Cyclic structure and dynamics are of great interest in both the fields ofstochastic processes and nonequilibrium statistical physics. In this paper, wefind a new symmetry of the Brownian motion named as the quasi-time-reversalinvariance. It turns out that such an invariance of the Brownian motion is thekey to prove the cycle symmetry for diffusion processes on the circle, whichsays that the distributions of the forming times of the forward and backwardcycles, given that the corresponding cycle is formed earlier than the other,are exactly the same. With the aid of the cycle symmetry, we prove the stronglaw of large numbers, functional central limit theorem, and large deviationprinciple for the sample circulations and net circulations of diffusionprocesses on the circle. The cycle symmetry is further applied to obtainvarious types of fluctuation theorems for the sample circulations, netcirculation, and entropy production rate.
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