Analysis and approximation of infinite dimensional sticky reflected Ornstein-Uhlenbeck processes

• Applicant: Professor Dr. Martin Grothaus
• Subject Area: Mathematics
• Term: from 2016 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 324039129

• Final Report Year: 2023

Final Report Abstract

In this project, an integration by parts formula for the modulus of the Brownian bridge is derived. We get an explicit expression for the reflection term as a Hida distribution. Integration by parts formulas are an essential ingredient in the theory of Dirichlet forms to analyse the associated Markov processes. Our formula provides a means to analyse infinite dimensional sticky Ornstein-Uhlenbeck processes. The methods in white noise analysis provided in its development can also be applied to construct the stochastic current of a Brownian motion. As this transfer results in a significant improvement of the current state of research on stochastic currents, we included it in the project. Another objective, the approximation of infinite dimensional sticky Ornstein-Uhlenbeck processes has to remain an open problem for now. However, promising new insights into the problem of approximating infinite dimensional Ornstein-Uhlenbeck processes whose invariant measures are non-log-concave are gained. The non-log-concavity of the invariant measure turns out to be an essential technical challenge in this question. For some other physically relevant cases in the context of stochastic interface models, which pose that problem, the abstract convergence result for Dirichlet forms derived in this project can be applied indeed. In the course of this project Dr. Wittmann wrote a phD thesis on the large scale asymptotic for Markov processes.

Publications

• Integration by parts on the law of the modulus of the Brownian bridge. Stochastics and Partial Differential Equations: Analysis and Computations, 6(3), 335-363.
  Grothaus, Martin & Vosshall, Robert
  
• A white noise approach to stochastic currents of Brownian motion. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 26(01).
  Grothaus, Martin; Suryawan, Herry Pribawanto & Da, Silva José Luís
  
• Large scale asymptotics for Markov processes in the analytic framework of Mosco-Kuwae-Shioya. PhD thesis, Technical University of Kaiserslautern, 2022
  S. Wittmann