## Journal of the Mathematical Society of Japan

### Penalising symmetric stable Lévy paths

#### Abstract

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index $1 < \alpha \le 2$. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $\sigma$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $\alpha = 2$.

#### Article information

Source
J. Math. Soc. Japan Volume 61, Number 3 (2009), 757-798.

Dates
First available in Project Euclid: 30 July 2009

http://projecteuclid.org/euclid.jmsj/1248961478

Digital Object Identifier
doi:10.2969/jmsj/06130757

Mathematical Reviews number (MathSciNet)
MR2552915

#### Citation

YANO, Kouji; YANO, Yuko; YOR, Marc. Penalising symmetric stable Lévy paths. J. Math. Soc. Japan 61 (2009), no. 3, 757--798. doi:10.2969/jmsj/06130757. http://projecteuclid.org/euclid.jmsj/1248961478.

#### References

• J. Bertoin, Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.
• R. M. Blumenthal, Excursions of Markov processes, Probability and its Applications, Birkhäuser Boston Inc., Boston, MA, 1992.
• R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, 29, Academic Press, New York, 1968.
• L. Chaumont, Excursion normalisée, méandre et pont pour les processus de Lévy stables, Bull. Sci. Math., 121 (1997), 377–403.
• Z.-Q. Chen, M. Fukushima and J. Ying, Extending Markov processes in weak duality by Poisson point processes of excursions, (eds. F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal and T. Zhang), Stochastic Analysis and Applications, The Abel Symposium 2005, Springer, Heidelberg, 2007, pp. 153–196.
• P. J. Fitzsimmons and R. K. Getoor, Excursion theory revisited, Illinois J. Math., 50 (2006), 413–437.
• K. Itô, Poisson point processes attached to Markov processes, In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), III: Probability theory, Univ. California Press, Berkeley, Calif., 1972, pp. 225–239.
• J. Najnudel, Pénalisations de l'araignée brownienne, Ann. Inst. Fourier (Grenoble), 57 (2007), 1063–1093.
• J. Najnudel, B. Roynette and M. Yor, A remarkable $\sigma$-finite measure on $\mathcal{C}(\mbi{R}_{+},\mbi{R})$ related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris, 345 (2007), 459–466.
• J. Najnudel, B. Roynette and M. Yor, A global view of Brownian penalisations, MSJ Memoirs, 19, Mathematical Society of Japan, Tokyo, 2009.
• D. Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984.
• D. Revuz and M. Yor, Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, third edition, 1999.
• B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II, Studia Sci. Math. Hungar., 43 (2006), 295–360.
• B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights, I, Studia Sci. Math. Hungar., 43 (2006), 171–246.
• B. Roynette, P. Vallois and M. Yor, Some penalisations of the Wiener measure, Jpn. J. Math., 1 (2006), 263–290.
• B. Roynette and M. Yor, Penalising Brownian paths: Lecture Notes in Math., 1969, Springer, Berlin, 2009.
• P. Salminen, On last exit decompositions of linear diffusions, Studia Sci. Math. Hungar., 33 (1997), 251–262.
• P. Salminen and M. Yor, Tanaka formula for symmetric Lévy processes, In Séminaire de Probabilités, XL, Lecture Notes in Math., 1899, Springer, Berlin, 2007, pp. 265–285.
• K. Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge, 1999.
• K. Yano, Excursions away from a regular point for one-dimensional symmetric Lévy processes without Gaussian part, submitted, preprint, arXiv:0805.3881, 2008.
• K. Yano, Y. Yano and M. Yor, On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes, Séminaire de Probabilités XLII, Lecture Notes in Math., 1979, Springer, to appear in 2009.