One-side Liouville theorems under an exponential growth condition for Kolmogorov operators (2405.03410v1)
Abstract: It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator $$ L= \frac{1}{2}\text{ tr} (QD2 ) + \langle Ax, D \rangle = \frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in RN, $$ 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., $s(A) \le 0)$. We show that if $Q$ is positive definite and $s(A) \le 0$, 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.
- A. E. Kogoj and E. Lanconelli. Liouville theorems for a class of linear second-order operators with non-negative characteristic form. Bound. Value Probl., Art. ID 48232, 16 pages, 2007.
- L. P. Kupcov. The fundamental solutions of a certain class of elliptic-parabolic second order equations. Differ. Uravn., 8:1649–1660, 1716, 1972.
- E. Lanconelli and S. Polidoro. On a class of hypoelliptic evolution operators. Rend. Sem. Mat. Univ. Politec. Torino, 52(1):29–63, 1994. Partial differential equations, II (Turin, 1993).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Collections
Sign up for free to add this paper to one or more collections.
