On the Liouville and strong Liouville properties for a class of non-local operators. (English) Zbl 07572348
Summary: We prove a necessary and sufficient condition for the Liouville and strong Liouville properties of the infinitesimal generator of a Lévy process and subordinate Lévy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of) the set of zeros of the characteristic exponent of the Lévy process.
References:
| [1] | Alibaud, N., del Teso, F., Endal, J., and Jakobsen, E. R., The Liouville theorem and linear operators satisfying the maximum principle, J. Math. Pures Appl. (9) 142 (2020), 229-242. https://doi.org/10.1016/j.matpur.2020.08.008 Berg, C., and Forst, G., Potential theory on locally compact Abelian groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 87. Springer-Verlag, New York-Heidelberg, 1975. Bourbaki, N., General topology. Chapters 5-10, Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1989. Böttcher, B., Schilling, R. L., and Wang, J., Lévy-Type processes: construction, approximation and sample path properties, Lecture Notes in Mathematics 2099 (Lévy Matters vol. III), Springer, Cham, 2013. https://doi.org/10.1007/978-3-319-02684-8 Choquet, G., Deny, J., Sur l’equation de convolution \(mu =mu *sigma \), C. R. Acad. Sci. Paris 250 (1960), 799-801. Deny, J., Sur l’equation de convolution \(mu =mu *sigma \), Seminaire Théorie du Potentiel Brelot, Choquet, Deny, Paris 4ème année (1959-60) exposé no. 5, 11 pp. Doob, J. L., Snell, J. L., and Williamson, R. E., Application of boundary theory to sums of independent random variables, In: Olkin, I. et al. (eds.): Contributions to Probability and Statistics; essays in honor of Harold Hotelling. Stanford University Press, Stanford 1960, 182-197. Hardy, G. H., Weierstrass’s non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), no. 3, 301-325. https://doi.org/10.2307/1989005 Jacob, N., Pseudo differential operators and Markov processes. Vol. 1, Imperial College Press, London 2001. https://doi.org/10.1142/9781860949746 Khoshnevisan, D., and Schilling, R. L., From Lévy-type processes to parabolic SPDEs, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser/Springer, Cham 2016. https://doi.org/10.1007/978-3-319-34120-0 Kühn, F., Existence and estimates of moments for Lévy-type processes, Stochastic Process. Appl. 127 (2017), no. 3, 1018-1041. https://doi.org/10.1016/j.spa.2016.07.00 Kühn, F., A Liouville theorem for Lévy generators, Positivity 25 (2021), no. 3, 997-1012. https://doi.org/10.1007/s11117-020-00800-7 Lukacs, E., Characteristic functions, Second edition, Griffin, London 1970. Sato, K., Lévy processes and infinitely divisible distributions, Revised edition, Cambridge University Press, Cambridge 2013. Schilling, R. L., Dirichlet operators and the positive maximum principle, Integral Equations Operator Theory 41 (2001), no. 1, 74-92. https://doi.org/10.1007/BF01202532 Schilling, R. L., Measures, integrals and martingales, Second edition, Cambridge University Press, Cambridge 2017. Schilling, R. L., Song, R., and Vondraček, Z., Bernstein functions: theory and applications, Second edition, Walter de Gruyter & Co., Berlin, 2012. https://doi.org/10.1515/9783110269338 Trèves, F., Topological vector spaces, distributions and kernels, Academic Press, New York-London 1967. Ying, J., Invariant measures of symmetric Lévy processes, Proc. Amer. Math. Soc. 120 (1994), no. 1, 267-273. https://doi.org/10.2307/2160195 · doi:10.1016/j.matpur.2020.08.008{\par}Berg, |
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