  Since its introduction, some sixty years ago, the Montroll-Weiss continuous
time random walk has found numerous applications due its ease of use and
ability to describe both regular and anomalous diffusion. Yet, despite its
broad applicability and generality, the model cannot account for effects coming
from random diffusivity fluctuations which have been observed in the motion of
asset prices and molecules. To bridge this gap, we introduce a doubly
stochastic version of the model in which waiting times between jumps are
replaced with a fluctuating jump rate. We show that this newly added layer of
randomness gives rise to a rich phenomenology while keeping the model fully
tractable -- allowing us to explore general properties and illustrate them with
examples. In particular, we show that the model presented herein provides an
alternative pathway to Brownian yet non-Gaussian diffusion which has been
observed and explained via diffusing diffusivity approaches.
