## The Annals of Probability

### Large deviations for renormalized self-intersection local times of stable processes

#### Abstract

We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index β∈(2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether β<d or β=d.

#### Article information

Source
Ann. Probab., Volume 33, Number 3 (2005), 984-1013.

Dates
First available in Project Euclid: 6 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1115386716

Digital Object Identifier
doi:10.1214/009117904000001099

Mathematical Reviews number (MathSciNet)
MR2135310

Zentralblatt MATH identifier
1087.60060

Subjects
Primary: 60J55: Local time and additive functionals
Secondary: 60G52: Stable processes

#### Citation

Bass, Richard; Chen, Xia; Rosen, Jay. Large deviations for renormalized self-intersection local times of stable processes. Ann. Probab. 33 (2005), no. 3, 984--1013. doi:10.1214/009117904000001099. https://projecteuclid.org/euclid.aop/1115386716

#### References

• Bass, R. F. (1995). Probabilistic Techniques in Analysis. Springer, New York.
• Bass, R. F. and Chen, X. (2004). Self intersection local time: Critical exponent, large deviations and laws of the iterated logarithm. Ann. Probab. 32 3221–3247.
• Bass, R. F. and Khoshnevisan, D. (1993). Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré Probab. Statist. 29 419–452.
• Bass, R. F. and Levin, D. A. (2002). Harnack inequalities for jump processes. Potential Anal. 17 375–388.
• Chen, X. (2004). Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 3248–3300.
• Chen, X. and Li, W. (2004). Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 213–254.
• Chen, X., Li, W. and Rosen, J. (2005). Large deviations for local times of stable processes and stable random walks in 1 dimension. Electron. J. Probab. To appear.
• Chen, X. and Rosen, J. (2005). Exponential asymptotics for intersection local times of stable processes and random walk. Ann. Inst. H. Poincaré. To appear.
• Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Springer, New York.
• Dynkin, E. B. (1988). Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 1–57.
• Gine, E. and de la Pena, V. (1999). Decoupling. Springer, Berlin.
• König, W. and Mörters, P. (2002). Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 1605–1656.
• Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
• Le Gall, J.-F. (1986). Proprietes d'intersection des marches aleatoires. Comm. Math. Phys. 104 471–507.
• Le Gall, J.-F. (1988). Fluctuation results for the Wiener sausage. Ann. Probab. 16 991–1018.
• Le Gall, J.-F. (1990). Some properties of planar Brownian motion. École d'Été de Probabilités de Saint-Flour XX. Lecture Notes in Math. 1527 112–234. Springer, Berlin.
• Le Gall, J.-F. (1994). Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 172–180. Springer, Berlin.
• Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705.
• Marcus, M. and Rosen, J. (1994). Laws of the iterated logarithm for the local times of symmetric Lévy processes and recurrent random walks. Ann. Probab. 22 626–659.
• Marcus, M. and Rosen, J. (1994). Laws of the iterated logarithm for the local times of recurrent random walks on $Z^2$ and of Lévy processes and random walks in the domain of attraction of Cauchy random variables. Ann. Inst. H. Poincaré Probab. Statist. 30 467–499.
• Marcus, M. and Rosen, J. (1999). Renormalized self-intersection local times and Wick power chaos processes. Mem. Amer. Math. Soc. 142.
• Rosen, J. (1988). Continuity and singularity of the intersection local time of stable processes in $R^{2}$. Ann. Probab. 16 75–79.
• Rosen, J. (1988). Limit laws for the intersection local time of stable processes in $R^{2}$. Stochastics 23 219–240.
• Rosen, J. (1990). Random walks and intersection local time. Ann. Probab. 18 959–977.
• Rosen, J. (1992). The asymptotics of stable sausages in the plane. Ann. Probab. 20 29–60.
• Rosen, J. (1996). Joint continuity of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 32 671–700.
• Rosen, J. (2001). Dirichlet processes and an intrinsic characterization for renormalized intersection local times. Ann. Inst. H. Poincaré 37 403–420.
• Varadhan, S. R. S. (1969). Appendix to “Euclidian quantum field theory” by K. Symanzik. In Local Quantum Theory (R. Jost, ed.). Academic Press, Reading, MA.