Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems


In this paper, we study differential operators of the form

$$\left[\mathcal{L}_{\infty}v\right](x) = A\triangle v(x) + {\langle}Sx,\nabla v(x) {\rangle} - Bv(x),\,x \in \mathbb{R}^d,\,d\geqslant 2,$$

for matrices \({A,B \in \mathbb{C}^{N,N}}\), where the eigenvalues of A have positive real parts. The sum \({A\triangle v(x)+ {\langle} Sx,\nabla v(x) \rangle}\) is known as the Ornstein–Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix \({S \in \mathbb{R}^{d,d}}\). Differential operators such as \({\mathcal{L}_{\infty}}\) arise as linearizations at rotating waves in time-dependent reaction–diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that A and B can be diagonalized simultaneously, we construct a heat kernel matrix \({H(x,\xi,t)}\) of \({\mathcal{L}_{\infty}}\) that solves the evolution equation \({v_t=\mathcal{L}_{\infty}v}\). In the following, we study the Ornstein–Uhlenbeck semigroup

$$\left[T(t)v\right](x) =\int_{\mathbb{R}^d}H(x,\xi,t)v(\xi){\mathrm{d}} \xi,\,x \in \mathbb{R}^d,\,t > 0,$$

in exponentially weighted function spaces. This is used to derive resolvent estimates for \({\mathcal{L}_{\infty}}\) in exponentially weighted L p-spaces \({L^p_{\theta}({\mathbb{R}}^d,\mathbb{C}^N)}\), \({1\leqslant p < \infty}\), as well as in exponentially weighted C b -spaces \({C_{b, \theta}({\mathbb{R}}^d,\mathbb{C}^N)}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Aarão J.: Fundamental solutions for some partial differential operators from fluid dynamics and statistical physics. SIAM Rev. 49(2), 303–314 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  2. 2.

    H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank. Matrix Riccati equations. Systems & Control: Foundations & Applications. Birkhäuser Verlag, Basel, 2003. In control and systems theory.

  3. 3.

    Addona D.: Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions. Semigroup Forum 87(3), 509–536 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    H. W. Alt. Lineare Funktionalanalysis. Springer-Verlag Berlin Heidelberg, Berlin, Heidelberg, 2006.

  5. 5.

    H. Amann and J. Escher. Analysis. III. Grundstudium Mathematik. [Basic Study of Mathematics]. Birkhäuser Verlag, Basel, 2001.

  6. 6.

    M. Arató, S. Baran, and M. Ispány. Functionals of complex Ornstein-Uhlenbeck processes. Comput. Math. Appl., 37(1):1–13, 1999.

  7. 7.

    Beals R.: A note on fundamental solutions. Comm. Partial Differential Equations 24(1–2), 369–376 (1999)

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Beyn W.-J, Lorenz J: Nonlinear stability of rotating patterns. Dyn. Partial Differ. Equ. 5(4), 349–400 (2008)

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    M. Bramanti, G. Cupini, E. Lanconelli, and E. Priola. Global L p estimates for degenerate Ornstein-Uhlenbeck operators. Math. Z., 266(4):789–816, 2010.

  10. 10.

    M. Bramanti, G. Cupini, E. Lanconelli, and E. Priola. Global L p estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients. Math. Nachr., 286(11–12):1087–1101, 2013.

  11. 11.

    O. Calin, D.-C. Chang, K. Furutani, and C. Iwasaki. Heat kernels for elliptic and sub-elliptic operators. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2011. Methods and techniques.

  12. 12.

    Cerrai S.: A Hille-Yosida theorem for weakly continuous semigroups. Semigroup Forum 49(3), 349–367 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    G. Da Prato and A. Lunardi. On the Ornstein-Uhlenbeck operator in spaces of continuous functions. J. Funct. Anal., 131(1):94–114, 1995.

  14. 14.

    G. Da Prato and J. Zabczyk. Second order partial differential equations in Hilbert spaces, volume 293 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2002.

  15. 15.

    Delmonte S, Lorenzi L: On a class of weakly coupled systems of elliptic operators with unbounded coefficients. Milan J. Math. 79(2), 689–727 (2011)

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

  17. 17.

    D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Springer, Berlin [u.a.], 2010.

  18. 18.

    E. Harboure, J. L. Torrea, and B. Viviani. On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup. Math. Ann., 318(2):341–353, 2000.

  19. 19.

    L. Lorenzi and M. Bertoldi. Analytical methods for Markov semigroups, volume 283 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007.

  20. 20.

    Lunardi A.: On the Ornstein-Uhlenbeck operator in L 2 spaces with respect to invariant measures. Trans. Amer. Math. Soc. 349(1), 155–169 (1997)

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    A. Lunardi and G. Metafune. On the domains of elliptic operators in L 1. Differential Integral Equations, 17(1–2):73–97, 2004.

  22. 22.

    X. Mao. Stochastic differential equations and their applications. Horwood, Chichester, 2. ed. edition, 2008.

  23. 23.

    Mauceri G, Noselli L: The maximal operator associated to a nonsymmetric Ornstein–Uhlenbeck semigroup. J. Fourier Anal. Appl. 15(2), 179–200 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  24. 24.

    G. Metafune. L p-spectrum of Ornstein-Uhlenbeck operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30(1):97–124, 2001.

  25. 25.

    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(1):40–60, 2002.

  26. 26.

    G. Metafune, D. Pallara, and V. Vespri. L p-estimates for a class of elliptic operators with unbounded coefficients in \({\mathbf{R}^N}\). Houston J. Math., 31(2):605–620 (electronic), 2005.

  27. 27.

    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. Sc. Norm. Super. Pisa Cl. Sci. (5), 1(2):471–485, 2002.

  28. 28.

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX).

  29. 29.

    D. Otten. Spatial decay and spectral properties of rotating waves in parabolic systems. PhD thesis, Bielefeld University, 2014, https://www.math.uni-bielefeld.de/~dotten/files/diss/Diss_DennyOtten.pdf. Shaker Verlag, Aachen.

  30. 30.

    D. Otten. The identification problem for Ornstein-Uhlenbeck operators in \({L^p(\mathbb{R}^d,\mathbb{C}^N)}\) and a new antieigenvalue condition. Sonderforschungsbereich 701, Preprint 14067, 2014 (submitted), https://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb14067.pdf.

  31. 31.

    A. Pascucci. Kolmogorov equations in physics and in finance. In Elliptic and parabolic problems, volume 63 of Progr. Nonlinear Differential Equations Appl., pages 353–364. Birkhäuser, Basel, 2005.

  32. 32.

    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(2):563–576 (electronic), 2006.

  33. 33.

    M. Sobajima and T. Yokota. A direct approach to generation of analytic semigroups by generalized Ornstein-Uhlenbeck operators in weighted L p spaces. J. Math. Anal. Appl., 403(2):606–618, 2013.

  34. 34.

    G. E. Uhlenbeck and L. S. Ornstein. On the theory of the Brownian motion. Phys. Rev., 36:823–841, Sep 1930.

  35. 35.

    S. Zelik and A. Mielke. Multi-pulse evolution and space–time chaos in dissipative systems. Mem. Amer. Math. Soc., 198(925):vi+97, 2009.

Download references

Author information



Corresponding author

Correspondence to Denny Otten.

Additional information

D. Otten was supported by CRC 701 ‘Spectral Structures and Topological Methods in Mathematics’.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Otten, D. Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems. J. Evol. Equ. 15, 753–799 (2015). https://doi.org/10.1007/s00028-015-0279-1

Download citation

Mathematics Subject Classification

  • 35K45 (35J47, 35K08, 35B65, 35B45, 35B40)


  • Heat kernel matrix
  • Ornstein–Uhlenbeck semigroup
  • Exponentially weighted function spaces
  • Resolvent estimates