The Robinson-Schensted (RS) correspondence and its variants naturally give rise to integrable dynamics of non-intersecting particle systems. In previous work, the author exhibited a RS correspondence for geometric crystals by constructing a Littelmann path model, in general Lie type. Since this representation-theoretic map takes as input a continuous path on a Euclidian space, the natural starting measure is the Wiener measure. In this paper, we characterize the measures induced by Brownian motion through the RS map. On the one hand, the highest weight in the output is a remarkable Markov process (the Whittaker process), and can be interpreted as a weakly non-intersecting particle system which deforms Brownian motion in a Weyl chamber. One the other hand, the measure induced on geometric crystals is given by the Landau-Ginzburg potential for complete flag manifolds which appear in mirror symmetry. The measure deforms the uniform measure on string polytopes. Whittaker functions, which appear as volumes of geometric crystals, play the role of characters in the theory.