Introduction and Main Results

  • Kazuaki Taira
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

In this monograph we solve the problem of the existence of Feller semigroups associated with strong Markov processes. More precisely, we prove the unique solvability of boundary value problems for Waldenfels integro-differential operators with general Ventcel’ (Wentzell) boundary conditions, and construct Feller semigroups corresponding to the diffusion phenomenon where a Markovian particle moves chaotically in the state space, incessantly changing its direction of motion until it “dies” at the time when it reaches the set where the particle is definitely absorbed. This monograph provides a careful and accessible exposition of the functional analytic approach to the problem of constructing strong Markov processes with Ventcel’ boundary conditions in probability. Our approach here is distinguished by the extensive use of ideas and techniques characteristic of recent developments in the theory of partial differential equations.

Keywords

Manifold Radon 

References

  1. [ADN]
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math. 12, 623–727 (1959)CrossRefMATHMathSciNetGoogle Scholar
  2. [AV]
    Agranovich, M.S., Vishik, M.I.: Elliptic problems with a parameter and parabolic problems of general type. Uspehi Mat. Nauk 19(3)(117), 53–161 (1964, in Russian); English translation: Russ. Math. Surv. 19(3), 53–157 (1964)Google Scholar
  3. [Ap]
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 116, 2nd edn. Cambridge University Press, Cambridge (2009)Google Scholar
  4. [Ba1]
    Bass, R.F.: Probabilistic Techniques in Analysis. Probability and its Applications. Springer, New York (1995)MATHGoogle Scholar
  5. [Ba2]
    Bass, R.F.: Diffusions and Elliptic Operators. Probability and its Applications. Springer, New York (1998)MATHGoogle Scholar
  6. [Bi]
    Bichteler, K.: Stochastic Integration with Jumps. Encyclopedia of Mathematics and its Applications, vol. 89. Cambridge University Press, Cambridge (2002)Google Scholar
  7. [BCP]
    Bony, J.-M., Courrège, P., Priouret, P.: Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum. Ann. Inst. Fourier (Grenoble) 18, 369–521 (1968)CrossRefMATHGoogle Scholar
  8. [Bd]
    Bourdaud, G.: L p-estimates for certain non-regular pseudo-differential operators. Commun. Partial Differ. Equ. 7, 1023–1033 (1982)CrossRefMATHMathSciNetGoogle Scholar
  9. [Bo]
    Boutet de Monvel, L.: Boundary problems for pseudo-differential operators. Acta Math. 126, 11–51 (1971)Google Scholar
  10. [CZ]
    Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)CrossRefMATHMathSciNetGoogle Scholar
  11. [Cn]
    Cancelier, C.: Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum. Commun. Partial Differ. Equ. 11, 1677–1726 (1986)CrossRefMATHMathSciNetGoogle Scholar
  12. [CM]
    Coifman, R.R., Meyer, Y.: Au-delà des opérateurs pseudo-différentiels. Astérisque, vol. 57. Société Mathématique de France, Paris (1978)Google Scholar
  13. [Dy1]
    Dynkin, E.B.: Foundations of the theory of Markov processes. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1959) (in Russian); English translation: Pergamon Press, Oxford/London/New York/Paris (1960); German translation: Springer, Berlin/Göttingen/Heidelberg (1961); French translation: Dunod, Paris (1963)Google Scholar
  14. [Dy2]
    Dynkin, E.B.: Markov Processes I, II. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/Göttingen/Heidelberg (1965)CrossRefMATHGoogle Scholar
  15. [DY]
    Dynkin, E.B., Yushkevich, A.A.: Markov Processes, Theorems and Problems. Plenum Press, New York (1969)CrossRefGoogle Scholar
  16. [Fe1]
    Feller, W.: The parabolic differential equations and the associated semigroups of transformations. Ann. Math. 55, 468–519 (1952)CrossRefMATHMathSciNetGoogle Scholar
  17. [Fe2]
    Feller, W.: On second order differential equations. Ann. Math. 61, 90–105 (1955)CrossRefMATHMathSciNetGoogle Scholar
  18. [GM2]
    Garroni, M.G., Menaldi, J.-L.: Second Order Elliptic Integro-Differential Problems. Chapman & Hall/CRC Research Notes in Mathematics, vol. 430. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  19. [Ho1]
    Hörmander, L.: Pseudo-differential operators and non-elliptic boundary problems. Ann. Math. 83, 129–209 (1966)CrossRefMATHGoogle Scholar
  20. [Ho4]
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Pseudo-Differential Operators. Reprint of the 1994 edition, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin/Heidelberg/New York/Tokyo (2007)Google Scholar
  21. [IM]
    Itô, K., McKean, H.P. Jr.: Diffusion Processes and Their Sample Paths. Grundlehren der Mathematischen Wissenschaften, Second printing. Springer, Berlin/New York (1974)Google Scholar
  22. [Ja]
    Jacob, N.: Pseudo Differential Operators and Markov Processes. Fourier Analysis and Semigroups, vol. I. Imperial College Press, London (2001); Generators and Their Potential Theory, vol. II. Imperial College Press, London (2002); Markov Processes and Applications, vol. III. Imperial College Press, London (2005)Google Scholar
  23. [OR]
    Oleĭnik, O.A., Radkevič, E.V.: Second Order Equations with Nonnegative Characteristic Form. Itogi Nauki, Moscow (1971, in Russian); English translation: American Mathematical Society/Plenum Press, Providence, New York/London (1973)Google Scholar
  24. [Ra]
    Ray, D.: Stationary Markov processes with continuous paths. Trans. Am. Math. Soc. 82, 452–493 (1956)CrossRefMATHGoogle Scholar
  25. [SU]
    Sato, K., Ueno, T.: Multi-dimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 14, 529–605 (1964, 1965)Google Scholar
  26. [Sr5]
    Schrohe, E.: A short introduction to Boutet de Monvel’s calculus. In: Gil, J., Grieser, D., Lesch, M. (eds.) Approaches to Singular Analysis. Operator Theory, Advances and Applications, vol. 125, pp. 85–116. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  27. [SS1]
    Schrohe, E., Schulze, B.-W.: Boundary value problems in Boutet de Monvel’s calculus for manifolds with conical singularities I. In: Michael Demuth, Elmar Schrohe and Bert-Wolfgang Schulze (eds.) Pseudo-Differential Calculus and Mathematical Physics. Mathematical Topics, vol. 5, pp. 97–209. Akademie Verlag, Berlin (1994)Google Scholar
  28. [Se1]
    Seeley, R.T.: Refinement of the functional calculus of Calderón and Zygmund. Nederl. Akad. Wetensch. Proc. Ser. A 68, 521–531 (1965)MATHMathSciNetGoogle Scholar
  29. [Se2]
    Seeley, R.T.: Singular integrals and boundary value problems. Am. J. Math. 88, 781–809 (1966)CrossRefMATHMathSciNetGoogle Scholar
  30. [Sk]
    Skubachevskii, A.L.: Elliptic Functional-Differential Equations and Applications, Operator Theory: Advances and Applications, vol. 91. Birkhäuser Verlag, Basel (1997)Google Scholar
  31. [St]
    Stroock, D.W.: Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 209–244 (1975)CrossRefMATHMathSciNetGoogle Scholar
  32. [Ta2]
    Taira, K.: Sur le problème de la dérivée oblique II. Ark. Mat. 17, 177–191 (1979)CrossRefMATHMathSciNetGoogle Scholar
  33. [Ta3]
    Taira, K.: Sur l’existence de processus de diffusion. Ann. Inst. Fourier (Grenoble) 29, 99–126 (1979)CrossRefMATHMathSciNetGoogle Scholar
  34. [Ta4]
    Taira, K.: Un théorème d’existence et d’unicité des solutions pour des problèmes aux limites non-elliptiques. J. Funct. Anal. 43, 166–192 (1981)CrossRefMATHMathSciNetGoogle Scholar
  35. [Ta5]
    Taira, K.: Diffusion Processes and Partial Differential Equations. Academic, Boston (1988)MATHGoogle Scholar
  36. [Ta6]
    Taira, K.: On the existence of Feller semigroups with boundary conditions. Mem. Am. Math. Soc. 99(475) (1992). American Mathematical Society, Providence, vii+65Google Scholar
  37. [Ta7]
    Taira, K.: Analytic Semigroups and Semilinear Initial-Boundary Value Problems. London Mathematical Society Lecture Note Series, vol. 223. Cambridge University Press, Cambridge (1995)Google Scholar
  38. [Ta8]
    Taira, K.: Boundary value problems for elliptic integro-differential operators. Math. Z. 222, 305–327 (1996)CrossRefMathSciNetGoogle Scholar
  39. [Ta9]
    Taira, K.: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics, vol. 1499, 2nd edn. Springer, Berlin (2009)Google Scholar
  40. [Ta10]
    Taira, K.: On the existence of Feller semigroups with discontinuous coefficients. Acta Math. Sin. (English Series), 22, 595–606 (2006)Google Scholar
  41. [Ta11]
    Taira, K.: On the existence of Feller semigroups with discontinuous coefficients II. Acta Math. Sin. (English Series), 25, 715–740 (2009)Google Scholar
  42. [Wa]
    von Waldenfels, W.: Positive Halbgruppen auf einem n-dimensionalen Torus. Archiv der Math. 15, 191–203 (1964)CrossRefMATHGoogle Scholar
  43. [Wb]
    Watanabe, S.: Construction of diffusion processes with Wentzell’s boundary conditions by means of Poisson point processes of Brownian excursions. In: Probability Theory, Banach Center Publications, vol. 5, pp. 255–271. PWN, Warsaw (1979)Google Scholar
  44. [We]
    Wentzell (Ventcel’), A.D.: On boundary conditions for multidimensional diffusion processes. Teoriya Veroyat. i ee Primen. 4, 172–185 (1959, in Russian); English translation: Theory Prob. Appl. 4, 164–177 (1959)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Kazuaki Taira
    • 1
  1. 1.Institute of MathematicsUniversity of TsukubaTsukubaJapan

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