SPDEs

Mathematical challenges

Stochastic partial differential equations describe spatially extended systems under random forcing. Our work ranges from conservative and degenerate equations to singular and supercritical regimes in which classical solution concepts no longer apply.

We develop notions of solution, a priori estimates and compactness methods that respect conservation laws and nonlinear geometry. We study well-posedness, regularity, invariant behaviour and scaling limits, as well as situations in which noise regularizes an otherwise unstable or ill-posed evolution. A recurring objective is to connect SPDEs to the microscopic particle systems from which their fluctuations emerge and to devise numerical approximations that preserve the structure of the underlying equation.

Current directions

  • Conservative, degenerate and singular SPDEs
  • Well-posedness and regularity
  • Regularization and stabilization by noise
  • Supercritical equations
  • Fluctuation limits and invariant behaviour

Related people

Related projects

Selected publications

  • 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…

  • A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

    The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key…

  • Conservative stochastic PDEs on the whole space

    Published · Stochastics and partial differential equations : analysis and computations, 14 (2026) 1, pp. 350-388. The purpose of this paper is to establish a well-posedness theory for conservative stochastic partial differential equations on the whole space. This class of stochastic PDEs arises in fluctuating hydrodynamics, and includes…

  • Higher Order Fluctuation Expansions for Nonlinear Stochastic Heat Equations in Singular Limits

    Published · Stochastic processes and their applications, 193 (2026), p. 104847. Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint scaling regimes of…

  • Long-time behaviour of stochastic Hamilton-Jacobi equations

    Published · Journal of functional analysis, 286 (2024) 4, p. 110269. The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them…

  • Lyapunov exponents and synchronisation by noise for systems of SPDEs

    Published · The annals of probability, 52 (2024) 5, pp. 1903-1953. Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the…

  • 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…