Conference Agenda

We derive quantitative estimates proving the conditional propagation of chaos for large stochastic systems of interacting particles subject to both idiosyncratic and common noise. We obtain explicit bounds on the relative entropy between the conditional Liouville equation and the stochastic Fokker--Planck equation with an bounded and square integrable interaction kernel, extending far beyond the Lipschitz case. Our method relies on reducing the problem to the idiosyncratic setting, which allows us to utilize the exponential law of large numbers.

We study stochastic differential equations with additive noise and distributional drift on $\mathbb{T}^d$ or $\mathbb{R}^d$ and $d \geqslant 2$. We work in a scaling-supercritical regime using energy solutions and recent ideas for generators of singular stochastic partial differential equations. We mainly focus on divergence-free drift, but allow for scaling-critical non-divergence free perturbations.

Roughly speaking we prove weak well-posedness of energy solutions $X$ with initial law $\mu \ll \text{Leb}$ for drift $b \in L^p_T B^{- \gamma}_{p, 1}$ with $p \in (2, \infty]$ and $p \geqslant \frac{2}{1 - \gamma}$. We can extend this by allowing a blow-up of $\| b \|_{L^p_T B^{- \gamma}_{p, 1}}$ around some space-time singularity set, but have to assume $X$ to be of a certain Hoelder regularity to make the equation well-posed. In this way we can find for any $p > 2$ a dimension $d$ and $b \notin B^{- 1}_{p, 2}$ such that weak well-posedness holds for (Hoelder-regular) energy solutions with drift $b$.

This talk is based on joint work with Nicolas Perkowski.

In particle systems, flocking refers to the phenomenon where particles’ individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behaviour. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. The simulation of the MFL equation requires solving a high dimensional PDE, motivating the derivation of reduced PDEs. In this talk, we present a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations. This is a joint work with Federico Cornalba, Natasa Djurdjevac Conrad and Ana Djurdjevac.

The absence of a Wiener-Ito chaos decomposition in general non-Gaussian analysis results in the absence of a Hida-Malliavin derivative and hence of a general Malliavin-Watanabe regularity theory. In this project we will develop a white noise calculus for processes with conditional independent increments and apply it to a the class of non-Gaussian processes represented by randomly time-changed Brownian motion. In this setting, we obtain a conditioned Wiener-Ito chaos decomposition, which allows us to introduce spaces of regular test and generalized functions w.r.t. time-changed white noise. On these spaces we define a non-Gaussian Malliavin-Watanabe regularity theory using a similar characterisation result to the Gaussian case from Grothaus et al.. We then apply the results to a stochastic transport equation with Skorokhod type noise.