Skip to main content
SpringerLink
Log in

Some Aspects of Large Time Behavior of the Heat Kernel: An Overview with Perspectives

  • Conference paper
  • First Online:
Book cover Mathematical Physics, Spectral Theory and Stochastic Analysis

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 232))

Abstract

We discuss a variety of developments in the study of large time behavior of the positive minimal heat kernel of a time-independent (not necessarily symmetric) second-order parabolic operator defined on a domain, or more generally, on a noncompact Riemannian manifold. Our attention is mainly focused on general results in general settings.

Mathematics Subject Classification (2010). Primary 35K08; secondary 35B09, 47D07, 47D08.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Agmon, On positivity and decay of solutions of second-order elliptic equations on Riemannian manifolds, in “Methods of Functional Analysis and Theory of Elliptic Equations”, pp. 19–52, (ed. D. Greco), Liguori, Naples, 1983.

    Google Scholar 

  2. S. Agmon, private communication.

    Google Scholar 

  3. A. Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997), 45–92.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.-P. Anker, P. Bougerol, and T. Jeulin, The infinite Brownian loop on symmetric space, Rev. Mat. Iberoamericana 18 (2002), 41–97.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Arendt, and C.J.K. Batty, Exponential stability of a diffusion equation with absorption, Differential Integral Equations 6 (1993), 1009–1024.

    MATH  MathSciNet  Google Scholar 

  6. D.G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607–694.

    Google Scholar 

  7. R. Bañuelos, and B. Davis, Heat kernel, eigenfunctions, and conditioned Brownian motion in planner domains, J. Funct. Anal. 84 (1989), 188–200.

    Article  MATH  MathSciNet  Google Scholar 

  8. N.H. Bingham, C.M. Goldie, and J.L. Teugels, “Regular Variation”, Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  9. K. Burdzy and T.S. Salisbury, On minimal parabolic functions and time-homogeneous parabolic ℎ-transforms, Trans. Amer. Math. Soc. 351 (1999), 3499–3531.

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  11. I. Chavel, “Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives”, Cambridge Tracts in Mathematics 145, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  12. I. Chavel and L. Karp, Large time behavior of the heat kernel: the parabolic λ- potential alternative, Comment. Math. Helv. 66 (1991), 541–556.

    Article  MATH  MathSciNet  Google Scholar 

  13. K.L. Chung, “Markov Chains with Stationary Transition Probabilities” 2nd ed., Grundlehren der mathematischen Wissenschaften, 104, Springer-Verlag, New York, 1967.

    Google Scholar 

  14. P. Collet, S. Martínez, and J. San Martín, Ratio limit theorems for a Brownian motion killed at the boundary of a Benedicks domain, Ann. Probab., 27 (1999), 1160–1182.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Collet, Asymptotic of the heat kernel in general Benedicks domains, Probab. Theory Related Fields 125 (2003), 350–364.

    Google Scholar 

  16. C. Constantinescu, and A. Cornea, “Potential Theory on Harmonic Spaces”, Grundlehren der Mathematischen Wissenschaften 158, Springer-Verlag, New York- Heidelberg, 1972.

    Google Scholar 

  17. T. Coulhon, Heat kernels on non-compact Riemannian manifolds: a partial survey, in “Séminaire de Théorie Spectrale et Géométrie”, pp. 167–187, Sémin. Théor. Spectr. Géom. 15, Univ. Grenoble I, Saint-Martin-d’HÈres, 1997.

    Google Scholar 

  18. T. Coulhon, Large time behaviour of heat kernels on Riemannian manifolds: fast and slow decays, Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1998), Exp. No. II, 12 pp., Univ. Nantes, Nantes, 1998.

    Google Scholar 

  19. T. Coulhon, and A. Grigor’yan, On-diagonal lower bounds for heat kernels on noncompact manifolds and Markov chains, Duke Math. J. 89 (1997), 133–199.

    Google Scholar 

  20. E.B. Davies, “Heat Kernels and Spectral Theory”, Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  21. E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99–119.

    Google Scholar 

  22. E.B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), 105–125.

    Google Scholar 

  23. J.L. Doob, “Classical potential theory and its probabilistic counterpart”, Grundlehren der Mathematischen Wissenschaften 262, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  24. M. Fraas, D. Krejčiřík, and Y. Pinchover, On some strongr atio limit theorems for heat kernels, Disc. Contin. Dyn. Syst. 28 (2010), 495–509.

    Article  MATH  Google Scholar 

  25. A. Grigor’yan, Estimates of heat kernels on Riemannian manifolds, in “Spectral Theory and Geometry”, pp. 140–225, London Math. Soc. Lecture Note Ser. 273, Cambridge Univ. Press, Cambridge, 1999.

    Google Scholar 

  26. A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135– 249.

    Google Scholar 

  27. A. Grigor’yan, “Heat Kernel and Analysis on Manifolds”, AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society, Providence, RI, International Press, Boston, MA, 2009.

    Google Scholar 

  28. A. Grigor’yan, and L. Saloff-Coste, Heat kernel on manifolds with ends, Ann. Inst. Fourier (Grenoble) 59 (2009), 1917–1997.

    Google Scholar 

  29. R.Z. Has’minski\( \mathrm{\breve{i}} \), “Stochastic Stability of Differential Equations”, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.

    Google Scholar 

  30. K. Ishige, and M. Murata, Uniqueness of nonnegative solutions of the Cauchy problem for parabolic equations on manifolds or domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 171–223.

    Google Scholar 

  31. P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), 1–21.

    Google Scholar 

  32. M. Keller, D. Lenz, H. Vogt, and R. Wojciechowski, Note on basic features of large time behaviour of heat kernels. arXiv: 1101.0373.

    Google Scholar 

  33. J.F.C. Kingman, Ergodic properties of continuous-time Markov processes and their discrete skeletons, Proc. London Math. Soc. (3) 13 (1963), 593–604.

    Google Scholar 

  34. W. Kirsch, and B. Simon, Approach to equilibrium for a forced Burgers equation, J. Evol. Equ. 1 (2001), 411–419.

    Article  MATH  MathSciNet  Google Scholar 

  35. G. Kozma, A graph counterexample to Davies’ conjecture. arXiv:1111.4593.

    Google Scholar 

  36. D. Krejčiřík, and E. Zuazua, The Hardy inequality and the heat equation in twisted tubes, J. Math. Pures Appl. (9) 94 (2010), 277–303.

    Google Scholar 

  37. P. Li, “Geometric Analysis” (Cambridge Studies in Advanced Mathematics; 134), Cambridge University Press, Cambridge, 2012.

    Google Scholar 

  38. V. Lin and Y. Pinchover, Manifolds with group actions and elliptic operators, Memoirs Amer. Math. Soc. 112, (1994), 1–78.

    Article  MathSciNet  Google Scholar 

  39. V. Liskevich, and Y. Semenov, Two-sided estimates of the heat kernel of the Schrödinger operator, Bull. London Math. Soc., 30 (1998), 596–602.

    Article  MATH  MathSciNet  Google Scholar 

  40. P.D. Milman, and Yu. A. Semenov, Heat kernel bounds and desingularizing weights, J. Funct. Anal., 202 (2003), 1–24.

    Article  MATH  MathSciNet  Google Scholar 

  41. M. Murata, Positive solutions and large time behaviors of Schrödinger semigroups, Simon’s problem, J. Funct. Anal., 56 (1984), 300–310.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. Murata, Structure of positive solutions to (−Δ+V )u = 0 in ℝn, Duke Math. J. 53 (1986), 869–943.

    Google Scholar 

  43. M. Murata, Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math., 102 (1997), 29–60.

    Google Scholar 

  44. M. Murata, Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations, J. Funct. Anal. 194 (2002), 53–141.

    Google Scholar 

  45. M. Murata, Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries, J. Funct. Anal. 245 (2007), 177–212.

    Google Scholar 

  46. Y. Pinchover, Criticality and ground states for second-order elliptic equations, J. Differential Equations, 80 (1989), 237–250.

    Article  MATH  MathSciNet  Google Scholar 

  47. Y. Pinchover, On criticality and ground states of second-order elliptic equations. II, J. Differential Equations 87 (1990), 353–364.

    Google Scholar 

  48. Y. Pinchover, Large time behavior of the heat kernel and the behavior of the Green function near criticality for nonsymmetric elliptic operators, J. Funct. Anal. 104 (1992), 54–70.

    Google Scholar 

  49. Y. Pinchover, On nonexistence of any λ 0-invariant positive harmonic function, a counterexample to Stroock’s conjecture, Comm. Partial Differential Equations 20 (1995), 1831–1846.

    Google Scholar 

  50. Y. Pinchover, Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann., 314 (1999), 555–590.

    Google Scholar 

  51. Y. Pinchover, Large time behavior of the heat kernel, J. Funct. Anal. 206 (2004), 191–209.

    Google Scholar 

  52. Y. Pinchover, On Davies’ conjecture and strongra tio limit properties for the heat kernel, in “Potential Theory in Matsue”, Proceedings of the International Workshop on Potential Theory, 2004, pp. 339–352, (eds. H. Aikawa, et al.), Advanced Studies in Pure Mathematics 44, Mathematical Society of Japan, Tokyo, 2006.

    Google Scholar 

  53. Y. Pinchover, A Liouville-type theorem for Schrödinger operators, Comm. Math. Phys., 272 (2007), 75–84.

    Google Scholar 

  54. R.G. Pinsky, “Positive Harmonic Function and Diffusion”, Cambridge University Press, Cambridge, 1995.

    Book  Google Scholar 

  55. F.O. Porper, and S.D. Èǐdel′ man, Two-sided estimates of the fundamental solutions of second-order parabolic equations and some applications of them, Uspekhi Mat. Nauk 39 (1984), 107–156.

    Google Scholar 

  56. V.C. Repnikov, S.D. Èǐdel′man, Necessary and sufficient conditions for establishing a solution to the Cauchy problem, Soviet Math. Dokl. 7 (1966), 388–391.

    Google Scholar 

  57. D.W. Robinson, “Elliptic Operators and Lie Groups”, Oxford Mathematical Monographs, Oxford University Press, New York, 1991.

    MATH  Google Scholar 

  58. B. Simon, “Functional integration and quantum physics”, Pure and Applied Mathematics, 86, Academic Press, Inc., New York-London, 1979.

    Google Scholar 

  59. B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.

    Google Scholar 

  60. B. Simon, Large time behavior of the heat kernel: on a theorem of Chavel and Karp, Proc. Amer. Math. Soc. 118 (1993), 513–514.

    Google Scholar 

  61. U. Stadtmüller, and R. Trautner, Ratio Tauberian theorems for Laplace transforms without monotonicity assumptions, Quart. J. Math. Oxford Ser. (2) 36 (1985), 363– 381.

    Google Scholar 

  62. S.R.S. Varadhan, “Lectures on Diffusion Problems and Partial Differential Equations”, Tata Institute of Fundamental Research 64, Springer-Verlag, Berlin, 1980.

    Google Scholar 

  63. N.Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, “Analysis and Geometry on Groups”, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992.

    Google Scholar 

  64. W. Woess, “Random Walks on Infinite Graphs and Groups”, Cambridge Tracts in Mathematics 138, Cambridge University Press, Cambridge, 2000.

    Book  Google Scholar 

  65. K.Wong, Large time behavior of Dirichlet heat kernels on unbounded domains above the graph of a bounded Lipschitz function, Glas. Mat. Ser. III 41 (61) (2006), 177– 186.

    Article  MATH  MathSciNet  Google Scholar 

  66. Q.S. Zhang, Large time behavior of Schrödinger heat kernels and applications, Comm. Math. Phys. 210 (2000), 371–398.

    Article  MATH  MathSciNet  Google Scholar 

  67. Q.S. Zhang, A sharp comparison result concerningSc hrödinger heat kernels, Bull. London Math. Soc., 35 (2003), 461–472.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, 32000, Israel

    Yehuda Pinchover

Authors
  1. Yehuda Pinchover
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yehuda Pinchover .

Editor information

Editors and Affiliations

  1. Institut für Mathematik, TU Clausthal, Clausthal-Zellerfeld, Germany

    Michael Demuth

  2. Fakultät für Mathematik und Informatik, FernUniversität Hagen, Hagen, Germany

    Werner Kirsch

Additional information

Dedicated to Professor Michael Demuth on the occasion of his 65th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this paper

Cite this paper

Pinchover, Y. (2013). Some Aspects of Large Time Behavior of the Heat Kernel: An Overview with Perspectives. In: Demuth, M., Kirsch, W. (eds) Mathematical Physics, Spectral Theory and Stochastic Analysis. Operator Theory: Advances and Applications(), vol 232. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0591-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation