Fields of research
The research group focuses on:
- Functional Analysis (operator semigroups, Dirichlet forms)
- Stochastic Analysis (construction, ergodicity and scaling limits of stochastic dynamics)
- White Noise Analysis
- Mathematical Physics (Statistical Mechanics, Quantum Mechanics, Quantum Field Theory, polymer models)
Operator semigroups and Dirichlet forms
Operator semigroups are a very useful tool for analysing solutions of (stochastic, partial) differential equations. We use operator semigroups associated with Dirichlet forms for the construction of solutions of stochastic differential equations (SDE), for the analysis of their properties and to determine their long time behaviour. We also apply Dirichlet forms in combination with concepts like Gamma or Mosco convergence to consider scaling limits of solutions of SDEs. Many of those analyses can be transferred to the solutions of the partial differential equations associated via the Itô formula. Also stochastic partial differential equations are in the focus of our research. One approach is via Dirichlet forms with infinite dimensional state space, another uses the concept of mild solutions in the context of operator semigroups. In addition to conceptual research we are also always interested in applications. In this regard there were and still are several projects concerning problems from Statistical Physics (construction and analysis of stochastic dynamics in continuous particle systems; scaling limits of continuous, infinite particle systems; wetting models and their scaling limits) and Industrial Mathematics (fiber lay down models, stochastic partial differential equations for fluid dynamics with an algebraic constraint).
White Noise Analysis
White Noise Analysis or more general Gaussian Analysis allows an Analysis on infinite dimensional spaces. Due to the nonexistence of a locally finite, translation invariant measure (Lebesgue measure) on infinite dimensional vector spaces, Gaussian measures become central. In Gaussian Analysis on conuclear, infinite dimensional vector spaces, a rich Analysis has been developed. Concepts like Fourier transform, differential operators and distribution spaces (spaces of generalized functions) are included. We work on a further development of this Analysis where the direction is oriented towards applications. Those are located in Quantum Mechanics (Feynman Path Integrals), Quantum Field Theory (Wightman functions), polymer models (Edwards model) and Stochastic Partial Differential Equations. Recently we also started to work on the development of a non Gaussian Grey Noise Analysis. From this new research direction we expect a generalisation of the Feynman Kac formula to time fractional heat equations and the possibility to analyse time fractional Schrödinger equations.