The aim of this paper is to give a stochastic representation for the solution
to a natural extension of the Caputo-type evolution equation. The
nonlocal-in-time operator is defined by a hypersingular integral with a
(possibly time-dependent) kernel function, and it results in a model which
serves a bridge between normal diffusion and anomalous diffusion. We derive the
stochastic representation for the weak solution of the nonlocal-in-time problem
in case of nonsmooth data. We do so by starting from an auxiliary Caputo-type
evolution equation with a specific forcing term. Numerical simulations are also
provided to support our theoretical results.