| Abstract: |
| We consider the heat equation on a bounded interval with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints. Although boundary white-noise forcing is usually associated with very low regularity, we show that the corresponding stochastic evolution in fact exhibits a sharp holomorphic regularity. More precisely, for every positive time the solution extends holomorphically to a rhombus in the complex plane having the original interval as one of its diagonals, and the resulting process admits a continuous version with values in a scale of weighted Bergman spaces on that domain.
We determine the precise range of parameters for which this phenomenon holds, indexed by the parameters $\delta\in(0,1)$ and $\Theta\in\left(0,\frac{\pi}{4}\right)$, and prove that it is optimal: the conclusion fails at the critical values $\delta=0$ and $\Theta=\frac{\pi}{4}$.
Our approach combines recent advances on reachability theory for boundary-controlled systems with techniques from Bergman-space theory. In this way, we extend to the stochastic setting the sharp regularity of time continuous trajectories recently developed for deterministic heat equations. |
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