TRR 388:  Rough Analysis, Stochastic Dynamics and Related Fields

• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 516748464

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 renormalizations, 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. (i) Singular Dynamics - How to approach long-term/large scale stochastic effects in singular dynamics? (ii) Robustification - How do complex stochastic systems depend on specified noise?(iii) 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.

• DFG Programme: CRC/Transregios

Current projects

• A01 - Energy solutions, nonlinear analysis and stochastic homogenization (Project Heads Otto, Felix ;  Perkowski, Nicolas )
• A02 - Optimal transport meets rough analysis (Project Heads Friz, Peter Karl ;  Liero, Matthias )
• A03 - Random interface models, scaling limits and large deviations (Project Heads Deuschel, Jean-Dominique ;  König, Wolfgang ;  Zwicknagl, Barbara )
• A04 - Algebra and geometry of path signatures (Project Heads Améndola Cerón, Carlos ;  Preiß, Rosa ;  Sturmfels, Ph.D., Bernd )
• A05 - Signatures at the threshold of analysis, algebra and geometry: new perspectives (Project Heads Friz, Peter Karl ;  Paycha, Ph.D., Sylvie )
• A06 - Dynamics and bifurcation theory of pathwise SPDEs (Project Heads Blessing, Alexandra ;  Perkowski, Nicolas )
• A07 - Rough backward stochastic differential equations (Project Heads Becherer, Dirk ;  Friz, Peter Karl )
• A08 - Thin films and entropic repulsion for Gaussian Free Field (Project Heads Deuschel, Jean-Dominique ;  Otto, Felix ;  Zwicknagl, Barbara )
• A09 - Dormant populations in heterogeneous random environments (Project Heads König, Wolfgang ;  Perkowski, Nicolas ;  Wilke-Berenguer, Maite )
• A10 - Optimal control of McKean-Vlasov stochastic partial differential equations (Project Heads Perkowski, Nicolas ;  Stannat, Wilhelm )
• B01 - Statistical learning from path observations (Project Heads Améndola Cerón, Carlos ;  Bayer, Christian ;  Reiß, Markus )
• B02 - Microstructural foundations of rough volatility models (Project Heads Bayer, Christian ;  Horst, Ulrich ;  Kreher, Dörte )
• B03 - Signature methods for optimal control in finance (Project Heads Bank, Peter ;  Bayer, Christian )
• B04 - Rough analysis and mean field games (Project Heads Friz, Peter Karl ;  Horst, Ulrich )
• B05 - Rough analysis in stochastic control (Project Heads Bank, Peter ;  Horst, Ulrich )
• B06 - (S)PDEs on time-dependent random domains (Project Heads Djurdjevac, Ana ;  Schillings, Claudia )
• B07 - Statistics for population models with dormancy (Project Heads Reiß, Markus ;  Wilke-Berenguer, Maite )
• B08 - Bayesian inference and mean field approximation for nonlinear (S)PDE (Project Heads Schillings, Claudia ;  Spokoiny, Vladimir ;  Wang, Sven )
• B09 - Mean field theories and scaling limits of nonlinear stochastic evolution systems (Project Heads Stannat, Wilhelm ;  Thomas, Marita )
• Z01 - Central project (Project Head Friz, Peter Karl )

• Applicant Institution: Technische Universität Berlin
• Co-Applicant Institution: Freie Universität Berlin; Humboldt-Universität zu Berlin
• Participating University: Universität Konstanz; Universität Potsdam
• Participating Institution: Max-Planck-Institut für Mathematik in den Naturwissenschaften (MIS); Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS); Leibniz-Institut im Forschungsverbund Berlin e. V.
• Spokesperson: Professor Dr. Peter Karl Friz