Course overview -- Wintersemester 2024/2025

In this talk, we present a new perspective on the comparison principle for viscosity solutions of Hamilton-Jacobi (HJ), HJ-Bellman, and HJ-Isaacs equations. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test-function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to both continuous and discrete operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to derive well-posedness results for partial integro-differential operators. In the context of spatially dependent Lévy operators, we show that the comparison principle is implied by a Wasserstein-contractivity property on the Lévy jump measures.

Joint work with Serena Della Corte (TU Delft), Richard Kraaij (TU Delft) and Max Nendel (University of Bielefeld)

Traditional central limit theorems (CLTs) have limited applicability to non-stationary sequences, restricting their use in real-world, dynamic data contexts. We introduce relative weak convergence, a generalisation of classical weak convergence that provides a natural framework for CLTs in non-stationary settings. Within this framework we can prove multivariate, uniform, and sequential relative CLTs for dynamic sequences under general assumptions. These results bridge a critical gap in asymptotic statistics, enabling researchers to rigorously study the asymptotic behaviour of dynamic sequences that do not satisfy traditional assumptions of stationarity.

Hopfield proposed in 1982 as a simple model capable to store and successively retrieve a number of high dimensional patterns, which represents nowadays a cornerstone of artificially intelligence (he was awarded the Nobel prize in Physics 2024 for this work). I will present some old and new mathematical results about pattern retrieval in the Hopfield model along with some open problems.