Summary
LetX t be a symmetric stable process of index α, 0<α<2, inR d (d≧2). In this paper we deal with the perturbation ofX t by a multiplicative functional of the following form:
withF being a function onR d×R d satisfying certain conditions. First we prove the following gauge theorem: IfD is a bounded open domain ofR d, then the functiong(x)=E x{M(τ D )} is either identically infinite onD or bounded onD, where τ D is the first exit time fromD. Then we formulate the Dirichlet problem associated with the perturbed symmetric stable process by using Dirichlet form theory. Finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the Dirichlet problem mentioned above.
References
Benveniste, A., Jacod, J.: Systèmes de Lévy de Markov. Invent. Math.21, 183–198 (1973)
Blanchard, Ph., Ma, Z.: New results on the Schrodinger semigroups with potentials given by signed smooth measures. (Lect. Notes Math. vol. 1444) Berlin Heidelberg New York: Springer 1990
Blumenthal, R. M., Getoor, R. K.: Markov processes and potential theory. New York: Academic Press 1968
Carmona, R., Masters W. C., Simon, B.: Reletivistic Schrödinger operators: asymptotic behavior of the eigenvalues. J. Funct. Anal.91 (1990)
Chung, K. L., Rao, K. M.: Feynman-Kac functional and the Schrödinger equation. Seminar on stochastic processes. Boston: Birkhäuser 1981
Chung, K. L., Rao, K. M.: General gauge theorem for multiplicative functional. Trans. Am. Math. Soc.306, 819–836 (1988)
Doleáns-Dade, C.: Queleqes applications de la formule de changement de variables pour les semimartingales. Z. Wahrscheinlichkeitstheor. Verw. Geb.16, 181–194 (1970)
Dynkin, E. B.: Markov processes, vol. 1. Berlin Heidelberg New York: Springer 1965
Fukushima, M.: Dirichlet forms and Markov processes. Amsterdam: North-Holland 1980
Elliot, J.: Dirichlet spaces associated with integro-differential operators, part 1. Ill. J. Math.9, 87–98 (1965)
Ikeda N., Watanabe, S.: On some relations between he harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ.2, 79–95 (1962)
Landkof, N. S.: Foundations of modern potential theory. Berlin Heidelberg New York: Springer 1972
Liao, M.: The Dirichlet problem of a discontinuous Markov process. Acta Math. Sinica, New Ser.5, 9–15 (1989)
Ma, Z., Song, R.: Probabilistic methods in Schrödinger equations. Seminar on stochastic processes. Boston: Birkhäuser 1990
Port S., Stone, C.: Infinitely divisible processes and their potential theory, part 1. Ann. Inst. Fourier21 (2), 157–275; (1971) Part 2. Ann. Inst. Fourier21(4), 179–265 (1971)
Port, S., Stone, C.: Brownian motion and classical potential theory. New York: Academic Press 1978
Sharpe, M.: General theory of Markov processes. San Diego: Academic Press 1988
Sturm, K. T.: Gauge theorems for resolvents with applications to Markov processes. Probab. Theory Relat. Fields89, 387–406 (1991)
Zhao, Z.: A probabilistic principle and generalized Schrödinger perturbation. J. Funct. Anal.101, 162–176 (1991)
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Song, R. Probabilistic approach to the Dirichlet problem of perturbed stable processes. Probab. Th. Rel. Fields 95, 371–389 (1993). https://doi.org/10.1007/BF01192170
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DOI: https://doi.org/10.1007/BF01192170
Mathematics Subject Classification (1991)
- 60J30
- 35S15
- 60J57