The Annals of Probability

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

Richard Bass, Xia Chen, and Jay Rosen

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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

Keywords
Large deviations stable processes intersection local time law of the iterated logarithm self-intersections

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


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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.