  By observing that the fractional Caputo derivative can be expressed in terms
of a multiplicative convolution operator, we introduce and study a class of
such operators which also have the same self-similarity property as the Caputo
derivative. We proceed by identifying a subclass which is in bijection with the
set of Bernstein functions and we provide several representations of their
eigenfunctions, expressed in terms of the corresponding Bernstein function,
that generalize the Mittag-Leffler function. Each eigenfunction turns out to be
the Laplace transform of the right-inverse of a non-decreasing self-similar
Markov process associated via the so-called Lamperti mapping to this Bernstein
function. Resorting to spectral theoretical arguments, we investigate the
generalized Cauchy problems, defined with these self-similar multiplicative
convolution operators. In particular, we provide both a stochastic
representation, expressed in terms of these inverse processes and an explicit
representation, given in terms of the generalized Mittag-Leffler functions, of
the solution of these self-similar Cauchy problems.

Probability;Analysis of PDEs