  We study a linear quadratic problem for a system governed by the heat
equation on a halfline with Dirichlet boundary control and Dirichlet boundary
noise. We show that this problem can be reformulated as a stochastic evolution
equation in a certain weighted L2 space. An appropriate choice of weight allows
us to prove a stronger regularity for the boundary terms appearing in the
infinite dimensional state equation. The direct solution of the Riccati
equation related to the associated non-stochastic problem is used to find the
solution of the problem in feedback form and to write the value function of the
problem.
