Stochastic Dynamics

Mathematical challenges

Stochastic dynamics asks how random systems evolve over long times and how individual sample paths organize into coherent behaviour. We study random dynamical systems generated by stochastic differential equations and SPDEs.

Our questions include ergodicity, mixing, stability and the formation of random attractors. Particular attention is paid to synchronization by noise, where common fluctuations can bring initially different states together, and to regularization mechanisms that change the qualitative dynamics. We combine probabilistic, analytical and geometric techniques to identify Lyapunov behaviour, quantify convergence and understand how microscopic stochasticity affects macroscopic stability. These tools also inform the study of learning algorithms and large interacting systems.

Current directions

  • Random dynamical systems generated by SPDEs
  • Synchronization by noise
  • Mixing and ergodicity
  • Random attractors and stability
  • Long-time qualitative behaviour

Related people

Related projects

Selected publications

  • Conservative SPDEs as fluctuating mean field limits of stochastic gradient descent

    Published · Probability theory and related fields, 192 (2025) 3/4, pp. 1447-1515. The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a…

  • Ergodicity and random dynamical systems for conservative SPDEs

    The dynamics of the solutions to a class of conservative SPDEs are analysed from two perspectives: Firstly, a probabilistic construction of a corresponding random dynamical system is given for the first time. Secondly, the existence and uniqueness of invariant measures, as well…