Advertisement

Processus de diffusion gouverne par la forme de dirichlet de l'operateur de Schrödinger

Processus De Markov, Etc
Part of the Lecture Notes in Mathematics book series (LNM, volume 721)

Abstract

It is well known that Schrödinger operator is unitary equivalent to the Dirichlet operator of the ground state measure, whenever the infimum of its spectrum is actually an eigenvalue. Moreover, the Markov process governed by this Dirichlet operator plays a crucial role in the so-called Feynes-Nelson stochastic mechanics. Unfortunately, none of the various attempts to construct this process seems to be satisfactory. Indeed, either the dimension is restricted to one, or non-explosion is not proved, or the actual state space is not completely known, or the assumptions are too restrictive. The aim of the present note is to give a construction of the desired diffusion process that works in full generality, and to give some properties of the transition functions. We would like to put the emphasis on the fact that all the probabilistic techniques we use are elementary or standard. The only new ingredients are recently discovered regularity properties of Schrödinger operator.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. ALBEVERIO, R. HOEGH-KROHN and L. STREIT: Energy Forms, Hamiltonians and Distorted Brownian Paths. J.Math.Phys. 18 (1977) 907–917MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A.M. BERTHIER et B. GAVEAU: Critère de Convergence des Fonctionnelles de Kac et Application en Mécanique Quantique et en Géométrie. J.Funct.Analysis 29 (1978) 416–424.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    R.N. BHATTACHARYA: Criteria for Reccurence and Existence of Invariant Measures for Multidimensional Diffusions. Ann. Proba. 6 (1978) 541–553.CrossRefzbMATHGoogle Scholar
  4. [4]
    G. BROSAMLER: Quadratic Variation of Potentials and Harmonic Functions. Trans. Amer. Math. Soc. 149 (1970) 243–257.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    R.CARMONA: Regularity Properties of Schrödinger and Dirichlet Semigroups. J. Funct. Analysis (à paraitre)Google Scholar
  6. [6]
    W.G.FARIS: Self-Adjoint Operators. Lect. Notes in Math. # 433 (1975) Springer Verlag.Google Scholar
  7. [7]
    M. FUKUSHIMA: On the Generation of Markov Processes by Symmetric Forms. Proc. 2nd Japan-USSR Symp. Proba. Theory. Lect. Notes in Math. # 330 (1973) 46–79 Springer Verlag.MathSciNetzbMATHGoogle Scholar
  8. [8]
    M. FUKUSHIMA: Local Properties of Dirichlet Forms and Continuity of Sample Paths. Z. Wahrscheinlich. verw. Gebiete 29 (1974) 1–6.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M.FUKUSHIMA: Dirichlet Spaces and Additive Functionals of Finite Energy. Conf. Inter. Math. Helsinki (1978)Google Scholar
  10. [10]
    I.V. GIRSANOV: On Transforming a Class of Stochastic Processes by Absolutely Continuous Substitution of Measures. Theor. Prob. Appl. 5 (1960) 285–301.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R.Z. KHASMINSKII: Ergodic Properties of Reccurent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations. Theor. Prob. Appl. 5 (1960) 179–196.CrossRefGoogle Scholar
  12. [12]
    H.P.Mc KEAN: Stochastic Integrals. Academic Press (1969).Google Scholar
  13. [13]
    P.A. MEYER: La Formule de Ito pour le Mouvement Brownien d'apres G.Brosamler. Sem. Proba. Strasbourg 1976–77 Lect.Notes in Math. # 649 (1978) 763–769 Springer Verlag.MathSciNetGoogle Scholar
  14. [14]
    E.NELSON: Dynamical Theories of Brownian Motion. Princeton Univ. Press (1967)Google Scholar
  15. [15]
    S. OREY: Conditions for the Absolute Continuity of two Diffusions. Trans. Amer. Math. Soc. 193 (1974) 413–426.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    N.I. PORTENKO: Diffusion Processes with Unbounded Drift Coefficient. Theor. Prob. Appl. 20 (1975) 27–37.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P. PRIOURET et M. YOR: Processus de Diffusion à Valeurs dans ℝ et Mesures Quasi-invariantes sur C(ℝ, ℝ). Astérisque 22–23 (1975) 247–290.MathSciNetzbMATHGoogle Scholar
  18. [18]
    B.SIMON: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton Series in Physics (1971) Priceton Univ. Press.Google Scholar
  19. [19]
    B.SIMON: Functional Integration and Quantum Mechanics. Academic Press (livre à paraitre).Google Scholar
  20. [20]
    D.W. STROOCK and S.R.S. VARADHAN: Diffusion Processes with Continuous Coefficients. Comm. Pure Appl. Math. 22 (1969) 345–400.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    C. TUDOR: Diffusions avec Explosion Construites à l'aide des Martingales Exponentielles. Rev. Roum. Math. Pures et Appl. 20 (1975) 1187–1199.MathSciNetzbMATHGoogle Scholar
  22. [22]
    A.T. WANG: Generalized Ito's Formula and Additive Functionals of Brownian Motion. Z.Wahrscheinlich. verw. Gebiete 41 (1977) 153–159.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Saint-EtienneSaint Etienne

Personalised recommendations