论文标题
Harnack不平等和liouville型定理，用于Ornstein-Uhlenbeck和Kolmogorov操作员
Harnack inequality and Liouville-type theorems for Ornstein-Uhlenbeck and Kolmogorov operators
论文作者
论文摘要
我们用纯粹的分析技术证明了一个单方面的liouville定理，用于Ornstein-uhlenbeck操作员$ {\ Mathcal l_0} $ in $ \ m arthbb {r}^n $， 相应的Kolmogorov操作员$ {\ Mathcal l_0} - \ partial_t $ in $ \ Mathbb {r}^{n+1} $ in liouville定理at“ $ t = - \ infty $”。反过来，最后结果被证明是针对$（{\ Mathcal l_0} - \ partial_t）u = 0 $的全球harnack不平等的推论，这似乎对其本身具有独立的利益。 我们强调的是，如果$ n> 2 $，则无法通过基于复发的概率方法获得$ {\ mathcal l_0} $的liouville定理。 我们提供了一个涉及附录中的uhlenbeck随机过程的liouville定理的独立证明。
We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at "$t=- \infty$" for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.
