Strong convergence of solutions to nonautonomous Kolmogorov equations

Lorenzi, Luca; Lunardi, Alessandra; Schnaubelt, Roland

doi:10.1090/proc/13031

Strong convergence of solutions to nonautonomous Kolmogorov equations
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by Luca Lorenzi, Alessandra Lunardi and Roland Schnaubelt PDF

Proc. Amer. Math. Soc. 144 (2016), 3903-3917 Request permission

Abstract:

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb {R}^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to +\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.

References

Luciana Angiuli, Pointwise gradient estimates for evolution operators associated with Kolmogorov operators, Arch. Math. (Basel) 101 (2013), no. 2, 159–170. MR 3089772, DOI 10.1007/s00013-013-0542-z
Luciana Angiuli, Luca Lorenzi, and Alessandra Lunardi, Hypercontractivity and asymptotic behavior in nonautonomous Kolmogorov equations, Comm. Partial Differential Equations 38 (2013), no. 12, 2049–2080. MR 3169770, DOI 10.1080/03605302.2013.840790
D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
D. G. Aronson and P. Besala, Parabolic equations with unbounded coefficients, J. Differential Equations 3 (1967), 1–14. MR 208160, DOI 10.1016/0022-0396(67)90002-2
D. G. Aronson and P. Besala, Uniqueness of positive solutions of parabolic equations with unbounded coefficients, Colloq. Math. 18 (1967), 125–135. MR 219900, DOI 10.4064/cm-18-1-125-135
W. Bodanko, Sur le problème de Cauchy et les problèmes de Fourier pour les équations paraboliques dans un domaine non borné, Ann. Polon. Math. 18 (1966), 79–94 (French). MR 197998, DOI 10.4064/ap-18-1-79-94
V. I. Bogachev, N. V. Krylov, and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations 26 (2001), no. 11-12, 2037–2080. MR 1876411, DOI 10.1081/PDE-100107815
Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR 1707332, DOI 10.1090/surv/070
Giuseppe Da Prato and Beniamin Goldys, Elliptic operators on ${\Bbb R}^d$ with unbounded coefficients, J. Differential Equations 172 (2001), no. 2, 333–358. MR 1829633, DOI 10.1006/jdeq.2000.3866
Giuseppe Da Prato and Alessandra Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), no. 1, 94–114. MR 1343161, DOI 10.1006/jfan.1995.1084
Giuseppe Da Prato and Alessandra Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ. 7 (2007), no. 4, 587–614. MR 2369672, DOI 10.1007/s00028-007-0321-z
Giuseppe Da Prato and Michael Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, Progr. Probab., vol. 59, Birkhäuser, Basel, 2008, pp. 115–122. MR 2401953, DOI 10.1007/978-3-7643-8458-6_{7}
G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996. MR 1417491, DOI 10.1017/CBO9780511662829
William Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1–31. MR 63607, DOI 10.1090/S0002-9947-1954-0063607-6
Mark Freidlin, Some remarks on the Smoluchowski-Kramers approximation, J. Statist. Phys. 117 (2004), no. 3-4, 617–634. MR 2099730, DOI 10.1007/s10955-004-2273-9
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Matthias Geissert and Alessandra Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc. (2) 77 (2008), no. 3, 719–740. MR 2418301, DOI 10.1112/jlms/jdn009
Matthias Geissert and Alessandra Lunardi, Asymptotic behavior and hypercontractivity in non-autonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85–106. MR 2472135, DOI 10.1112/jlms/jdn057
N. V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal. 183 (2001), no. 1, 1–41. MR 1837532, DOI 10.1006/jfan.2000.3728
Mirosław Krzyżański, Sur la solution fondamentale de l’équation linéaire normale du type parabolique dont le dernier coefficient est non borné. I, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962), 326–330 (French). MR 158176
Mirosław Krzyżański, Sur la solution fondamentale de l’équation linéaire normale du type parabolique dont le dernier coefficient est non borné. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 32 (1962), 471–476 (French). MR 158177
Markus Kunze, Luca Lorenzi, and Alessandra Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc. 362 (2010), no. 1, 169–198. MR 2550148, DOI 10.1090/S0002-9947-09-04738-2
Luca Lorenzi and Marcello Bertoldi, Analytical methods for Markov semigroups, Pure and Applied Mathematics (Boca Raton), vol. 283, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2313847
Luca Lorenzi, Alessandra Lunardi, and Alessandro Zamboni, Asymptotic behavior in time periodic parabolic problems with unbounded coefficients, J. Differential Equations 249 (2010), no. 12, 3377–3418. MR 2737435, DOI 10.1016/j.jde.2010.08.019
Luca Lorenzi and Alessandro Zamboni, Cores for parabolic operators with unbounded coefficients, J. Differential Equations 246 (2009), no. 7, 2724–2761. MR 2503020, DOI 10.1016/j.jde.2008.12.015
G. Metafune, D. Pallara, and M. Wacker, Feller semigroups on $\mathbf R^N$, Semigroup Forum 65 (2002), no. 2, 159–205. MR 1911723, DOI 10.1007/s002330010129
Jan Prüss, Abdelaziz Rhandi, and Roland Schnaubelt, The domain of elliptic operators on $L^p(\Bbb R^d)$ with unbounded drift coefficients, Houston J. Math. 32 (2006), no. 2, 563–576. MR 2293873