  Stochastic methods for minimizing a convex integral functional, as initiated
by Robbins and Monro in the early 1950s, rely on the evaluation of a gradient
(or subgradient if the function is not smooth) and moving in the corresponding
direction. In contrast, we use a variational technique resulting in an implicit
stochastic minimization method, which has recently appeared in several diverse
contexts. Such an approach is desirable whenever the underlying space does not
have a differentiable structure and moreover it exhibits better stability
properties which makes it preferable even in linear spaces. Our results are
formulated in locally compact Hadamard spaces, but they are new even in
Euclidean space, the main novelty being more general growth conditions on the
functional. We verify that the assumptions of our convergence theorem are
satisfied in a few classical minimization problems.
