Title: Aging Feynman-Kac Equation,
Abstract: 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}$.