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We consider an overdamped Brownian particle moving in a confiningasymptotically logarithmic potential, which supports a normalized Boltzmannequilibrium density. We derive analytical expressions for the two-timecorrelation function and the fluctuations of the time-averaged position of theparticle for large but finite times. We characterize the occurrence of agingand nonergodic behavior as a function of the depth of the potential, andsupport our predictions with extensive Langevin simulations. While theBoltzmann measure is used to obtain stationary correlation functions, we showhow the non-normalizable infinite covariant density is related to thesuper-aging behavior.

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials