  We consider possibly degenerate parabolic operators in the form $$
  \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated
to a suitable family of stochastic differential equations, and satisfying the
H\"ormander condition. Note that, under this assumption, the operators in the
form $\L$ has a smooth fundamental solution that agrees with the density of the
corresponding stochastic process. We describe a method based on Harnack
inequalities and on the construction of Harnack chains to prove lower bounds
for the fundamental solution. We also briefly discuss PDE and SDE methods to
prove analogous upper bounds. We eventually give a list of meaningful examples
of operators to which the method applies.
