Harnack inequalities and Bounds for Densities of Stochastic Processes

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.