Skip to main content
Log in

Quasi-analytic solutions of linear parabolic equations

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Uniqueness of solutions of the Cauchy problem of a parabolic equation, and the related question of analyticity with respect to time, depend on global properties of the solution. We demonstrate that if the growth of the initial function (and, if relevant, of the inhomogeneous part and its derivatives) is not too great, solutions of non-stationary parabolic equations whose coefficients belong to quasi-analytic classes are quasi-analytic with respect to all variables and hence are unique. This study is motivated by the problem of endogenous completeness in continuous-time financial markets.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Agmon, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math. 18 (1965), 627–664.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623–727.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. M. Anderson and R. C. Raimondo, Equilibrium in continuous-time financial markets: endogenously dynamically complete markets, Econometrica 76 (2008), 841–907.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Bierstone and P. D. Milman, Resolution of singularities in Denjoy-Carleman classes, Selecta Math. (N.S.) 10 (2004), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. E. Browder, Real analytic functions on product spaces and separate analyticity, Canad. J. Math. 13 (1961), 650–656.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. S. D. Èĭdel’man, Parabolic Systems, North-Holland, Amsterdam-London, 1969.

    MATH  Google Scholar 

  8. A. Friedman, On classes of solutions of elliptic linear partial differential equations, Proc. Amer. Math. Soc. 8 (1957), 418–427.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Friedman, Classes of solutions of linear systems of partial differential equations of parabolic type, Duke Math. J. 24 (1957), 433–442.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Friedman, On the uniqueness of the Cauchy problem for parabolic equations, Amer. J. Math. 81 (1959), 503–511.

    Google Scholar 

  11. A. Friedman, Differentiability of solutions of ordinary differential equations in Hilbert space, Pacific J. Math. 16 (1966), 267–271.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.

    MATH  Google Scholar 

  13. A. Gorny, Contribution à l’etude des fonctions dérivables d’une variable réelle, Acta Math. 71 (1939), 317–358.

    Article  MathSciNet  Google Scholar 

  14. L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671–704.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964.

    Google Scholar 

  16. L. Hörmander, The Analysis of Linear Partial Differential Operators I, 2nd ed., Springer-Verlag, Berlin, 1990.

    Book  MATH  Google Scholar 

  17. Y. Kannai and R. C. Raimondo, Endogenously dynamically complete equilibria for financial markets: the general case, preprint.

  18. I. Karatzas and S. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998.

    MATH  Google Scholar 

  19. T. Kotaké, Analyticité du noyau élémentaire de l’opérateur parabolique, in Les Équations aux Dérivées Partielles, Éditions du Centre National de la Recherche Scientifique, Paris, 1963, pp. 53–60.

    Google Scholar 

  20. M. Krzyżański, Sur les solutions non négatives de l’équation linéare normale parabolique, Rev. Roumaine Math. Pures Appl. 9 (1964), 393–408.

    MathSciNet  MATH  Google Scholar 

  21. J. L. Lions and E. Magenes, Espaces de fonctions et distributions du type de Gevrey et problèmes aux limites paraboliques, Ann. Mat. Pura Appl. (4) 68 (1965), 341–417.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, 2nd ed., Springer-Verlag, Berlin, 2005.

    MATH  Google Scholar 

  23. C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math 10 (1957), 271–290.

    Article  MathSciNet  MATH  Google Scholar 

  24. F. O. Porper and S. D. Èĭdel’man, Two-sided estimates of the fundamental solutions of secondorder parabolic equations and some applications of them, Uspekhi Mat. Nauk 39 (1984), no. 3(237), 107–156; English translation: Russian Math. Surveys 39 (1984), 119–179.

    Google Scholar 

  25. L. I. Ronkin, Quasi-analytic classes of functions of several variables, Dokl. Akad. Nauk SSSR 146 (1962), 546–549; English translation: Soviet Math. Dokl. 3 (1962), 1360–1364.

    MathSciNet  Google Scholar 

  26. S. Täcklind, Sur les classes quasianalytiques des solutions des equations aux dérivées partielles du type parabolique, Nova Acta Soc. Sci. Upsal.(4) 10 (1936), 1–57.

    Google Scholar 

  27. H. Tanabe, On differentiability and analyticity of solutions of weighted elliptic boundary value problems, Osaka J. Math. 2(1965), 163–190.

    MathSciNet  MATH  Google Scholar 

  28. H. Tanabe, On regularity of solutions of abstract differential equations of parabolic type in Banach space, J. Math. Soc. Japan 19 (1967), 521–542.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Tanabe, Equations of Evolution, Pitman, Boston-London, 1979.

    MATH  Google Scholar 

  30. D. V. Widder, The Heat Equation, Academic Press, New York, 1975.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yakar Kannai.

Additional information

Parts of the research reported on in this paper were performed while Y. K. was visiting the University of Melbourne.

Raimondo’s work was supported by grant DP0558187 from the Australian Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kannai, Y., Raimondo, R.C. Quasi-analytic solutions of linear parabolic equations. JAMA 119, 115–145 (2013). https://doi.org/10.1007/s11854-013-0004-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-013-0004-3

Keywords

Navigation