Holomorphic regularity of processes generated by the heat equation with white noise boundary data (joint work with Sorin Micu and Ionel Roventa)

We begin by recalling some recent results for the one dimensional heat equation with non-homogeneous Dirichlet or Neuman boundary conditions. We give in particular the smallest Hilbert space $V$ for which all the solutions with boundary data in $L^2(0,\infty)$ are continuous in time with values in $V$. It turns out that $V$ is a Hilbert space of analytic functions (of Bergman type).

Our main new results concern the heat equation on a bounded interval, say $[0,\pi]$, with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints. Classical results show that such systems admit solutions only in Sobolev spaces of negative order or in weighted $L^p$ spaces. In contrast, we prove that the solution process takes values in a Hilbert space of holomorphic functions of weighted Bergman type. This means that for each positive time the solution (apriori defined on $[0,\pi]$) can be extended holomorphically to a rhombus in the complex plane having $[0,\pi]$ as one of its diagonals. Moreover, we show that this solution admits a version that is continuous in time with values in weighted Bergman spaces of functions defined this rhombus. These Bergman spaces form a scale depending on two parameters $\delta\in (0,1)$ and $\Theta\in \left(0,\frac{\pi}{4}\right)$. Our results, besides identifying natural state spaces of holomorphic functions, reveal a sharp threshold phenomenon: the result fails at the critical parameters $\delta=0$ and $\Theta=\pi/4$. Our approach combines admissibility and reachability theories for boundary controlled systems with tools from the theory of holomorphic function spaces, thereby extending to the stochastic setting the reachable-space framework previously developed for deterministic heat equations.