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Compact symmetry groups and generators for sub-Markovian semigroups

  • Palle E. T. Jørgensen
Article

Summary

We consider closed infinitesimal operators L with dense domain D in the algebra of all continuous functions, vanishing at infinity, on a given locally compact Hausdorff space X. Necessary and sufficient conditions are given for L to be the infinitesimal generator of a strongly continuous sub-Markovian semigroup on C0(X). The conditions involve: (1) commutativity with a compact group G acting continuously on X (i.e., spatial G-symmetry), (2) a tangential notion defined in terms of the G-action, and finally, (3) a dispersive estimate for L. The dispersive estimate is satisfied on a subalgebra which is a core for L, and contained in the domain D. If G is also a Lie group, we explicitly construct such a core algebra in the domain.

Keywords

Continuous Function Stochastic Process Probability Theory Symmetry Group Mathematical Biology 
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References

  1. 1.
    Bratteli, O., Jørgensen, P.E.T.: Unbounded derivations, and infinitesimal generators. Amer. Math. Soc. Proc. vol. 38 of the 1980 Summer Institute on Operator Algebras (Kingston, Canada)Google Scholar
  2. 2.
    Bratteli, O., Jørgensen, P.E.T.: Unbounded derivations tangential to compact groups of automorphisms. J. Functional Anal. 48, 107–133 (1982)Google Scholar
  3. 3.
    Bratteli, O., Robinson, D.W.: Positive C 0-semigroups on C *-algebras. Math. Scand. 49, 259–274 (1981)Google Scholar
  4. 4.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Berlin-Heidelberg-New York: Springer 1979Google Scholar
  5. 5.
    Bredon, G.E.: Introduction to Compact Transformation Groups. New York: Academic Press 1972Google Scholar
  6. 6.
    Davies, E.B.: Semigroups of Operators. London: Academic Press 1981Google Scholar
  7. 7.
    Duflo, M.: Semigroups of complex measures on a locally compact group in Non-commutative Hatmonic Analysis, ed. Carmona, J. et al. Lecture Notes in Mathematics 466. Berlin-Heidelberg-New York: Springer 1975Google Scholar
  8. 8.
    Evans, D., Hanche-Olsen, H.: The generators of positive semigroups. J. Functional Anal. 32, 207–212 (1979)Google Scholar
  9. 9.
    Goodman, F., Jørgensen, P.E.T.: Unbounded derivations commuting with compact group actions. Comm. Math. Phys. 82, 399–405 (1981)Google Scholar
  10. 10.
    Heyer, H.: Probability Measures on Locally Compact Groups. Ergebnisse Mathematik 94. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  11. 11.
    Hirsch, F.: Famille résolvantes, générateurs, cogénérateurs, potentiels. Ann. Inst. (Fourier (Grenoble) 22, 1, 89–210 (1972)Google Scholar
  12. 12.
    Hunt, G.A.: Semigroup of measures on a Lie group. Trans. Amer. Math. Soc. 81, 264–293 (1956)Google Scholar
  13. 13.
    Ito, K.: Brownian motion on a Lie group. Proc. Japan Acad. 26, 4–10 (1950)Google Scholar
  14. 14a.
    Jørgensen, P.E.T.: Monotone convergence of operator semigroups and the dynamics of infinite particle systems. Preprint 1980. [To appear in Adv. in Math].Google Scholar
  15. 14b.
    Jørgensen, P.E.T.: Approximately invariant subspaces for unbounded linear operators. II. Math. Ann. 227, 177–182 (1977)Google Scholar
  16. 15.
    Levy, P.: Sur les intégrales don les éléments sont des variables aléatories indépendantes. Ann. R. Scu. Norm. Sup. Pisa (sn2) 3, 337–366 (1934)Google Scholar
  17. 16.
    Lumer, G., Phillips, R.S.: Dissipative operators in Banach space. Pac. J. Math. 11, 679–698 (1961)Google Scholar
  18. 17.
    McKean, H.P.: Stochastic Integrals. New York: Academic Press 1969Google Scholar
  19. 18.
    Nagel, R., Uhlig, H.: An abstract Kato inequality for generators of positive operator semigroups on Banach lattices. J. Operator Th. 6, 113–123 (1981)Google Scholar
  20. 19a.
    Nelson, E.: Representation of a Markovian semigroup and its infinitesimal generator. J. Math. Mech. 7, 977–987 (1959)Google Scholar
  21. 19b.
    Nelson, E.: Dynamical theories of Brownian motion. Math. Notes of the Princeton University Press, Princeton, N.J. 1967Google Scholar
  22. 20.
    Phillips, R.S.: Semigroups of positive contraction operators. Cech. Math. 3. 12, 294–313 (1962)Google Scholar
  23. 21.
    Roth, J.-P.: Operateurs dissipatifs et semi-groupes dans les espaces de fonctions continue. Ann. Inst. Fourier (Grenoble) 26, 4, 1–97 (1976)Google Scholar
  24. 22.
    Sakai, S.: The theory of unbounded derivations in C *-algebras. Lecture Notes, University of Copenhagen, and University of Newcastle upon Tyne 1977Google Scholar
  25. 23.
    Simon, B.: An Abstract Kato's Inequality for Generators of Positivity Preserving Semigroups. Indiana Univ. Math. J. 26, 1067–1073 (1977)Google Scholar
  26. 24.
    Williams, D.: Diffusions, Markov processes, and martingales (vol. 1). New York: J. Wiley, 1979Google Scholar
  27. 25a.
    Yosida, K.: An operator-theoretic treatment of temporally homogeneous Markoff processes. J. Math. Soc. Japan 1, 224–253 (1949)Google Scholar
  28. 25b.
    Yosida, K.: Functional Analysis. (5th Edition), Berlin-Heidelberg-New York: Springer 1978Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Palle E. T. Jørgensen
    • 1
  1. 1.Department of Mathematics ElUniversity of PennsylvaniaPhiladelphiaUSA

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