Let be a -contraction semigroup on a real Banach space . A -exit law is a -valued function satisfying the functional equation: , . Let be a Bochner subordinator and let be the subordinated semigroup of (in the Bochner sense) by means of . Under some regularity assumption, it is proved in this paper that each -exit law is subordinated to a unique -exit law. 1. Introduction Let be a -contraction semigroup on a real Banach space with generator . A - exit law is a -valued function satisfying the functional equation: Exit laws are introduced by Dynkin (cf. [1]). They play an important role in the framework of potential theory without Green function. Indeed, they allow in this case, an integral representation of potentials and explicit energy formulas. Moreover, this notion was investigated in many papers (cf. [2–13]).In particular, the following theorem is proved in our paper[10]. Theorem 1.1. If a -exit law is Bochner integrable at (shortly zero-integrable), this is equivalent to, then is of the form where The present paper is devoted to investigate the subordinated abstract case where we study the zero-integrable solution of the exit equation (1.1) after Bochner subordination. More precisely, let be a Bochner subordinator, that is, a vaguely continuous convolution semigroup of subprobability measures on and let be the subordinated -semigroup of in the sense of Bochner by means of , that is, It can be seen that, for each exit law for , the function defined by is an exit law for The function is said to be subordinated to by means of . Conversely, it is natural to ask if any -exit law is subordinated to some -exit law. In general, we do not have a positive answer (see Example 5.3 below or [2, page 1922]). However, this problem was solved (cf. [2, 4–6]) for and positive -exit laws , and under some regularity assumptions on , , and . Basing on our paper[10, Theorem ],we consider in this paper the zero-integrable -exit laws in the abstract case. Namely, we prove the following. Theorem 1.2. Let be a zero-integrable -exit law satisfying the following conditions: There exist a constant such that: where and is the associated generator to . Then, is subordinated to a unique -exit law . Moreover, is explicitly given by The conditions in Theorem 1.2 are fulfilled for the closed -exit laws . This is always the case for the zero-integrable -exit laws in the bounded case. As application, we consider the holomorphic case and we prove the following result: Theorem 1.3. We suppose that is a -contraction holomorphic semigroup on and be a Bochner subordinator satisfying Then
References
[1]
E. B. Dynkin, “Green's and Dirichlet spaces associated with fine Markov processes,” Journal of Functional Analysis, vol. 47, no. 3, pp. 381–418, 1982.
[2]
S. Ben Othman, S. Bouaziz, and M. Hmissi, “On subordination of convolution semigroups,” International Journal of Bifurcation and Chaos, vol. 13, no. 7, pp. 1917–1922, 2003.
[3]
P. J. Fitzsimmons, “Markov process and non symmetric Dirichlet forms without regularity,” Journal of Functional Analysis, vol. 85, pp. 287–306, 1989.
[4]
F. Hmissi, “On energy formulas for symmetric semigroups,” Annales Mathematicae Silesianae, no. 19, pp. 7–18, 2005.
[5]
F. Hmissi, M. Hmissi, and W. Maaouia, “On subordinated exit laws for densities,” in Iteration Theory (ECIT '06), vol. 351 of Grazer Mathematische Berichte, pp. 52–65, Institut für Mathematik, Karl-Franzens-Universit？t Graz, Graz, Austria, 2007.
[6]
F. Hmissi and W. Maaouia, “On Bochner subordination of contraction semigroups with sector condition,” International Journal of Applied Mathematics, vol. 18, no. 4, pp. 429–445, 2005.
[7]
M. Hmissi, “Lois de sortie et semi-groupes basiques,” Manuscripta Mathematica, vol. 75, no. 3, pp. 293–302, 1992.
[8]
M. Hmissi, “Sur la représentation par les lois de sortie,” Mathematische Zeitschrift, vol. 213, no. 4, pp. 647–656, 1993.
[9]
M. Hmissi and H. Mejri, “On representation by exit laws for some Bochner subordinated semigroups,” Annales Mathematicae Silesianae, vol. 22, pp. 7–26, 2008.
[10]
M. Hmissi, H. Mejri, and E. Mliki, “On the abstract exit equation,” Grazer Mathematische Berichte, vol. 354, pp. 84–98, 2009.
[11]
M. Hmissi, H. Mejri, and E. Mliki, “On the fractional powers of semidynamical systems,” in Iteration Theory (ECIT '06), vol. 351 of Grazer Mathematische Berichte, pp. 66–78, Institut für Mathematik, Karl-Franzens-Universit？t Graz, Graz, Austria, 2007.
[12]
M. Hmissi and E. Mliki, “On exit law for subordinated semigroups by means of -subordinators,” submitted to Commentationes Mathematicae Universitatis Carolinae.
[13]
H. Mejri and E. Mliki, “On the exit laws for semidynamical systems and Bochner subordination,” IAENG International Journal of Applied Mathematics, vol. 40, no. 1, pp. 6–12, 2010.
[14]
W. Arendt, A. Grabosch, and A. Grabosch, One-Parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1986.
[15]
K. Yosida, Functional Analysis, Springer, Berlin, Germany, 1965.
[16]
K. J. Engel and N. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 2000.
[17]
N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I: Fourier Analysis and Semigroup, Imperial College Press, London, UK, 2003.
[18]
C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 8, Springer, New York, NY, USA, 1975.
[19]
A. S. Carasso and T. Kato, “On subordinated holomorphic semigroups,” Transactions of the American Mathematical Society, vol. 327, no. 2, pp. 867–878, 1991.
