A stochastic adaptive control problem is formulated and solved for some unknown linear, stochastic distributed parameter systems that are described by analytic semigroups. The control occurs on the boundary. The "highest-order" operator is assumed to be known but the "lower-order" operators contain unknown parameters. Furthermore, the linear operators of the state and the control on the boundary contain unknown parameters. The noise in the system is a cylindrical white Gaussian noise. The performance measure is an ergodic, quadratic cost functional. For the identification of the unknown parameters a diminishing excitation is used that has no effect on the ergodic cost functional but ensures sufficient excitation for strong consistency. The adaptive control is the certainty equivalence control for the ergodic, quadratic cost functional with switchings to the zero control.