On a long standing conjecture: positive Liouville Theorem for hypoelliptic Ornstein-Uhlenbeck operators

The linear second order partial differential operator in R^n L = div (A D ) + < B x, D>, where A and B are n x n matrices with constant real entries with A non-negative definite, is usually called the finite dimensional Ornstein-Uhlenbeck operator related to the pair (A, B). This possibly degenerate operator is hypoelliptic if and only if rank [ Q, BQ, .... B^{n-1} Q] = n, being Q the square root of A. This condition, in fact, is equivalent to the celebrated Hormander rank condition for L. In 2004 Priola and Zabczyk proved the following Liouville-type Theorem: if L is hypoelliptic, every bounded solution to Lu = 0 in R^n is constant if and only if (*) each eigenvalue of B has real part less than or equal to zero. This remarkable result raised the following problem, which is still not completely solved today. (PB) Suppose L hypoelliptic and assume condition (*) is satisfied. Then, is it true that every non-negative solution to Lu=0 in R^n is constant? Here, together with a survey on the state of the art about this problem, we present some recent new results obtained in collaboration with Alessia E. Kogoj and Giulio Tralli.