We provide sharp bounds in the supremum- and Hölder norm for parameter-dependent stochastic integrals. As an application we obtain novel long-term bounds for stochastic partial differential equations as well as novel bounds on the space-time modulus of continuity of the stochastic heat equation. This concerns joint work with Joris van Winden (TU Delft).

The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information as a duality pairing between $H^{-1}$ and $H^{1}$, which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions. This is a joint work with J. A. Carrillo and J. Warnett. 

The Riemann problem is an IVP having simple piecewise constant initial data that is invariant under scaling. In 1D, the problem was originally considered by Riemann during the 19th century in the context of gas dynamics, and the general theory was more or less completed by Lax and Glimm in the mid-20th century. In 2D and MD, the situation is much more complicated, and very few analytic results are available. We discuss a shock reflection problem for the Euler equations for potential flow, with initial data that generates four interacting shockwaves. After reformulating the problem as a free boundary problem for a nonlinear PDE of mixed hyperbolic-elliptic type, the problem is solved via a sophisticated iteration procedure. The talk is based on joint work with G-Q Chen (Oxford) et. al. arXiv:2305.15224, to appear in JEMS (2025).

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. 

Minimal discrete energy problems arise in a variety of scientific contexts – such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few.     Our goal is to analyze the structure of configurations generated by optimal (and near optimal)-point configurations that minimize the Riesz s-energy over a sphere in Euclidean space R^d and, more generally, over a bounded manifold. The Riesz s-energy potential, which is a generalization of the Coulomb potential, is simply given by 1/r^s, where r denotes the distance between pairs of points. We show how such potentials for s>d and their minimizing point configurations are ideal for use in sampling surfaces.

Connections to the results by Field's medalist M. Viazovska and her collaborators on best-packing and universal optimality in 8 and 24 dimensions will be discussed. Finally we analyze the minimization of a "k-nearest neighbor" truncated version of Riesz energy that reduces the order N^2 computation for energy minimization to order N log N , while preserving global and local properties.