We study the effective behavior of a Brownian motion in both
one and two dimensional comb like domains. This problem arises in a
variety of physical situations such as transport in tissues, and linear
porous media. We show convergence to a limiting process when both
the spacing between the teeth, and the probability of entering a tooth
vanish at the same rate. This limiting process exhibits an anomalous
diffusive behavior, and can be described as a Brownian motion
time-changed by the local time of an independent sticky Brownian motion.
At the PDE level, this leads to equations that have fractional time
derivatives and are similar to the Bassett differential equation.