×

Continuous differentiability of renormalized intersection local times in \(\mathbb{R}^{1}\). (English) Zbl 1210.60084

Summary: We study \(\gamma_k(x_{2}, \ldots , x_k; t)\), the \(k\)-fold renormalized self-intersection local time for a Brownian motion in \(\mathbb{R}^{1}\). Our main result says that \(\gamma _k(x_{2}, \ldots , x_k; t)\) is continuously differentiable in the spatial variables, with probability 1.

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML

References:

[1] R. Bass and D. Khoshnevisan. Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 419-452. · Zbl 0798.60072
[2] E. B. Dynkin. Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 (1988) 1-57. · Zbl 0638.60081
[3] J.-F. Le Gall. Propriétés d’intersection des marches aléatoires, I. Comm. Math. Phys. 104 (1986) 471-507. · Zbl 0609.60078
[4] J.-F. Le Gall. Fluctuation results for the Wiener sausage. Ann. Probab. 16 (1988) 991-1018. · Zbl 0665.60080
[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. · Zbl 0779.60068
[6] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion . Springer, Berlin, 1998. · Zbl 1087.60040
[7] J. Rosen. Tanaka’s formula for multiple intersections of planar Brownian motion. Stochastic Process. Appl. 23 (1986) 131-141. · Zbl 0612.60070
[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. · Zbl 0611.60065
[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. · Zbl 1063.60110
[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. · Zbl 0922.60072
[11] J. Rosen. Joint continuity of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 671-700. · Zbl 0867.60049
[12] J. Rosen. Dirichlet processes and an intrinsic characterization of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 403-420. · Zbl 0981.60072
[13] J. Rosen. A stochastic calculus proof of the CLT for the L 2 modulus of continuity of local time. · Zbl 1219.60028
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.