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Register a new userWiener-Landis criterion for Kolmogorov-type operators
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We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepys Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
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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.
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