  We construct classes of stochastic differential equations with fluctuating
friction forces that generate a dynamics correctly described by Tsallis
statistics and nonextensive statistical mechanics. These systems generalize the
way in which ordinary Langevin equations underly ordinary statistical mechanics
to the more general nonextensive case. As a main example, we construct a
dynamical model of velocity fluctuations in a turbulent flow, which generates
probability densities that very well fit experimentally measured probability
densities in Eulerian and Lagrangian turbulence. Our approach provides a
dynamical reason why many physical systems with fluctuations in temperature or
energy dissipation rate are correctly described by Tsallis statistics.
