Sensitivity Analysis of Nonlocal Operators with Applications to Jump Processe

• Applicant: Professor Dr. René Leander Schilling
• Subject Area: Mathematics
• Term: from 2019 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 426577679

Many real-world phenomena can be thought of as transportation of particles in space. The transport may be continuous, i.e. diffusive, similar to the erratic movement of pollen suspended in a medium (like Brownian motion) or discontinuous, with jumps gradually arriving in time. The latter occurs, e.g., when we observe trapping or tunneling of the particles. Another typical example are changes of a system with a finite number of states or observations, which cannot be continuous. There are good arguments that that many real-life phenomena, like weather patterns or stock prices exhibit jump-type behaviour. From a mathematical perspective, we may use Lévy-type processes which generalize continuous `diffusive’ phenomena since they allow for jumps and their behaviour depends locally on the state of the system. We want to capture fundamental features of jump dynamics and resolve essential difficulties by developing an abstract but flexible mathematical framework. This aims at offering a unifying perspective for complicated phenomena. It turns out that specifying the intensity of jumps does not necessarily (uniquely) define the target dynamic. We need to put considerable effort into the construction of the mechanism and describe its qualitative and quantitative properties before they can be applied in theory and modelling.Among the applications, we focus (i) on inner-mathematical applications studying qualitative properties of Lévy-type processes (Do the processes return? How quickly do they move to infinity? Can we identify trends?) but also on (ii) the question of ergodicity (is observing many particles for a short time as good as observing one particle for a long time?) which is paramount for experimental sciences, as well as (iii) applied questions on the statistics and approximation of the processes.A key feature of our proposal is that we will use advanced methods from partial differential equations to the theory of Lévy-type processes, and vice versa. Therefore, we combine the expertise of two internationally acknowledged teams, at TU Wroclaw (Bogdan) and TU Dresden (Schilling), working on the complementary fields of analysis and analytic methods in probability, resp., jump processes and stochastic analysis.

• DFG Programme: Research Grants
• International Connection: Poland
• Partner Organisation: Narodowe Centrum Nauki (NCN)
• Co-Investigator: Dr. Viktoriya Knopova
• Cooperation Partner: Professor Krzysztof Bogdan, Ph.D.