The heat equation is well posed in Bergman spaces

Abstract

We show that the one-dimensional heat equation with boundary inputs admits a natural well-posed formulation in Hilbert spaces of holomorphic functions. In the deterministic case, the minimal state space is a Bergman space on a rhombus adapted to the interval. For boundary white-noise forcing, weighted Bergman spaces yield almost sure continuity of the mild solution. The approach connects reachability, semigroup theory, complex analysis, and stochastic PDEs.