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We consider stochastic motion of a particle on a cyclic graph witharbitrarily periodic time dependent kinetic rates. We demonstrate dualityrelations for statistics of currents in this model and in its continuousversion of a diffusion in one dimension. Our duality relations are valid beyonddetailed balance constraints and lead to exact expressions that relatestatistics of currents induced by dual driving protocols. We also show thatpreviously known no-pumping theorems and some of the fluctuation relations,when they are applied to cyclic graphs or to one dimensional diffusion, arespecial consequences of our duality.

Duality and fluctuation relations for statistics of currents on cyclic graphs