InvitedSpeakers

We study the stochastic Navier–Stokes equations on a 2D torus with the noise degenerated to a single Fourier mode. The aim is to complement the results by Hairer and Mattingly [1] on the uniqueness of the invariant measure for the Navier–Stokes equation. When the noise acts on the lowest Fourier mode, we prove the uniqueness of the invariant measure. We then characterise all possible 3-dimensional Galerkin approximations of the Navier–Stokes equation. For a very similar 3-dimensional system, we show the existence of nontrivial invariant measures, a result similar to Coti Zelati and Hairer [2]. More precisely, we prove that a nontrivial invariant measure exists if and only if the corresponding deterministic system admits a nontrivial fixed point. The presented results were obtained jointly with Prof. Szymon Peszat. [1] M. Hairer and J. Mattingly, Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing, Annals of Mathematics, 164 (2006), 993–1032. [2] M. Coti Zelati and M. Hairer, A noise-induced transition in the Lorenz system, Communications in Mathematical Physics, 383 (2021), 2243–2274.

We consider the Foias-Prodi estimates as a tool to investigate the asymptotic behavior of the 2D Navier-Stokes equations. First, we revise the result in the deterministic setting. Then we consider the stochastic Navier-Stokes equations. With an additive noise the uniqueness of the invariant measure by means of coupling technique has been proved by Glatt-Holtz, Mattingly and Richards (2017). This is based on pathwise Foias-Prodi estimates. We show how to modify this technique when the noise is multiplicative, requiring the Foias-Prodi estimates in expectation. Finally, if time permits, we show how to use this strategy to deal with the Continuous Data Assimilation (Azouani-Olson-Titi algorithm) to 2D stochastic Navier-Stokes equations. The talk is based on joint work with M. Zanella, and in the last part also with H. Bessaih and O. Landoulsi.

In this talk, I will discuss new results on large deviations for stochastic evolution equations. Starting with the variational setting, I will present a large deviation principle (LDP) for a class of SPDEs that includes many new examples with gradient noise and unbounded spatial domains. We will then move to non-variational settings and explore LDPs for the stochastic 3D primitive equations and reaction-diffusion equations. While our method is based on the well-known weak convergence approach, the main novel ingredients come from the theory of critical spaces and maximal regularity techniques. Based on joint work with Antonio Agresti and Mark Veraar.

We study the stochastic Burgers equation driven by a rough noise $(-\Delta)^{\gamma} dW_t$, where $\Delta$ is the Laplacian in one dimension with Dirichlet boundary conditions, and $\gamma \in [0,1/4)$. We prove exponential estimates for the solution $X_t^x$, starting from $x \in L^2(0,1)$, by showing that there exists some constant $\lambda > 0$ for which \[ \mathbb{E} \left[ \exp\left( \lambda \sup_{t \in [0,T]} \|X_t^x\|_{L^2(0,1)}^2 \right) \right] < \infty. \] This estimate was known only in the case of trace class noise when $-1/2 < \gamma < -1/4$, since in that case one can use the Itô formula. To prove the result, we combine the Boué–Dupuis method with an argument used in Da Prato–Debussche, Potential Analysis, 2007. These estimates have interesting applications. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup. This is joint work with Francesco C. De Vecchi and Josef Janák.

Sublinear expectations arise naturally in models involving volatility uncertainty. In this talk, I will discuss the existence of invariant sublinear expectations for Markov semigroups on sublinear expectation spaces, as well as their ergodic properties. We first establish a weighted nonlinear Harris–oscillation contraction theorem, which yields large-time convergence of sublinear expectation operators in a suitably metrized complete space. We then develop probabilistic and analytic tools for studying nondegenerate (G)-diffusions with weakly dissipative drifts. As an application, we prove that the associated sublinear Markov systems are ergodic in the sense of the (0,0)-regime introduced by Feng and Zhao (2021): every invariant set of the corresponding canonical dynamical system either has capacity zero or has a complement of capacity zero. The talk is based on two joint works, one with Chunrong Feng and the other with Danyang Zhao.

We will discuss preliminary results on the well-posedness of the rough Euler equations with rough transport noise in a general smooth bounded domain. Our approach is based on [Bourguignon, Brezis: Remarks on the Euler equation, J. Funct. Anal., 15:4, 1974]], that is, we investigate a Lagrangian reformulation of the Euler equations as a rough differential equation in a suitable Banach space. This is a joint work with Zdzis{l}aw Brze'{z}niak and Yuanke Chen.

We establish the strong Feller property and irreducibility for the transition semigroup associated with a class of nonlinear stochastic partial differential equations with multiplicative degenerate noise. As a by-product, we prove the uniqueness of the invariant measure. The drift of the equation diverges exactly where the noise coefficient vanishes, resulting in a competition between the dissipative effects and the degeneracy of the noise. The main idea is to introduce a mathematical method to measure the accumulation of the solution towards the potential barriers, allowing us to give rigorous meaning to the inverse of the noise operator even in the degenerate case. If the singularity of the drift and the degeneracy of the noise are suitably balanced, the dynamics are shown to stabilise for large times. From the mathematical point of view, the results provide a first generalisation of the classical work by Peszat and Zabczyk [1] to the case of degenerate multiplicative diffusions. From the application perspective, the models cover interesting scenarios in physics, particularly the evolution of relative concentrations of mixtures under the influence of thermodynamically relevant potentials of the Flory–Huggins type. The talk is based on joint work with L. Scarpa. [1] Peszat, S., Zabczyk, J.: Strong Feller property and irreducibility for diffusions on Hilbert spaces. The Annals of Probability, 157–172, (1995).

We consider an advection-diffusion model with a random Gaussian, white-in-time velocity field, in both compressible and incompressible settings. In the diffusive limit, the solution is known to converge to a Brownian motion with enhanced diffusivity. Here, we show that the fluctuations have a Gaussian distribution and satisfy an additive stochastic heat equation. Based on joint work with Sotirios Kotitsas and Dejun Luo. https://arxiv.org/pdf/2509.07956

In this talk, I will present first the location uncertainty (LU) formalism, which allows to derive stochastic interpretations of physical conservation laws systematically (such as of mass, momentum and energy). This approach typically involves transport noises, which have been widely studied by the mathematical community. Next, the LU formalism will serve to interpret the primitive equations in the stochastic context. In particular, the arising interpretation allows to infer a relaxation of the classical hydrostatic balance through the noise. Due to the new stochastic non-hydrostatic terms, well-posedness appears significantly more intricate to study, therefore these terms will be regularised using a low-pass filter. The proof of pathwise global existence and uniqueness will be presented breifly in this context. Eventually, I will discuss the link between the LU relaxed primitive and the 3D LU Navier-Stokes equations. Namely the aforementioned stochastic non-hydrostatic terms can be formally recovered from the Navier-Stokes equations, using a specific noise scaling. Moreover, I will present a few error estimates that establish the convergence of the 3D LU Navier-Stokes towards the LU primitive equations in the limit of vanishing aspect ratio.

We establish the well-posedness of SDEs and SPDEs with constraints in the Marcus canonical form driven by a Levy process. As a byproduct, we prove a Wong-Zakai type approximation theorem showing that solutions to these SDEs and SPDEs can be approximated by solutions to the corresponding equations driven by compound Levy processes. This is a joint work with Z. Brzezniak (York) and S. Peszat (Krakow).

Starting from the seminal work of H. Jia and V. Šverák (2015), it is now well understood that the existence of linearly unstable forward self-similar solutions can be used to establish non-uniqueness of Leray–Hopf solutions to the unforced incompressible 3D Navier--Stokes equations. Several works have since built upon the ideas of Jia and Šverák to prove non-uniqueness for various fluid dynamics models with non-zero forcing terms and, very recently, also for the unforced equations via computer-assisted methods. In this talk, we consider the parabolic–elliptic Keller–Segel equation in the mass-supercritical regime, in dimensions $3leq dleq 9$, and rigorously implement the Jia--Šverák mechanism to establish non-uniqueness of local mild solutions in the full range $1leq q<d/2$ of supercritical Lebesgue spaces. In particular, we give a fully rigorous, pen-and-paper proof of the spectral instability condition underlying the Jia--Šverák mechanism. This is joint work with U. Pappalettera.

I will present a framework for obtaining global-in-time strong solutions to a class of nonlinear evolution equations by introducing a suitably designed stochastic perturbation. The approach encompasses viscous and inviscid, compressible and incompressible models, including stochastically controlled versions of the three-dimensional Navier–Stokes and Euler equations and the rotating shallow water equations. The perturbation takes the form of superlinear multiplicative Itô noise, with an amplitude depending on a suitable norm of the solution. Its stabilising action is a second-order effect: applying a scale function to the solution norm reveals a contribution from the quadratic variation that counteracts nonlinear growth. A central analytical difficulty is that the stochastic integrals need only be local martingales, and the solution norm is not guaranteed to have finite moments. I will also discuss a selective control strategy in which the noise is activated only when the solution norm reaches a prescribed upper threshold and switched off once it returns to a lower level. This allows the system to evolve according to its original deterministic dynamics between stochastic interventions. This talk is based on joint work with Oana Lang and is based on the paper: Lang, O., Crisan, D. Global solutions for stochastically controlled fluid dynamics models. Stoch PDE: Anal Comp (2025).

We study the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Under sufficiently strong damping, we establish a unique invariant measure and exponential mixing for the associated discrete time Markov semigroup. The key ingredient is a global in time $W^{1,infty}$ estimate for the vorticity, which yields tightness for the construction of an invariant measure and enable a synchronous argument for exponential convergence to equilibrium. We also obtain a corresponding result for sufficiently small noise at any fixed positive damping. This is joint work with Chunrong Feng and Huaizhong Zhao.

We show that a bounded H00 calculus is not enough to ensure stochastic maximal regularity for the generator A of an analytic semigroup on a UMD space with type 2. The problem even occurs for spaces X such as L_2(L_r) or L_2 + L_r for r>2 and A the Laplace operator L_p(R,X). These are joint results with E. Lorist, J van Neerven and M. Veraar from Delft University of Technology.

Abstract: The talk is based on joint work with Zdzisław Brzeźniak. First, we introduce several classes of SDEs on $mathbb{R}^{d+1}$ preserving the sphere $mathbb{S}^d$, whose restrictions to $mathbb{S}^d$ are Markov processes generated by the Laplace-Beltrami operator $Delta_{mathbb{S}^d}$. Then, we are interested in the jump diffusions preserving $mathbb{S}^d$. We argue that they should be considered in the Marcus canonical form. Special emphasis will be placed on rotationally invariant diffusions. In this case, we will calculate the generator kernels.

We study invariant measures for three-dimensional stochastic primitive-equation models driven by projected horizontal Stratonovich transport noise.  We assume that the transport field is horizontal, independent of the vertical variable, and horizontally divergence-free.  Under this structure the projected noise is skew-adjoint in the hydrostatic $L^2$ space; after taking the baroclinic projection or one vertical derivative, the stochastic hydrostatic pressure disappears and the transport becomes an exact divergence-free derivation.  Consequently the high-power baroclinic and vertical-derivative balances have exact Stratonovich cancellations. Adapting the logarithmic strong-moment mechanism of Glatt-Holtz--Kukavica--Vicol--Ziane, we obtain a time-averaged logarithmic $H^2$ bound.  A localized continuous-dependence estimate yields the Feller property on the mean-zero $H^1$ phase space, and the Krylov--Bogoliubov procedure then gives an invariant probability measure.

We consider the three-dimensional incompressible Navier–Stokes equations on the torus driven by a finite family of high-frequency pseudo-transport noises. We prove that arbitrary fixed initial data at the scaling-critical (H^{1/2}) regularity can be globally regularized with arbitrarily high probability by sufficiently strong, high-frequency pseudo-transport noise. The proof relies on a high-frequency scaling limit in the critical space. The Itô corrector converges to an additional fractional dissipation, while the stochastic contribution vanishes in a global-in-time critical norm. This yields convergence to a deterministic equation with enhanced dissipation, whose global well-posedness for sufficiently large noise intensity leads to the high-probability global regularity result.

We consider the defocusing stochastic Hartree equation with finite-dimensional linear multiplicative noise in the mass- and energy-critical cases in dimensions . The proof is based on a Doss–Sussmann-type transformation, which converts the stochastic equation into a random dispersive equation with lower-order coefficients. We establish local well-posedness and a blow-up alternative in for the mass-critical case and in for the energy-critical case, using random Strichartz estimates and Hartree-type nonlinear estimates. Global well-posedness is then obtained through stability arguments on random time intervals, treating the transformed equation as a perturbation of the corresponding deterministic critical Hartree equation.