# Probability and Mathematical Physics Seminar

#### Scaling limits for random growth driven by reflecting Brownian motion

**Speaker:**
Kevin Yang, Harvard University

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, February 16, 2024, 11:10 a.m.

**Synopsis:**

We discuss long-time asymptotics for a continuum version of origin-excited random walk. It is a growing submanifold in Euclidean space that is pushed outward from within by the boundary trace of a reflecting Brownian motion. We show that the leading-order behavior of the submanifold process is described by a flow-type PDE whose blow-ups correspond to changes in diffeomorphism class of the growth process. We then show that if we simultaneously smooth the submanifold as it grows, fluctuations of an associated height function are described by a regularized KPZ equation with noise modulated by a Dirichlet-to-Neumann operator. If the dimension of the manifold is 2, we show well-posedness of the singular limit of this regularized KPZ-type equation. Based on joint work with Amir Dembo.