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

2020/02/11 by Alessia E. Kogoj, Kogoj, Alessia E., Ermanno Lanconelli +3
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Spectral Theory in Mathematical Physics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2002.04718

Abstract

We prove, with a purely analytic technique, a one-side Liouville theorem for\na class of Ornstein--Uhlenbeck operators mathcal L₀ in \ℝ^N, as\na consequence of a\n Liouville theorem at "t=- \∞" for the corresponding Kolmogorov\noperators mathcal L₀ - \∂_t in \ℝ^N+1. In turn, this\nlast result is proved as a corollary of a global Harnack inequality for\nnon-negative solutions to ( mathcal L₀ - \∂_t) u = 0 which seems to\nhave an independent interest in its own right.\n We stress that our Liouville theorem for mathcal L₀ cannot be obtained\nby a probabilistic approach based on recurrence if N>2.\n We provide a self-contained proof of a Liouville theorem involving recurrent\nOrnstein--Uhlenbeck stochastic processes in the Appendix.\n

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