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The probability distributions, as well as the mean values of stochasticcurrents and fluxes, associated with a driven Langevin process, provide a goodand topologically protected measure of how far a stochastic system is drivenout of equilibrium. By viewing a Langevin process on a compact orientedmanifold of arbitrary dimension m as a theory of a random vector fieldassociated with the environment, we are able to consider stochastic motion ofhigher-dimensional objects, which allow new observables, calledhigher-dimensional currents, to be introduced. These higher dimensionalcurrents 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 expressthe mean fluxes in terms of the solutions of the Supersymmetric Fokker-Planck(SFP), thus generalizing the corresponding well-known expressions for theconventional currents.

Stochastic Dynamics of Extended Objects in Driven Systems: I. Higher-Dimensional Currents in the Continuous Setting