  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.
