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Harnack inequality and Liouville-type theorems for Ornstein-Uhlenbeck and Kolmogorov operators

1 Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Piazza della Repubblica 13, 61029 Urbino (PU), Italy
2 Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
3 Dipartimento di Matematica, Università degli Studi di Pavia, Via Adolfo Ferrata 5, 27100 Pavia, Italy

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: https://www.aimspress.com/newsinfo/1429.html

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

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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Keywords Liouville theorems; Harnack inequalities; Kolmogorov operators; Ornstein–Uhlenbeck operators; mean value formulae

Citation: Alessia E. Kogoj, Ermanno Lanconelli, Enrico Priola. Harnack inequality and Liouville-type theorems for Ornstein-Uhlenbeck and Kolmogorov operators. Mathematics in Engineering, 2020, 2(4): 680-697. doi: 10.3934/mine.2020031

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