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We develop the theory of discrete-time gradient flows for convex functions onAlexandrov spaces with arbitrary upper or lower curvature bounds. We employdifferent resolvent maps in the upper and lower curvature bound cases toconstruct such a flow, and show its convergence to a minimizer of the potentialfunction. We also prove a stochastic version, a generalized law of largenumbers for convex function valued random variables, which not only extendsSturm's law of large numbers on nonpositively curved spaces to arbitrary loweror upper curvature bounds, but this version seems new even in the Euclideansetting. These results generalize those in nonpositively curved spaces (partlyfor squared distance functions) due to Ba\v{c}\'ak, Jost, Sturm and others, andthe lower curvature bound case seems entirely new.

Discrete-time gradient flows and law of large numbers in Alexandrov spaces