Continuous differentiability of renormalized intersection local times in R1

Jay S. Rosen

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 4, page 1025-1041
  • ISSN: 0246-0203

Abstract

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We study γk(x2, …, xk; t), the k-fold renormalized self-intersection local time for brownian motion in R1. Our main result says that γk(x2, …, xk; t) is continuously differentiable in the spatial variables, with probability 1.

How to cite

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Rosen, Jay S.. "Continuous differentiability of renormalized intersection local times in R1." Annales de l'I.H.P. Probabilités et statistiques 46.4 (2010): 1025-1041. <http://eudml.org/doc/242562>.

@article{Rosen2010,
abstract = {We study γk(x2, …, xk; t), the k-fold renormalized self-intersection local time for brownian motion in R1. Our main result says that γk(x2, …, xk; t) is continuously differentiable in the spatial variables, with probability 1.},
author = {Rosen, Jay S.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {continuous differentiability; intersection local time; brownian motion; Brownian motion},
language = {eng},
number = {4},
pages = {1025-1041},
publisher = {Gauthier-Villars},
title = {Continuous differentiability of renormalized intersection local times in R1},
url = {http://eudml.org/doc/242562},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Rosen, Jay S.
TI - Continuous differentiability of renormalized intersection local times in R1
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 4
SP - 1025
EP - 1041
AB - We study γk(x2, …, xk; t), the k-fold renormalized self-intersection local time for brownian motion in R1. Our main result says that γk(x2, …, xk; t) is continuously differentiable in the spatial variables, with probability 1.
LA - eng
KW - continuous differentiability; intersection local time; brownian motion; Brownian motion
UR - http://eudml.org/doc/242562
ER -

References

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  1. [1] R. Bass and D. Khoshnevisan. Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 419–452. Zbl0798.60072MR1246641
  2. [2] E. B. Dynkin. Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 (1988) 1–57. Zbl0638.60081MR920254
  3. [3] J.-F. Le Gall. Propriétés d’intersection des marches aléatoires, I. Comm. Math. Phys. 104 (1986) 471–507. Zbl0609.60078MR840748
  4. [4] J.-F. Le Gall. Fluctuation results for the Wiener sausage. Ann. Probab. 16 (1988) 991–1018. Zbl0665.60080MR942751
  5. [5] J.-F. Le Gall. Some properties of planar Brownian motion. In École d’ Été de Probabilités de St. Flour XX, 1990. 111–235. Lecture Notes in Math. 1527 Springer, Berlin, 1992. Zbl0779.60068MR1229519
  6. [6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1998. Zbl0731.60002
  7. [7] J. Rosen. Tanaka’s formula for multiple intersections of planar Brownian motion. Stochastic Process. Appl. 23 (1986) 131–141. Zbl0612.60070MR866291
  8. [8] J. Rosen. A renormalized local time for the multiple intersections of planar Brownian motion. In Séminaire de Probabilités XX, 1984/85. 515–531. Lecture Notes in Math. 1204. Springer, Berlin, 1986. Zbl0611.60065MR942041
  9. [9] J. Rosen. Derivatives of self-intersection local times. In Séminaire de Probabilités, XXXVIII 263–281. Lecture Notes in Math. 1857. Springer, New York, 2005. Zbl1063.60110MR2126979
  10. [10] J. Rosen. Joint continuity and a Doob–Meyer type decomposition for renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 143–176. Zbl0922.60072MR1678521
  11. [11] J. Rosen. Joint continuity of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 671–700. Zbl0867.60049MR1422306
  12. [12] J. Rosen. Dirichlet processes and an intrinsic characterization of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 403–420. Zbl0981.60072MR1876838
  13. [13] J. Rosen. A stochastic calculus proof of the CLT for the L2 modulus of continuity of local time. Preprint. Zbl1219.60028MR2790369
  14. [14] S. R. S. Varadhan. Appendix to Euclidian quantum field theory by K. Symanzyk. In Local Quantum Theory. R. Jost (ed.). Academic Press, New York, 1969. 

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