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

  • Weak synchronisation for McKean–Vlasov SDEs

    Synchronisation by noise for McKean–Vlasov stochastic differential equations is investigated. A transfer principle is introduced by which synchronisation by noise and diagonal mixing can be transferred from an associated limiting frozen-diffusion SDE to a genuinely law-dependent McKean–Vlasov SDE. The usefulness…

  • Large spikes in SGD: a large-deviations view of catapults

    Workshop report · Oberwolfach Reports 23(1), 698–700 (2026). A large-deviations perspective on spikes and catapult behaviour in stochastic gradient descent.

  • Asymptotics of Multi-Scale McKean–Vlasov Diffusions with Super-Linear Kernels: a Lifted Semigroup Approach

    In this work, we establish the small-noise asymptotic behaviour (namely, the functional law of large numbers and the large deviation principle) for multi-scale McKean–Vlasov diffusions with super-linear kernels. In this setting, the interaction depends on the laws of both the slow component…

  • Large Spikes in Stochastic Gradient Descent: A Large-Deviations View

    Large loss spikes in stochastic gradient descent are studied through a rigorous large-deviations analysis for a shallow, fully connected network in the NTK scaling. In contrast to full-batch gradient descent, the catapult phase is shown to split into inflationary and deflationary regimes,…

  • Ergodicity for SPDEs driven by divergence-free transport noise

    We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even…

  • Mixing at the Batchelor Scale for White-In-Time Flows

    We consider the mixing properties of solutions to the advection-diffusion equation of a white-in-time velocity field on the 2-dimensional torus with four forced modes. As the diffusivity parameter goes to zero, we show that the almost-sure exponential dissipation rate stays bounded from…

  • A Dynamical Systems Perspective on the Analysis of Neural Networks

    In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements.…

  • Random dynamical systems for McKean–Vlasov SDEs via rough path theory

    The existence of random dynamical systems for McKean–Vlasov SDEs is established. This is approached by considering the joint dynamics of the corresponding nonlinear Fokker-Planck equation governing the law of the system and the underlying stochastic differential equation (SDE) as a dynamical system…