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

Alessia E. Kogoj; Ermanno Lanconelli; Enrico Priola; Alessia E. Kogoj; Ermanno Lanconelli; Enrico Priola

doi:10.3934/mine.2020031

Mathematics in Engineering

2020, Volume 2, Issue 4: 680-697. doi: 10.3934/mine.2020031

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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

Received: 09 February 2020 Accepted: 14 April 2020 Published: 27 May 2020

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.
- 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[J]. Mathematics in Engineering, 2020, 2(4): 680-697. doi: 10.3934/mine.2020031

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Abstract

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.

References

[1]	Bonfiglioli A, Lanconelli E, Uguzzoni F (2007) Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Berlin: Springer.
[2]	Cupini G, Lanconelli E (2020) On mean value formulas for solutions to second order linear PDEs. Ann Scuola Norm Sci, in press.
[3]	Cranston M, Orey S, Rösler U (1983) The Martin boundary of two-dimensional Ornstein-Uhlenbeck processes, In: Probability, Statistics and Analysis, Cambridge-New York: Cambridge University Press, 63-78.
[4]	Da Prato G, Zabczyk J (1996) Ergodicity for Infinite-Dimensional Systems, Cambridge: Cambridge University Press.
[5]	Dym H (1966) Stationary measures for the flow of a linear differential equation driven by white noise. T Am Math Soc 123: 130-164. doi: 10.1090/S0002-9947-1966-0198541-2
[6]	Dynkin EB (1965) Markov Processes Vols. I & II, Berlin-Göttingen-Heidelberg: Springer-Verlag.
[7]	Erickson RV (1971) Constant coefficient linear differential equations driven by white noise. Ann Math Statist 42: 820-823. doi: 10.1214/aoms/1177693440
[8]	Garofalo N, Lanconelli E (1989) Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients. Math Ann 283: 211-239. doi: 10.1007/BF01446432
[9]	Getoor RK (1980) Transience and recurrence of Markov processes, In: Seminar on Probability, XIV (Paris, 1978/1979) (French), Berlin: Springer, 397-409.
[10]	Kogoj AE, Lanconelli E (2007) Liouville theorems for a class of linear second-order operators with nonnegative characteristic form. Bound Value Probl 2007: 16.
[11]	Kupcov LP (1972) The fundamental solutions of a certain class of elliptic-parabolic second order equations. Differ Uravn 8: 1649-1660.
[12]	Lanconelli E, Polidoro S (1994) On a class of hypoelliptic evolution operators. Rend Semin Mat U Pad 52: 29-63.
[13]	Priola E, Wang FY (2006) Gradient estimates for diffusion semigroups with singular coefficients. J Funct Anal 236: 244-264. doi: 10.1016/j.jfa.2005.12.010
[14]	Priola E, Zabczyk J (2004) Liouville theorems for non-local operators. J Funct Anal 216: 455-490. doi: 10.1016/j.jfa.2004.04.001
[15]	Zabczyk J (1981/82) Controllability of stochastic linear systems. Syst Control Lett 1: 25-31.

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