  We study the random processes with non-local memory and obtain new solutions
of the Mori-Zwanzig equation describing non-markovian systems. We analyze the
system dynamics depending on the amplitudes $\nu$ and $\mu_0$ of the local and
non-local memory and pay attention to the line in the ($\nu$, $\mu_0$)-plane
separating the regions with asymptotically stationary and non-stationary
behavior. We obtain general equations for such boundaries and consider them for
three examples of the non-local memory functions. We show that there exist two
types of the boundaries with fundamentally different system dynamics. On the
boundaries of the first type, the diffusion with memory takes place, whereas on
borderlines of the second type, the phenomenon of noise-induced resonance can
be observed. A distinctive feature of noise-induced resonance in the systems
under consideration is that it occurs in the absence of an external regular
periodic force. It takes place due to the presence of frequencies in the noise
spectrum, which are close to the self-frequency of the system. We analyze also
the variance of the process and compare its behavior for regions of asymptotic
stationarity and non-stationarity, as well as for diffusive and
noise-induced-resonance borderlines between them.
