Infinite densities in continuous-time random walks

I discuss the occupation time statistics in thermal, ergodic continuous-time random walks. While the average occupation time is given by the canonical Boltzmann-Gibbs law from Statistical Physics, the finite-time fluctuations around this mean turn out to be large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations.

Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas.

Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and stable densities of fluctuations.