We consider the Cauchy problem $(mathbb D_{(k)} u)(t)=lambda u(t)$, $u(0)=1$, where $mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {bf 71} (2011), 583--600), $lambda >
0$. The solution is a generalization of the function $tmapsto E_alpha (lambda t^alpha)$ where $0<alpha <1$, $E_alpha$ is the Mittag-Leffler function. The asymptotics of this solution, as $tto infty$, is studied.
We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by th
e fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.
In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination
principle for the solutions to forward Kolmogorov equations. The class of subordinators for which asymptotic analysis may be realized is described.
