Global properties of invariant measures

We study global regularity properties of invariant measures associated with second order differential operators in $\R^N$. Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds. Many local regularity properties are known for invariant measures, even under very weak conditions on the coefficients, see e.g. [MR1876411 (2002m:60117)]. On the other hand, to our knowledge the only available results dealing with global regularity are [MR1351647 (96m:28015)] and [MR1391637 (98d:60120)], which have been the starting point of our investigation. The proofs relies upon Lyapunov functions and Moser's iteration techniques