# American Institute of Mathematical Sciences

June  2013, 6(3): 731-760. doi: 10.3934/dcdss.2013.6.731

## Nonautonomous Kolmogorov equations in the whole space: A survey on recent results

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

Received  March 2010 Revised  January 2011 Published  December 2012

In this paper we survey some recent results concerned with nonautonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator
Citation: Luca Lorenzi. Nonautonomous Kolmogorov equations in the whole space: A survey on recent results. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 731-760. doi: 10.3934/dcdss.2013.6.731
##### References:
 [1] L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations,, Available on arXiv (, ().   Google Scholar [2] A. A. Albanese and E. M. Mangino, Cores for Feller semigroups with an invariant measure, J. Differential Equations, 225 (2006), 361-377. doi: 10.1016/j.jde.2005.09.014.  Google Scholar [3] A. A. Albanese and E. M. Mangino, Corrigendum to: "Cores for Feller Semigroups with an Invariant Measure'', J. Differential Equations, 244 (2008), 2980-2982. doi: 10.1016/j.jde.2008.03.001.  Google Scholar [4] A. A. Albanese, L. Lorenzi and E. M. Mangino, $L^p$-uniqueness for elliptic operators with unbounded coefficients in $\mathbbR^N$, J. Funct. Anal., 256 (2009), 1238-1257. doi: 10.1016/j.jfa.2008.07.022.  Google Scholar [5] D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational. Mech. Anal., 25 (1967), 81-122.  Google Scholar [6] R. Azencott, Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. France, 102 (1974), 193-240.  Google Scholar [7] S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.  Google Scholar [8] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc., 357 (2005), 2627-2664. doi: 10.1090/S0002-9947-05-03781-5.  Google Scholar [9] M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," 283 of Pure and applied mathematics, Chapman Hall/CRC Press, 2006.  Google Scholar [10] V. I. Bogachev, G. Da Prato and M. Röckner, On parabolic equations for measures, Comm. Partial Differential equations, 33 (2008), 397-418. doi: 10.1080/03605300701382415.  Google Scholar [11] V. I. Bogachev, G. Da Prato, M. Röckner and W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., 39 (2007), 631-640. doi: 10.1112/blms/bdm046.  Google Scholar [12] V. I. Bogachev, N. V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusion under minimal conditions, Comm. Partial Differential equations, 26 (2001), 2037-2080. doi: 10.1081/PDE-100107815.  Google Scholar [13] V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Global regularity and bounds for solutions of parabolic equations for probability measures, Theory Probab. Appl., 50 (2006), 561-581. doi: 10.1137/S0040585X97981986.  Google Scholar [14] V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Theory Probab. Appl., 52 (2008), 209-236. doi: 10.1137/S0040585X97982967.  Google Scholar [15] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Amer. Math. Soc., Providence (RI), 1999.  Google Scholar [16] R. Chill, E. Fasangova, G. Metafune and D. Pallara, The sector of analyticity of the Ornstein-Uhlenbeck semigroup on $L^p$ spaces with respect to invariant measure, J. London Math. Soc., 71 (2005), 703-722. doi: 10.1112/S0024610705006344.  Google Scholar [17] G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal., 131 (1995), 94-114. doi: 10.1006/jfan.1995.1084.  Google Scholar [18] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Differential Equations, 198 (2004), 35-52. doi: 10.1016/j.jde.2003.10.025.  Google Scholar [19] G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ., 7 (2007), 587-614. doi: 10.1007/s00028-007-0321-z.  Google Scholar [20] G. Da Prato and M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Rend. Lincei Mat. Appl., 17 (2006), 397-403. doi: 10.4171/RLM/476.  Google Scholar [21] G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, in "Seminar on Stochastic Analysis, Random Fields and Applications V", pp. 115-122, Progr. Probab., 59, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-8458-6_7.  Google Scholar [22] E. B. Dynkin, Three classes of infinite-dimensional diffusions, J. Funct. Anal., 86 (1989), 75-110. doi: 10.1016/0022-1236(89)90065-7.  Google Scholar [23] S. Fornaro, N. Fusco, G. Metafune and D. Pallara, Sharp upper bounds for the density of some invariant measures, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1145-1161. doi: 10.1017/S0308210508000498.  Google Scholar [24] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Disc. Cont. Dyn. Syst. Series A, 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747.  Google Scholar [25] M. Geissert, L. Lorenzi and R. Schnaubelt, $L^p$-regularity for parabolic operators with unbounded time-dependent coefficients, Annali Mat. Pura Appl., 189 (2010), 303-333. doi: 10.1007/s10231-009-0110-0.  Google Scholar [26] M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc., 77 (2008), 719-740. doi: 10.1112/jlms/jdn009.  Google Scholar [27] M. Geissert and A. Lunardi, Asymptotic behavior and hypercontractivity in non-autonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc., 79 (2009), 85-106. doi: 10.1112/jlms/jdn057.  Google Scholar [28] S. Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Jap. J. Math., 27 (1957), 55-102.  Google Scholar [29] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: 10.1090/S0002-9947-09-04738-2.  Google Scholar [30] L. Lorenzi, Schauder estimates for the Ornstein-Uhlenbeck semigroup in spaces of functions with polynomial or exponential growth, Dynam. Systems Appl., 9 (2000), 199-219.  Google Scholar [31] L. Lorenzi, On a class of elliptic operators with unbounded time- and space-dependent coefficients in $\mathbbR^N$, in "Functional Analysis and Evolution Equations," 433-456, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-7794-6_28.  Google Scholar [32] L. Lorenzi, Optimal regularity for nonautonomous Kolmogorov equations, Discr. Cont. Dyn. Syst. Series S, 4 (2011), 169-191. doi: 10.3934/dcdss.2011.4.169.  Google Scholar [33] L. Lorenzi and A. Lunardi, Elliptic operators with unbounded diffusion coefficients in $L^2$ spaces with respect to invariant measures, J. Evol. Equ., 6 (2006), 691-709. doi: 10.1007/s00028-006-0283-6.  Google Scholar [34] L. Lorenzi, A. Lunardi and A. Zamboni, Asymptotic behavior in time periodic parabolic problems with unbounded coefficients, J. Differential Equations, 249 (2010), 3377-3418. doi: 10.1016/j.jde.2010.08.019.  Google Scholar [35] L. Lorenzi and A. Zamboni, Cores for parabolic operators with unbounded coefficients, J. Differential Equations, 246 (2009), 2724-2761. doi: 10.1016/j.jde.2008.12.015.  Google Scholar [36] A. Lunardi, On the Ornstein-Uhlenbeck Operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc., 349 (1997), 155-169. doi: 10.1090/S0002-9947-97-01802-3.  Google Scholar [37] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^N$, Studia Math., 128 (1998), 171-198.  Google Scholar [38] G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 30 (2001), 97-124.  Google Scholar [39] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\R^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar [40] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60. doi: 10.1006/jfan.2002.3978.  Google Scholar [41] G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures, J. Funct. Anal., 223 (2005), 396-424. doi: 10.1016/j.jfa.2005.02.001.  Google Scholar [42] G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1 (2002), 471-485.  Google Scholar [43] J. Prüss, A. Rhandi and R. Schnaubelt, The domain of elliptic operators on $L^p(\mathbb R^d)$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.  Google Scholar [44] W. Stannat, Time-dependent diffusion operators on $L^1$, J. Evol. Equ., 4 (2004), 463-495. doi: 10.1007/s00028-004-0147-x.  Google Scholar

show all references

##### References:
 [1] L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations,, Available on arXiv (, ().   Google Scholar [2] A. A. Albanese and E. M. Mangino, Cores for Feller semigroups with an invariant measure, J. Differential Equations, 225 (2006), 361-377. doi: 10.1016/j.jde.2005.09.014.  Google Scholar [3] A. A. Albanese and E. M. Mangino, Corrigendum to: "Cores for Feller Semigroups with an Invariant Measure'', J. Differential Equations, 244 (2008), 2980-2982. doi: 10.1016/j.jde.2008.03.001.  Google Scholar [4] A. A. Albanese, L. Lorenzi and E. M. Mangino, $L^p$-uniqueness for elliptic operators with unbounded coefficients in $\mathbbR^N$, J. Funct. Anal., 256 (2009), 1238-1257. doi: 10.1016/j.jfa.2008.07.022.  Google Scholar [5] D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational. Mech. Anal., 25 (1967), 81-122.  Google Scholar [6] R. Azencott, Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. France, 102 (1974), 193-240.  Google Scholar [7] S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.  Google Scholar [8] M. Bertoldi and L. Lorenzi, Estimates of the derivatives for parabolic operators with unbounded coefficients, Trans. Amer. Math. Soc., 357 (2005), 2627-2664. doi: 10.1090/S0002-9947-05-03781-5.  Google Scholar [9] M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," 283 of Pure and applied mathematics, Chapman Hall/CRC Press, 2006.  Google Scholar [10] V. I. Bogachev, G. Da Prato and M. Röckner, On parabolic equations for measures, Comm. Partial Differential equations, 33 (2008), 397-418. doi: 10.1080/03605300701382415.  Google Scholar [11] V. I. Bogachev, G. Da Prato, M. Röckner and W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., 39 (2007), 631-640. doi: 10.1112/blms/bdm046.  Google Scholar [12] V. I. Bogachev, N. V. Krylov and M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusion under minimal conditions, Comm. Partial Differential equations, 26 (2001), 2037-2080. doi: 10.1081/PDE-100107815.  Google Scholar [13] V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Global regularity and bounds for solutions of parabolic equations for probability measures, Theory Probab. Appl., 50 (2006), 561-581. doi: 10.1137/S0040585X97981986.  Google Scholar [14] V. I. Bogachev, M. Röckner and S. V. Shaposhnikov, Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Theory Probab. Appl., 52 (2008), 209-236. doi: 10.1137/S0040585X97982967.  Google Scholar [15] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Amer. Math. Soc., Providence (RI), 1999.  Google Scholar [16] R. Chill, E. Fasangova, G. Metafune and D. Pallara, The sector of analyticity of the Ornstein-Uhlenbeck semigroup on $L^p$ spaces with respect to invariant measure, J. London Math. Soc., 71 (2005), 703-722. doi: 10.1112/S0024610705006344.  Google Scholar [17] G. Da Prato and A. Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal., 131 (1995), 94-114. doi: 10.1006/jfan.1995.1084.  Google Scholar [18] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Differential Equations, 198 (2004), 35-52. doi: 10.1016/j.jde.2003.10.025.  Google Scholar [19] G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ., 7 (2007), 587-614. doi: 10.1007/s00028-007-0321-z.  Google Scholar [20] G. Da Prato and M. Röckner, Dissipative stochastic equations in Hilbert space with time dependent coefficients, Rend. Lincei Mat. Appl., 17 (2006), 397-403. doi: 10.4171/RLM/476.  Google Scholar [21] G. Da Prato and M. Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, in "Seminar on Stochastic Analysis, Random Fields and Applications V", pp. 115-122, Progr. Probab., 59, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-8458-6_7.  Google Scholar [22] E. B. Dynkin, Three classes of infinite-dimensional diffusions, J. Funct. Anal., 86 (1989), 75-110. doi: 10.1016/0022-1236(89)90065-7.  Google Scholar [23] S. Fornaro, N. Fusco, G. Metafune and D. Pallara, Sharp upper bounds for the density of some invariant measures, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1145-1161. doi: 10.1017/S0308210508000498.  Google Scholar [24] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces, Disc. Cont. Dyn. Syst. Series A, 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747.  Google Scholar [25] M. Geissert, L. Lorenzi and R. Schnaubelt, $L^p$-regularity for parabolic operators with unbounded time-dependent coefficients, Annali Mat. Pura Appl., 189 (2010), 303-333. doi: 10.1007/s10231-009-0110-0.  Google Scholar [26] M. Geissert and A. Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc., 77 (2008), 719-740. doi: 10.1112/jlms/jdn009.  Google Scholar [27] M. Geissert and A. Lunardi, Asymptotic behavior and hypercontractivity in non-autonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc., 79 (2009), 85-106. doi: 10.1112/jlms/jdn057.  Google Scholar [28] S. Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Jap. J. Math., 27 (1957), 55-102.  Google Scholar [29] M. Kunze, L. Lorenzi and A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362 (2010), 169-198. doi: 10.1090/S0002-9947-09-04738-2.  Google Scholar [30] L. Lorenzi, Schauder estimates for the Ornstein-Uhlenbeck semigroup in spaces of functions with polynomial or exponential growth, Dynam. Systems Appl., 9 (2000), 199-219.  Google Scholar [31] L. Lorenzi, On a class of elliptic operators with unbounded time- and space-dependent coefficients in $\mathbbR^N$, in "Functional Analysis and Evolution Equations," 433-456, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-7794-6_28.  Google Scholar [32] L. Lorenzi, Optimal regularity for nonautonomous Kolmogorov equations, Discr. Cont. Dyn. Syst. Series S, 4 (2011), 169-191. doi: 10.3934/dcdss.2011.4.169.  Google Scholar [33] L. Lorenzi and A. Lunardi, Elliptic operators with unbounded diffusion coefficients in $L^2$ spaces with respect to invariant measures, J. Evol. Equ., 6 (2006), 691-709. doi: 10.1007/s00028-006-0283-6.  Google Scholar [34] L. Lorenzi, A. Lunardi and A. Zamboni, Asymptotic behavior in time periodic parabolic problems with unbounded coefficients, J. Differential Equations, 249 (2010), 3377-3418. doi: 10.1016/j.jde.2010.08.019.  Google Scholar [35] L. Lorenzi and A. Zamboni, Cores for parabolic operators with unbounded coefficients, J. Differential Equations, 246 (2009), 2724-2761. doi: 10.1016/j.jde.2008.12.015.  Google Scholar [36] A. Lunardi, On the Ornstein-Uhlenbeck Operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc., 349 (1997), 155-169. doi: 10.1090/S0002-9947-97-01802-3.  Google Scholar [37] A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\R^N$, Studia Math., 128 (1998), 171-198.  Google Scholar [38] G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 30 (2001), 97-124.  Google Scholar [39] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\R^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar [40] G. Metafune, D. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60. doi: 10.1006/jfan.2002.3978.  Google Scholar [41] G. Metafune, D. Pallara and A. Rhandi, Global properties of invariant measures, J. Funct. Anal., 223 (2005), 396-424. doi: 10.1016/j.jfa.2005.02.001.  Google Scholar [42] G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1 (2002), 471-485.  Google Scholar [43] J. Prüss, A. Rhandi and R. Schnaubelt, The domain of elliptic operators on $L^p(\mathbb R^d)$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.  Google Scholar [44] W. Stannat, Time-dependent diffusion operators on $L^1$, J. Evol. Equ., 4 (2004), 463-495. doi: 10.1007/s00028-004-0147-x.  Google Scholar
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