Continuous differentiability of renormalized intersection local times in R 1
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1025-1041.

Nous étudions γk(x2, …, xk; t), le temps local renormalisé d'auto-intersection d'ordre k du mouvement brownien dans R1. Notre résultat principal montre que γk(x2, …, xk; t) est presque sûrement continûment différentiable dans les variables spatiales.

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.

DOI : 10.1214/09-AIHP338
Classification : 60J55, 60J65
Mots-clés : continuous differentiability, intersection local time, brownian motion 
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Rosen, Jay S. Continuous differentiability of renormalized intersection local times in $R^1$. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 4, pp. 1025-1041. doi : 10.1214/09-AIHP338. http://www.numdam.org/articles/10.1214/09-AIHP338/

[1] R. Bass and D. Khoshnevisan. Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 419-452. | Numdam | MR | Zbl

[2] E. B. Dynkin. Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 (1988) 1-57. | MR | Zbl

[3] J.-F. Le Gall. Propriétés d'intersection des marches aléatoires, I. Comm. Math. Phys. 104 (1986) 471-507. | MR | Zbl

[4] J.-F. Le Gall. Fluctuation results for the Wiener sausage. Ann. Probab. 16 (1988) 991-1018. | MR | Zbl

[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. | MR | Zbl

[6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1998. | Zbl

[7] J. Rosen. Tanaka's formula for multiple intersections of planar Brownian motion. Stochastic Process. Appl. 23 (1986) 131-141. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[11] J. Rosen. Joint continuity of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 671-700. | Numdam | MR | Zbl

[12] J. Rosen. Dirichlet processes and an intrinsic characterization of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 403-420. | Numdam | MR | Zbl

[13] J. Rosen. A stochastic calculus proof of the CLT for the L2 modulus of continuity of local time. Preprint. | MR | Zbl

[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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