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We consider continuous time random walks (CTRW) and discuss situationspertinent to aging. These correspond to the case when the initial state of thesystem is known not at preparation (at $t=0$) but at the later instant of time$t_1>0$ (intermediate-time initial condition). We derive the generalized agingdiffusion equation for this case and express it through a single memory kernel.The results obtained are applied to the practically relevant case of theequilibrated random walks. We moreover discuss some subtleties in the setup ofthe aging subdiffusion problem and show that the behavior of the system dependson what was taken as the intermediate-time initial condition: whether it wascoordinate of one particle given by measurement or the whole probabilitydistribution. The two setups lead to different predictions for the evolution ofa system. This fact stresses the necessity of a precise definition of agingstatistical ensembles.

Fractional diffusion equation for aging and equilibrated random walks