probability theory together with closely related applied topics, such as statistics, robust modeling under uncertainty, and stochastic control theory and mathematical finance, our overarching goal is to foster mutually beneficial interactions with the new field of rough analysis. To achieve this we have identified the following central questions that will guide our investigations.
- Singular Dynamics – How to approach long-term/large scale stochastic effects in singular dynamic?
- Robustification – How do complex stochastic systems depend on specified noise?
- How do we, and how should we, understand paths?(iv) What is the role of Markovianity in rough, stochastic and singular dynamics?
Our answers to these overarching questions pass through outward-looking investigations of rough and stochastic (partial) differential equations (e.g. understanding universal objects in statistical physics, ‘KPZ fixed point’, robustness and uncertainty quantification, relations to optimal transport), the study of related algebraic structures for statistics and high-dimensional probability (e.g. rough path induced signatures), robust and efficient statistics for dynamically specified non-linear stochastic processes, as well as the emergence and use of rough structures in stochastic control theory and financial mathematics (e.g. rough volatility).
The area of rough analysis has, to a large extent, progressed as a theory in its own right. With our motto “Repeat the success of Itô calculus!” we envision a future where these ideas profoundly influence the vast community of probability, including financial mathematics and statistics, and beyond. Our TRR team offers the ideal complementary scientific expertise and the firm commitment, through the development of new important applications and in combination with significant outreach work, to contribute substantially to this goal.
