Boundary white noise in the 1D heat equation yields holomorphic extensions to a rhombus and continuous versions in weighted Bergman spaces, with the result shown to be optimal.
Fattorini,Reachable states in boundary control of the heat equation are independent of time, Proceedings of the Royal Society of Edinburgh Section A: Mathematics81(1978), no
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Bergman-space regularity for the heat equation with white-noise boundary forcing
Boundary white noise in the 1D heat equation yields holomorphic extensions to a rhombus and continuous versions in weighted Bergman spaces, with the result shown to be optimal.