Electronic Journal of Probability

An Extended Generator and Schrödinger Equations

Ronald Getoor

Full-text: Open access

Abstract

The generator of a Borel right processis extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrödinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.

Article information

Source
Electron. J. Probab., Volume 4 (1999), paper no. 19, 23 pp.

Dates
Accepted: 16 November 1999
First available in Project Euclid: 4 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1457125528

Digital Object Identifier
doi:10.1214/EJP.v4-56

Mathematical Reviews number (MathSciNet)
MR1741538

Zentralblatt MATH identifier
0936.60066

Subjects
Primary: 60J40: Right processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Keywords
Markov processes Schrödinger equations generators smooth measures

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Getoor, Ronald. An Extended Generator and Schrödinger Equations. Electron. J. Probab. 4 (1999), paper no. 19, 23 pp. doi:10.1214/EJP.v4-56. https://projecteuclid.org/euclid.ejp/1457125528


Export citation

References

  • S. Albeverio, J. Brasche and M. Röckner, Dirichlet forms and generalized Schrödinger operators, Lecture Notes in Physics 345, Springer, 1989, pp. 1-42.
  • J. Azéma, Théorie générale des processus et retournement du temps, Ann. Sci. de l'Ecole Norm. Sup. 6 (1973), 459-519.
  • R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968.
  • A. Boukricha, W. Hansen and H. Hueber, Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expo. Math. 5 (1987), 97-135.
  • K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Springer, Berlin, Heidelberg, New York, 1995.
  • E. Çinlar, J. Jacod, P. Protter and M. J. Sharpe, Semimartingales and Markov processes, Z. Wahrsheinlichkeitstheorie 54, (1980), 161-219.
  • C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Ch. I-IV (1975), Ch. V-VIII (1980), Ch. IX-XI (1983), Ch. XII-XVI (1987) Hermann, Paris.
  • E. B. Dynkin, Markov Processes, Vol. 1, Springer, Berlin, Heidelberg, New York, 1965.
  • D. Feyel and A. de La Pradelle, Étude de l'équation où $mu$ est une mesure positive, Ann. Inst. Fourier, Grenoble 38 (1988), 199-218.
  • P. J. Fitzsimmons and R. K. Getoor, Revuz measures and time changes, Math. Z. 199 (1988), 233-256.
  • P. J. Fitzsimmons and R. K. Getoor, Smooth measures and continuous additive functionals of right Markov processes, Itò's Stochastic Calculus and Probability Theory, Springer, Berlin, Heidelberg, New York, 1996, pp. 31-49.
  • R. K. Getoor, Excessive Measures, Birkhauser, Boston, 1990.
  • R. K. Getoor, Measure perturbations of Markovian semigroups, Potential Analysis 11 (1999), 101-133.
  • R. K. Getoor and M. J. Sharpe, Naturality, standardness and weak duality, Z. Wahrsheinlichkeitstheorie verw. Geb. 67 (1984), 1-62.
  • R. K. Getoor and J. Steffens, The energy functional, balayage and capacity, Ann. Inst. Henri Poincaré 23 (1987), 321-357.
  • W. Hansen, A note on continuous solutions of the Schrödinger equation, Proc. AMS 117 (1993), 381-384.
  • H. Kunita, Absolute continuity of Markov processes and generators, Nagoya Math. J. 36 (1969), 1-26.
  • Z. Ma, Some new results concerning Dirichlet forms, Feynman-Kac semigroups and Schrödinger equations, Contemporary Math. 118 (1991), 239-254.
  • D. Revuz, Mesures associées aux fonctionnelles additive de Markov I, Trans. AMS 148 (1970), 501-531.
  • M. J. Sharpe, General Theory of Markov Processes, Academic Press, Boston, San Diego, New York, 1988.
  • P. Stollmann and J. Voigt, Perturbations of Dirichlet forms by measures, Potential Analysis 5 (1996), 109-139.