  In this article integro-differential Volterra equations whose convolution
kernel depends on the vector variable are considered and a connection of these
equations with a class of semi-Markov processes is established. The variable
order $\alpha(x)$-fractional diffusion equation is a particular case of our
analysis and it turns out that it is associated with a suitable
(non-independent) time-change of the Brownian motion. The resulting process is
semi-Markovian and its paths have intervals of constancy, as it happens for the
delayed Brownian motion, suitable to model trapping effects induced by the
medium. However in our scenario the interval of constancy may be position
dependent and this means traps of space-varying depth as it happens in a
disordered medium. The strength of the trapping is investigated by means of the
asymptotic behaviour of the process: it is proved that, under some technical
assumptions on $\alpha(x)$, traps make the process non-diffusive in the sense
that it spends a negligible amount of time out of a neighborhood of the region
$\text{argmin}(\alpha(x))$ to which it converges in probability under some more
restrictive hypotheses on $\alpha(x)$.

Probability