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 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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