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

Abstract

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck (OU) operator L=12tr(QD2)+⟨Ax,D⟩=12div(QD)+⟨Ax,D⟩,x∈RN,L=\frac{1}{2}\hspace{0.1em}\text{tr}\hspace{0.1em}\left(Q{D}^{2})+\langle Ax,D\rangle =\frac{1}{2}\hspace{0.1em}\text{div}\hspace{0.1em}\left(QD)+\langle Ax,D\rangle ,\hspace{0.33em}\hspace{0.33em}x\in {{\mathbb{R}}}^{N}, all (globally) bounded solutions of Lu=0Lu=0 on RN{{\mathbb{R}}}^{N} are constant if and only if all the eigenvalues of AA have non-positive real parts (i.e., s(A)≤0s\left(A)\le 0). We show that if QQ is positive definite and s(A)≤0s\left(A)\le 0, then any non-negative solution vv of Lv=0Lv=0 on RN{{\mathbb{R}}}^{N}, 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 AA. We also prove a related one-side Liouville theorem in the case of hypoelliptic OU operators.

Keywords

• one-side liouville theorems
• ornstein-uhlenbeck operators
• kolmogorov operators
• 31b05
• 47d07
• 60h30