  The probability distributions, as well as the mean values of stochastic
currents and fluxes, associated with a driven Langevin process, provide a good
and topologically protected measure of how far a stochastic system is driven
out of equilibrium. By viewing a Langevin process on a compact oriented
manifold of arbitrary dimension m as a theory of a random vector field
associated with the environment, we are able to consider stochastic motion of
higher-dimensional objects, which allow new observables, called
higher-dimensional currents, to be introduced. These higher dimensional
currents arise by counting intersections of a k-dimensional trajectory,
produced by a evolving (k-1)-dimensional cycle, with a reference cross section,
represented by a cycle of complimentary dimension (m - k). We further express
the mean fluxes in terms of the solutions of the Supersymmetric Fokker-Planck
(SFP), thus generalizing the corresponding well-known expressions for the
conventional currents.
