Conference Agenda

We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay between the distance towards the bifurcation and the noise intensity.

Our approach relies on a reduction of the infinite-dimensional dynamics to a finite-dimensional stochastic differential equation describing the dominant modes. This allows to carry over results for FTLE from the finite to the infinite dimensional setting. A technical challenge is to provide a suitable control of the nonlinear terms coupling the dominant and stable modes of the SPDE and of the corresponding linearization.

We illustrate the theory by applying it to the stochastic Burgers equation

We study a stochastic optimal control problem for McKean-Vlasov stochastic differential equations (SDEs) with common noise, where the dynamics depend on the conditional law of the state. We derive a stochastic maximum principle of Peng type without imposing convexity assumptions on the control domain. In comparison to the standard McKean-Vlasov case, the maximum principle for the common noise case contains a third adjoint state introduced in [1], which is needed to dualize all second-order Lions derivatives in the Taylor expansion of the cost functional. This additional adjoint state is given by a conditional McKean-Vlasov backward SDE. All three adjoint states together allow for a complete linearization of all contributions in the second-order expansion. As part of our analysis, we also prove a general well-posedness result for conditional McKean-Vlasov backward SDEs.

The talk is based on joint work with J. Spille (TU Berlin).

[1] J. Spille, W. Stannat: A Novel Approach to Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations, arXiv:2602.12006

[2] J. Spille, W. Stannat: Peng's Maximum Principle for McKean-Vlasov Stochastic Differential Equations with Common Noise, arXiv:2606.06193

It is known that global weak solutions to the stochastic 3D Navier-Stokes equations with various types of multiplicative noise exist. Capturing their regularity is one of the most fundamental open problems in fluid dynamics, even in the absence of noise. The set of singularities can be of a very intricate nature, even of fractal type. Partial regularity aims to provide sharp estimates on the fractal dimension (e.g., Hausdorff) for such sets. In this talk, we will discuss a way to estimate the set of singular times, that is, times at which a weak solution is not regular. In particular, we extend to the stochastic setting the well-known 1/2-bound on the fractal dimension of singular times, which goes back to the works by Leray and Scheffer. Interestingly, the bound is independent of the roughness of the noise. Our viewpoint is new even in the deterministic case, and can be applied to a wide class of stochastic PDEs, e.g., reaction-diffusion equations.

We introduce a carefully constructed stochastic perturbation whose diffusion coefficient grows super-linearly with the solution norm. The noise is designed to act selectively, becoming effective as the norm approaches potential blow-up, while its quadratic variation generates a stabilizing second-order mechanism. Under suitable structural assumptions on the drift term, we show that the resulting SPDE admits global strong solutions almost surely across three levels of initial regularity. In particular, the stochastic control prevents finite-time blow-up and extends the lifespan of solutions to all times, including regimes where global deterministic well-posedness remains open. This is joint work with Dan Crisan, based on the paper: Lang, O., Crisan, D. Global solutions for stochastically controlled fluid dynamics models. Stoch PDE: Anal Comp (2025).