  We analyze the influence of reflective boundary conditions on the statistics
of Poisson-Kac diffusion processes, and specifically how they modify the
Poissonian switching-time statistics. After addressing simple cases such as
diffusion in a channel, and the switching statistics in the presence of a
polarization potential, we thoroughly study Poisson-Kac diffusion in fractal
domains. Diffusion in fractal spaces highlights neatly how the modification in
the switching-time statistics associated with reflections against a complex and
fractal boundary induces new emergent features of Poisson-Kac diffusion leading
to a transition from a regular behavior at shorter timescales to emerging
anomalous diffusion properties controlled by walk dimensionality of the fractal
set.

Statistical Mechanics