Talks

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale \epsilon^2 and has spatial correlation length \delta; the critical regime \epsilon=\delta corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime \epsilon = o(\delta) we prove a law of large numbers in dimensions d=2,3 : the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption \epsilon = o(\delta^{1+\ell}) for some \ell >0, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

We develop a framework in which classical rough path techniques suffice to define and solve Gaussian-driven McKean–Vlasov equations, without the additional machinery of Bailleul, Catellier and Delarue. Our approach follows the standard strategy: freeze the law, solve the resulting equation, and find a fixed point. Gaussian integration by parts provides the gain in time regularity of the law needed to close this argument, yielding short-time existence and uniqueness under suitable regularity assumptions. Joint work with Anton Baëza.

We consider a system of $N$ particles, subject to a mean-field type pairwise interaction kernel $K$, each driven by an independent fractional Brownian motion (idiosyncratic noises). Previous works established that, for a large class of non-Lipschitz, possibly singular kernels, the associated McKean-Vlasov equation is well-posed, and the empirical measure converges to its law as $N\to\infty$, with rate of order $N^{-1/2}$ in suitable negative Sobolev norms. In this talk I will present results concerning the Gaussian fluctuations underlying this mean field convergence, validating the optimality of this rate; they are valid for both first order interactions and for kinetic systems. The proofs are based on the use of Girsanov transform and the method of U-statistics first introduced by Sznitman.

Based on joint work with Avi Mayorcas (Bath) and Johanna Weinberger (Technion).

We consider the problem of solving motion planning for bracket-generating control systems by means of a gradient flow on control space. A central difficulty is the possible degeneracy of the endpoint map. In joint work with F. Suciu (JMPA, 2026), we used ideas from Malliavin calculus to show that rough initial conditions can remove the relevant degeneracies. We also obtained a concrete convergence result: for step-2 nilpotent systems, a Brownian initial condition yields almost-sure convergence of the gradient flow.

We will present recent progress toward extending this result to more general systems. A key phenomenon is that, as brackets of greater length become relevant, the initial condition must be correspondingly rougher. We will explain how this requirement is related to the energy needed to shift the underlying rough path to zero by Young translation. The existence of such translations follows from recent joint work with C. Bellingeri and H. Nouaille.

We consider nonlinear Stochastic Wave Equations (SWE) and Stochastic Damped Wave Equations (SDWE), posed on $d$-dimensional bounded domains and perturbed by additive noise that is white in time and coloured in space. Leveraging regularisation-by-noise techniques, we show pathwise wellposedness of SWE and SDWE with Hölder-continuous nonlinearities $b$ for $d \leq 4$ and (probabilistically) weak existence of solutions for distributional $b$ in all dimensions. Our approach relies on a shifted version of Lê’s stochastic sewing lemma in Hilbert spaces along with novel temporal-regularity estimates for SWEs that are uniform in the spatial variable. Finally, we prove that weak solutions of SDWE converge strongly, in the small-mass limit, to strong solutions of a corresponding Stochastic Heat Equation. Thus we establish a Smoluchowski-Kramers approximation, with explicit rate, in the presence of irregular coefficients. Ongoing joint work with O. Butkovsky (HU and WIAS Berlin) and M. Salins (Boston University).

We revisit Milstein's second-order approximation scheme of stochastic differential equations that achieves strong convergence rate 1, as opposed to the 1/2 rate of the Euler-Maruyama scheme. Milstein's proof required four derivatives from drift and diffusion coefficients, which was later reduced to two by Wagner-Platen, relying on Ito-Taylor expansions.  Here we provide a strict improvement by showing that strong convergence rate 1 holds assuming nothing else than the standard Lipschitz condition on the drift, modulo an infinitesimal loss either in the rate or in the regularity. Additionally, we prove a central limit theorem that corrects a previously wrongly stated version in the literature. Joint work with Lukas Anzeletti.

We study a time discretization scheme for the stochastic heat equation driven by space-time white noise, with a (bounded and measurable) reaction term.

Our numerical approximation is a semi-implicit Euler scheme, in which the linear term is treated implicitly while the reaction term is treated explicitly. We focus on obtaining estimates in the spirit of avoiding any future CFL-type condition on the time step, while also keeping uniformity over all the increments u_n of the scheme, so as to avoid the typical issues of fully explicit schemes. We obtain strong convergence results with an explicit rate of 1/4-, uniform over the whole sequence of increments, allowing the visualization of trajectories over long time horizons. Moreover, combining a dissipative Lipschitz drift with a singular drift (of small Besov norm), the equation is known to admit a unique invariant measure, so we provide a numerical approximation of this invariant measure.
This is joint work (also based on previous work) with Charles-Édouard Brehier, El Mehdi Haress, Jonathan Naffrichoux, and Alexandre Richard.

When approximating solutions of SPDEs, a basic challenge is that the rate of convergence is limited due to the low time and space regularity of the solution. Considering semilinear SPDEs with additive space-time white noise in space dimension d=1, we introduce a novel numerical scheme which improves the spatial convergence rate from the classical rate 1/2 to rate 3/2. The temporal convergence rate is proven to be 1 for our scheme, which enhances the classical rate 1/4 using ideas of previous works (Jentzen, Kloeden ‘08, Jentzen ’11, Djurdjevac, Kremp, Gerencser '24) . The talk is based on a work in progress together with Lukas Anzeletti and Mate Gerencser.

In this talk, we use the fourth moment theorem to investigate a possible extension of probabilistic well-posedness theory of nonlinear dispersive PDEs with random initial data beyond variance blowup. As a model equation, we study the Benjamin-Bona-Mahony equation (BBM) with Gaussian random initial data. By introducing a suitable vanishing multiplicative renormalization constant on the initial data, we show that solutions to BBM with the renormalized Gaussian random initial data beyond variance blowup converge in law to a solution to the stochastic BBM forced by the derivative of a spatial white noise. If time permits, I will briefly discuss the results on wave equation. 

Based on the joint works with Guopeng Li (BIT), Shao Liu (Bonn), Tadahiro Oh (Edinburgh) and Nikolay Tzvetkov (ENS Lyon).

Under mild regularity assumptions, we prove that the martingale problem associated with the hybrid Young-Lyons-It\^o differential equation admits a unique solution, thereby establishing probabilistic weak well-posedness. Our proof relies on analysis of the associated Kolmogorov equations, which are Young-type parabolic PDEs with irregular coefficients. The resulting theory for such Young-type parabolic PDEs is also of independent interest.

A large body of recent work on regularisation by noise has focussed on equations with singular coefficients driven by highly oscillatory, typically non-Markovian noise, with fractional Brownian motion being a canonical example. These works leverage the aphorism that rougher noise leads to improved regularisation making non-Markovianity a common (if not strictly necessary) feature. With well-posedness and stability results in hand a natural direction to pursue is long time behaviour, stationarity and ergodicity of such equations. A robust and flexible approach to ergodic theory for equations driven by non-Markovian noise was provided by Hairer ’05 which introduced the notion of stochastic dynamical systems. In this talk I will survey the relevant aspects of this theory and the significant generalisations required to make it compatible with common regularisation by noise set-ups. As a motivating example we will focus on an SDE with singular drift in the class treated by Galeati-Gerenscér `25 and a Lipschitz confining potential.

Consider a stochastic differential equation with isotropic stable additive noise and non Lipschitz drift. For an appropriate corresponding Euler like discretization scheme, we will discuss some results for  the associated weak error. We will focus in particular on the pointwise error on densities as well as on the error associated to a suitable class of test functions related to the drift. We will particularly consider drifts belonging to Hölder, Lebesgue or Besov spaces with negative regularity index in their spatial variable. We will see how choosing a test function in the same space as the drift improves the convergence rate previously obtained on the densities for Hölder or Lebesgue drifts or allows to preserve the rate for possibly singular generalized test functions (for Besov spaces with negative regularity).

In this work we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only $K\geq1$ particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity $\chi>0$. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as $K\to \infty$ by weighting the individuals by $1/K$. Then, on the microscopic level when $K$ is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter $\chi$ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we categorize the traveling fronts as pushed or pulled according to the critical parameter $chi$.

Numerical errors in approximations of differential equations can often be traced to the underlying quadrature error. When the driving signal is irregular in time, as in time-irregular ODEs, or the coefficients are irregular, as in SDEs, quadrature becomes more subtle. Randomised quadrature provides a natural framework in these settings, with a close connection to the stochastic sewing lemma for rigorously controlling approximations under irregularity. The same principle extends beyond classical ODE and SDE solvers to finite element methods, where errors can similarly be reduced to localised quadrature-type approximations and randomised variants can help handle irregular data or coefficients.