Martingale Structure for General Thermodynamic Functionals of Diffusion Processes Under Second-Order Averaging

Novel hidden thermodynamic structures have recently been uncovered during the investigation of nonequilibrium thermodynamics for multiscale stochastic processes. Here we reveal martingale structure a general functional inhomogeneous singularly perturbed diffusion processes under second-order averaging, where is defined as logarithmic Radon–Nykodim derivative between laws original process and comparable (forward case) or its time reversal (backward case). In forward case, prove that regular anomalous parts are orthogonal martingales. backward while part may not be martingale, still martingale. With aid structure, integral fluctuation theorem satisfied by functional. Further extensions applications to also discussed, including theorems entropy production housekeeping heat in absence presence odd variables.