One-side Liouville theorems under an exponential growth condition for Kolmogorov operators

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator L= 1/2 tr (QD² ) + Ax, D = 1/2 div (Q D ) + Ax, D ,\;\; x ∈ RN, all (globally) bounded solutions of $Lu=0$ on RN are constant if and only if all the eigenvalues of A have non-positive real parts (i.e., s(A) ≤ 0). We show that if Q is positive definite and s(A) ≤ 0, then any non-negative solution v of $Lv=0$ on RN which has at most an exponential growth is indeed constant. Thus under a non-degeneracy condition we relax the boundedness assumption on the harmonic functions and maintain the sharp condition on the eigenvalues of A. We also prove a related one-side Liouville theorem in the case of hypoelliptic Ornstein-Uhlenbeck operators.