Invariance problems for stochastic partial differential equations

• Applicant: Dr. Stefan Tappe
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 444121509

Consider a stochastic partial differential equation. In this project, we are interested in investigating when a set of the state space is invariant for this equation. This means that for every starting point from the set the corresponding solution of the stochastic partial differential equations stays in the set; in this case we speak about stochastic invariance. The essential goals of my project consist of a theoretical part and an application part. The general goal of the theoretical part is to solve problems concerning stochastic invariance; here the given set can be, for example, a submanifold, or an arbitrary closed subset of the state space. In this regard, I also intend to investigate rough partial differential equations and robust stochastic partial differential equations with model uncertainty. Afterwards, the general goal of the application part is to apply the previous findings of the theoretical part to stochastic partial differential equations from various fields; this includes equations arising in natural sciences, equations in financial mathematics, and equations for modeling energy markets. Among other things, I intend to apply the findings about invariant submanifolds for numerical approximations of the solutions.

• DFG Programme: Research Grants