Semigroups Defined by Additive Processes

Authors

Bibliographical Reference

This paper was presented by the authors at the Second International Conference on Semigroup Theory and Evolution Equations, in the Dutch town of Delft.

The conference proceedings have appeared as volume 135 in the series "Lecture Notes in Pure and Applied Mathematics" by Marcel Dekker Inc., New York 1991, ISBN 0-8247-8545-2. The title is "Semigroup Theory and Evolution Equations," edited by Philippe Clément, Enzo Mitidieri and Ben de Pagter. The current paper is on pages 463-482 of the book.

Abstract

We study semigroups, or more general evolution families of operators on a Banach space of functions. The classical probabilistic view on such operators uses additive functionals of the Markov sample space. We show that some of the most imporant classical results on domains an integral kernels can be salvaged if the usual hypothesis on bounded variation is dropped. This is a necessary generalization if the theory must be applied to the magnetic case, because the Itô integrals involved do not have bounded variation.

Availability

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References

1. Aizenman, M. and Simon, Barry, Brownian Motion and Harnack Inequality for Schrödinger Operators, Comm. Pure Appl. Math. 35 (1982), 209-273.
2. Albeverio, Sergio and Ma, Z., Additive Functionals, Nowhere Radon and Kato Class Smooth Measures Associated with Dirichlet Forms, Preprint Nr. 66 (October 1989), Sonderforschungsbereich 237, Institut für Mathematik Ruhr-Universität Bochum, Germany.
3. Azencott, R., Grandes déviations et applications, in "Ecole d'été de probabilités de Saint-Flour VIII (1978)," Springer Lecture Notes in Mathematics 774.
4. Azencott, R. et al., "Géodésiques et diffusions en temps petit," Astérisque 84-85, Société mathématique de France, 1981.
5. Bauer, Heinz, "Probability Theory and Elements of Measure Theory," Holt, Reinhart and Winston, New York 1972.
6. Bismut, J.-M., "Large Deviations and the Malliavin Calculus," Birkhäuser, Basel 1984.
7. Bismut, J.-M., Probability and Geometry, in "Probability and Analysis," Springer Lecture Notes in Mathematics 1206, 1-60.
8. Blanchard, Ph. and Ma, Z., Semigroups of Schrödinger Operators with Potentials Given by Radon Measures, (preprint) BiBoS Nr. 262, 1987, to appear in the proceedings of the second Ascona-Locarno-Como meeting (International Conference of Stochastic Processes - Geometry and Physics), edited by Albeverio S., Casati S., Cattaneo U., Merlini D. and Moresi R., World Scientific, Signapore 1990.
9. Blumenthal, R.M. and Getoor, R.K., "Markov Processes and Potential Theory," Academic Press, New York 1968.
10. Carmona, René, Masters, W.C. and Simon, Barry, Relativistic Schrödinger Operators: Asymptotic Behaviour of the Eigenfunctions, to appear in J. Funct. Analysis.
11. Cranston, M. and Zhao, Z., Conditional Transformation of Drift Formula and Potential Theory for ½D + b(.).Ñ, Comm. Math. Phys. 112 (1987), 613-625.
12. Davies, I. and Truman, A., On the Laplace Asymptotic Expansion of Conditional Wiener Integrals and the Bender-Wu Formula for x^2N-anharmonic Oscillators, J. Math. Physics 24 (1983), 255-266.
13. De Angelis, G., Jona-Lasinio, G. and Sirugue, M., Probabilistic Solution of Pauli Type Equations, J. Phys. A: Math. Gen. 16 (1983), 2433-2444.
14. Dellacherie, C. and Meyer, P.-A., "Probabilités et potentiel V-VIII," Hermann, Paris 1980.
15. Durrett, Richard, "Brownian Motion and Martingales in Analysis," Wadsworth, Belmont CA 1984.
16. Feyel, D. and de La Pradelle, A., Etude de l'équation ½Du - um = 0, où m est une mesure positive, Annales Institut Fourier t. 38 (1988) f. 3, 199-218.
17. Fukushima, M., "Dirichlet Forms and Markov Processes," North-Holland Mathematical Library 23, North-Holland, Amsterdam 1980.
18. John, F. and Nirenberg, Louis, On Functions of Bounded Mean Oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426.
19. Khas'minskii, R., On Positive Solutions of Uu + Vu = 0, Theory Prob. Appl. Vol. IV, nr. 4 (1959), 309-318.
20. Molchanov, S.A., Diffusion Processes and Riemannian Geometry, Russ. Math. Survey 30, 1 (1975), 1-63.
21. Port, S.C. and Stone, C.J., "Brownian Motion and Potential Theory," Academic Press, New York 1978.
22. Simon, Barry, Schrödinger Semigroups, Bull. Amer. Math. Soc. 7 (1982), Nr. 3, 447-526.
23. van Casteren, Johannes A., "Generators of Strongly Continuous Semigroups," Pitman Research Notes in Mathematics 115, Pitman, London 1985.
24. van Casteren, Johannes A., On Non-symmetric Generalized Schödinger Semigroups, Stochastic Analysis and Applications 8 (2) (1990), 225-262.
25. Voigt, J., Lecture at the 2nd International Conference on Trends in Semigroup Theory and Evolution Equations, Delft, 25/9-29/9/1989: Schrödinger Operators with Singular Potentials.