In this second edition of the summer school on PDEs and Randomness, three speakers will present introductions to their areas of research at the interface of PDEs and probability:

Teaching Assistant: Ran Tao (MPI for Mathematics in the Sciences) rantao16.github.io

Title: An Analytic Perspective on KPZ: From Homogenization to Fluctuations

Abstract: The KPZ equation is a stochastic PDE modeling surface growth under random perturbations. Over the past quarter century, there has been tremendous progress in understanding this equation and the associated universality class. In this series of lectures, I will present several recent developments from a stochastic analytic perspective, with a focus on homogenization, fluctuations, and invariant measures.

Teaching Assistant: Pascal Hanigk (Universität Mainz) www.stochastik.mathematik.uni-mainz.de/pascal-hanigk/

Title: F-KPP equations, Feynman-Kac formulas, and spatial branching processes

Abstract: In this minicourse, I will explain how Feynman-Kac formulas can be used to solve Fisher-Kolmogorov-Petrovsky-Pikunov equations (F-KPP). Maury Bramson first used this approach in his seminal paper on the F-KPP equation about 50 years ago. We will revisit his approach and then also apply this technique to systems of F-KPP equations. Moreover, I will explain the duality between (certain) F-KPP equations and spatial branching processes (such as branching Brownian motion).

Teaching Assistant: Sophie Mildenberger (Universität Münster) www.uni-muenster.de/AMM/weber/mitarbeiter/team/Mildenberger/index.html

Title: Stochastic PDEs and Weak KPZ Universality

Abstract: The KPZ equation plays a fundamental role in the description of stochastic interface growth and is conjectured to arise as the universal scaling limit of many one-dimensional interacting particle systems. This so-called weak universality conjecture has been confirmed in a number of cases for microscopic models that satisfy strong symmetry assumptions, but the general case remains a major challenge. This mini-course presents a new approach to this problem based on the theory of singular stochastic PDEs. After introducing the KPZ equation and reviewing the main ideas of Hairer’s theory of regularity structures, I will explain how these techniques can be adapted to discrete systems. A key step is the development of a regularity-structure framework directly on the lattice, together with stochastic estimates for iterated martingales arising from particle systems. These ideas lead to a robust method for deriving the KPZ equation from microscopic dynamics. As an application, I will outline how this approach yields KPZ scaling limits for a large class of weakly asymmetric particle systems, substantially extending known results.

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