Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

Authors: Markus Kunze, Luca Lorenzi and Alessandra Lunardi
Journal: Trans. Amer. Math. Soc. 362 (2010), 169-198
MSC (2000): Primary 35K10, 35K15, 37L40
Published electronically: August 3, 2009
MathSciNet review: 2550148
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of elliptic operators $ A$ with unbounded coefficients defined in $ I\times\mathbb{R}^d$ for some unbounded interval $ I\subset\mathbb{R}$. We prove that, for any $ s\in I$, the Cauchy problem $ u(s,\cdot)=f\in C_b(\mathbb{R}^d)$ for the parabolic equation $ D_tu=Au$ admits a unique bounded classical solution $ u$. This allows to associate an evolution family $ \{G(t,s)\}$ with $ A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $ G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $ \{G(t,s)\}$ and we study the first properties of the extension of $ G(t,s)$ to the $ L^p$-spaces with respect to such measures.

References [Enhancements On Off] (What's this?)

  • 1. P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Diff. Int. Eqns. 1 (1988), 433-457. MR 945820 (90b:34094)
  • 2. S. Bernstein, Sur la généralisation du probléme de Dirichlet, I, Math. Ann. 62 (1906), 253-271. MR 1511375
  • 3. M. Bertoldi, L. Lorenzi, Analytical Methods for Markov Semigroups, Pure and applied mathematics 283, Chapman Hall/CRC Press, 2006. MR 2313847 (2009a:47090)
  • 4. C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surv. Monogr. 70, AMS, Providence RI, 1999. MR 1707332 (2001e:47068)
  • 5. G. Da Prato, M. 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 59, Birkhäuser, Basel (2008), 115-122. MR 2401953
  • 6. G. Da Prato, A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ. 7 (2007), 587-614. MR 2369672 (2008k:35200)
  • 7. E.B. Dynkin, Markov processes, Grundlehren der Mathematischen Wissenschaften 121, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. MR 0193671 (33:1887)
  • 8. S. Fornaro, G. Metafune, E. Priola, Gradient estimates for parabolic Dirichlet problems in unbounded domains, J. Differential Equations 205(2) (2004), 329-353. MR 2092861 (2005g:35131)
  • 9. A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, 1964. MR 0181836 (31:6062)
  • 10. M. Geissert, A. Lunardi, Invariant measures and maximal $ L^{2}$ regularity for nonautonomous Ornstein-Uhlenbeck equations, J. London Math. Soc. (2) 77 (2008), no. 3, 719-740. MR 2418301
  • 11. M. Geissert, A. Lunardi, Asymptotic behavior in nonautonomous Ornstein-Uhlenbeck equations, J. London Math. Soc. (2) 79 (2009), no 1, 85-106.
  • 12. I. Karatzas, S.E. Shreve, Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics 113, Springer-Verlag, New York, 1991. MR 1121940 (92h:60127)
  • 13. H. Kunita, Stochastic flows and stochastic differential equations. Reprint of the 1990 original. Cambridge Studies in Advanced Mathematics 24, Cambridge University Press, Cambridge, 1997. MR 1472487 (98e:60096)
  • 14. N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library 24, North-Holland Publishing Co., Amsterdam, 1989. MR 1011252 (90m:60069)
  • 15. O.U. Ladyženskaja, V.A. Solonnikov, N.N. Ural'ceva, Linear and quasilinear equations of parabolic type, Nauka, Moskow 1967 (Russian). English transl.: American Mathematical Society, Providence, 1968. MR 0241822 (39:3159b)
  • 16. G.M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, N.J., 1996. MR 1465184 (98k:35003)
  • 17. G. Metafune, D. Pallara, M. Wacker, Feller Semigroups on $ \mathbb{R}^N$, Semigroup Forum 65(2) (2002), 159-205. MR 1911723 (2003i:35170)
  • 18. D.W. Strook, S.R.S. Varadhan, Multidimensional diffusion processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K10, 35K15, 37L40

Retrieve articles in all journals with MSC (2000): 35K10, 35K15, 37L40

Additional Information

Markus Kunze
Affiliation: Graduiertenkolleg 1100, Ulm University, Helmholtzstrasse 18, 89069 Ulm, Germany
Address at time of publication: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands

Luca Lorenzi
Affiliation: Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti, 53/A, I-43100 Parma, Italy

Alessandra Lunardi
Affiliation: Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti, 53/A, I-43100 Parma, Italy

Keywords: Nonautonomous parabolic equations, evolution operators, gradient estimates, evolution systems of measures, evolution semigroups, invariant measures
Received by editor(s): July 5, 2007
Published electronically: August 3, 2009
Additional Notes: This work was supported by the M.I.U.R. research projects Prin 2004 and 2006 “Kolmogorov equations”
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society