Aging Feynman–Kac equation

Aging, the process of growing old or maturing, is one of the most widely seen natural phenomena in the world. For the stochastic processes, sometimes the influence of aging can not be ignored. For example, in this paper, by analyzing the functional distribution of the trajectories of aging particles performing anomalous diffusion, we reveal that for the fraction of the occupation time $T_+/t$ of strong aging particles, $\langle (T^+(t)^2)\rangle=\frac{1}{2}t^2$ with coefficient $\frac{1}{2}$, having no relation with the aging time $t_a$ and $\alpha$ and being completely different from the case of weak (none) aging. In fact, we first build the models governing the corresponding functional distributions, i.e., the aging forward and backward Feynman-Kac equations; the above result is one of the applications of the models. Another application of the models is to solve the asymptotic behaviors of the distribution of the first passage time, $g(t_a,t)$. The striking discovery is that for weakly aging systems, $g(t_a,t)\sim t_a^{\frac{\alpha}{2}}t^{-1-\frac{\alpha}{2}}$, while for strongly aging systems, $g(t_a,t)$ behaves as $ t_a^{\alpha-1}t^{-\alpha}$.