Recent applications of absolute stability methods to robustness analysis have led to a rebirth of interest in this topic. This paper is concerned with the robust stability of feedback distributed parameter systems against nonlinear and random disturbances. Our analysis is based on the Lyapunov direct method. To derive a stochastic Lyapunov function, we introduce the special stochastic infinite-dimensional counterparts of the Kalman-Yakubovich lemma. They are proved by use of dynamic programming methods in combination with the properties of Hilbert-space-valued Wiener processes. Examples exhibit extensions of the Popov and circle criteria to the feedback stochastic heat and delay equations, respectively.