We will discuss some Hardy inequalities and its consequences on the
large time behavior of diffusion processes. Roughly speaking, the
Hardy inequality ensures a further and faster decay rate.
Two differente situations will be addressed. First we shall consider
the heat equation with a singular square potential located both in the
interior of the domain and on the boundary, following a joint work
with J. L. Vazquez and a more recent one with C. Cazacu.
We shall also present the main results of a recent work in
collaboration with D. Krejcirik. In which we consider the case of
twisted domains. In this case the proof of the extra decay rate
requires of important analytical developments based on the theory of
self-similar scales. As we shall see, asymptotically, the twisting
ends up breaking the tube and adds a further Dirichlet condition, wich
eventually produces the increase of the decay rate.