  Von Neumann-Gale dynamical systems are defined in terms of multivalued
operators in spaces of random vectors, possessing certain properties of
convexity and homogeneity. A central role in the theory of such systems is
played by a special class of paths (trajectories) called rapid: they grow over
each time period t-1,t in a sense faster than others. The paper establishes
existence and characterization theorems for such paths showing, in particular,
that any trajectory maximizing a logarithmic functional over a finite time
horizon is rapid. The proof of this result is based on the methods of convex
analysis in spaces of measurable functions. The study is motivated by the
applications of the theory of von Neumann-Gale dynamical systems to the
modeling of capital growth in financial markets with frictions -- transaction
costs and portfolio constraints.
