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

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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

Luca Lorenzi ^1,

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

Abstract
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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

Keywords: Schauder estimates, asymptotic behaviour, compactness., evolution operators, evolution semigroups, Nonautonomous parabolic equations, gradient estimates, evolution systems of measures.

Mathematics Subject Classification: Primary: 35K10; Secondary: 35B65, 35K15, 35B40, 37L4.

Citation: Luca Lorenzi. Nonautonomous Kolmogorov equations in the whole space: A survey on recent results. Discrete and 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 (http://arxiv.org/abs/1203.1280).
[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.
[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.
[4]	A. A. Albanese, L. Lorenzi and E. M. Mangino, $L^p$-uniqueness for elliptic operators with unbounded coefficients in $\mathbb{R}^N2$, J. Funct. Anal., 256 (2009), 1238-1257. doi: 10.1016/j.jfa.2008.07.022.
[5]	D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational. Mech. Anal., 25 (1967), 81-122.
[6]	R. Azencott, Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. France, 102 (1974), 193-240.
[7]	S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.
[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.
[9]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," 283 of Pure and applied mathematics, Chapman Hall/CRC Press, 2006.
[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.
[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.
[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.
[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.
[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.
[15]	C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Amer. Math. Soc., Providence (RI), 1999.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	S. Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Jap. J. Math., 27 (1957), 55-102.
[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.
[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.
[31]	L. Lorenzi, On a class of elliptic operators with unbounded time- and space-dependent coefficients in $\mathbb{R}^N2$, in "Functional Analysis and Evolution Equations," 433-456, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-7794-6_28.
[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.
[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.
[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.
[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.
[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.
[37]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\mathbb{R}^{N}$, Studia Math., 128 (1998), 171-198.
[38]	G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 30 (2001), 97-124.
[39]	G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.
[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.
[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.
[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.
[43]	J. Prüss, A. Rhandi and R. Schnaubelt, The domain of elliptic operators on $L^p(\mathbb mathbb{R}^{d})$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.
[44]	W. Stannat, Time-dependent diffusion operators on $L^1$, J. Evol. Equ., 4 (2004), 463-495. doi: 10.1007/s00028-004-0147-x.

show all references

References:

[1]	L. Angiuli, L. Lorenzi and A. Lunardi, Hypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations, Available on arXiv (http://arxiv.org/abs/1203.1280).
[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.
[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.
[4]	A. A. Albanese, L. Lorenzi and E. M. Mangino, $L^p$-uniqueness for elliptic operators with unbounded coefficients in $\mathbb{R}^N2$, J. Funct. Anal., 256 (2009), 1238-1257. doi: 10.1016/j.jfa.2008.07.022.
[5]	D. G. Aronson and J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rational. Mech. Anal., 25 (1967), 81-122.
[6]	R. Azencott, Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. France, 102 (1974), 193-240.
[7]	S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann., 62 (1906), 253-271. doi: 10.1007/BF01449980.
[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.
[9]	M. Bertoldi and L. Lorenzi, "Analytical Methods for Markov Semigroups," 283 of Pure and applied mathematics, Chapman Hall/CRC Press, 2006.
[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.
[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.
[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.
[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.
[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.
[15]	C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Amer. Math. Soc., Providence (RI), 1999.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	S. Itô, Fundamental solutions of parabolic differential equations and boundary value problems, Jap. J. Math., 27 (1957), 55-102.
[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.
[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.
[31]	L. Lorenzi, On a class of elliptic operators with unbounded time- and space-dependent coefficients in $\mathbb{R}^N2$, in "Functional Analysis and Evolution Equations," 433-456, Birkhäuser, Basel, (2008). doi: 10.1007/978-3-7643-7794-6_28.
[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.
[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.
[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.
[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.
[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.
[37]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $\mathbb{R}^{N}$, Studia Math., 128 (1998), 171-198.
[38]	G. Metafune, $L^p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 30 (2001), 97-124.
[39]	G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $\mathbb{R}^{N}$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.
[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.
[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.
[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.
[43]	J. Prüss, A. Rhandi and R. Schnaubelt, The domain of elliptic operators on $L^p(\mathbb mathbb{R}^{d})$ with unbounded drift coefficients, Houston J. Math., 32 (2006), 563-576.
[44]	W. Stannat, Time-dependent diffusion operators on $L^1$, J. Evol. Equ., 4 (2004), 463-495. doi: 10.1007/s00028-004-0147-x.

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