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Non-equilibrium work relation beyond Boltzmann-Gibbs distribution

The presence of multiplicative noise can alter measurements of forces acting on nanoscopic objects. Taking into account of multiplicative noise, we derive a series of non-equilibrium thermodynamical equalities as generalization of the Jarzynski equality, the detailed fluctuation theorem and the Hatano-Sasa's relation. Our result demonstrates that the Jarzynski equality and the detailed fluctuation theorem remains valid only for systems with the Boltzmann-Gibbs distribution at the equilibrium state, but the Hatano-Sasa's relation is robust with respect to the ambiguity on multiplicative noise.