  The study of optimal control problems under uncertainty plays an important
role in scientific numerical simulations. This class of optimization problems
is strongly utilized in engineering, biology and finance. In this paper, a
stochastic gradient method is proposed for the numerical resolution of a
nonconvex stochastic optimization problem on a Hilbert space. We show that,
under suitable assumptions, strong or weak accumulation points of the iterates
produced by the method converge almost surely to stationary points of the
original optimization problem. Measurability and convergence rates of a
stationarity measure are handled, filling a gap for applications to nonconvex
infinite dimensional stochastic optimization problems. The method is
demonstrated on an optimal control problem constrained by a class of elliptic
semilinear partial differential equations (PDEs) under uncertainty.
