Geodesic
Local time as an element of the Sobolev space
Teoriâ slučajnyh processov, Tome 13 (2007) no. 3, pp. 65-79.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a centered Gaussian random field taking its values in d, we investigate the existence of a local time as a generalized functional, i.e an element of some Sobolev space. We give the sufficient condition for such an existence in terms of the field covariation and apply it in several examples: the self-intersection local time for a fractional Brownian motion and the intersection local time for two Brownian motions.
Keywords: Local time, Itô–Wiener expansion, Gaussian random field, fractional Brownian motion.
Mots-clés : Sobolev spaces 
@article{THSP_2007_13_3_a7,
     author = {Alexey V. Rudenko},
     title = {Local time as an element of the {Sobolev} space},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {65--79},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2007_13_3_a7/}
}
TY  - JOUR
AU  - Alexey V. Rudenko
TI  - Local time as an element of the Sobolev space
JO  - Teoriâ slučajnyh processov
PY  - 2007
SP  - 65
EP  - 79
VL  - 13
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2007_13_3_a7/
LA  - en
ID  - THSP_2007_13_3_a7
ER  - 
%0 Journal Article
%A Alexey V. Rudenko
%T Local time as an element of the Sobolev space
%J Teoriâ slučajnyh processov
%D 2007
%P 65-79
%V 13
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2007_13_3_a7/
%G en
%F THSP_2007_13_3_a7
Alexey V. Rudenko. Local time as an element of the Sobolev space. Teoriâ slučajnyh processov, Tome 13 (2007) no. 3, pp. 65-79. http://geodesic.mathdoc.fr/item/THSP_2007_13_3_a7/

[1] A. A. Dorogovtsev, V. V. Bakun, “Random mappings and a generalized additive functional of a Wiener process”, Theory of Stoch.Proc., 48:1 (2003)

[2] A. Rudenko, “Existence of generalized local times for Gaussian random fields”, Theory of Stoch. Proc., 12(28):1–2 (2006), 142-–154

[3] H. H. Kuo, Gaussian Measures in Banach Spaces, Springer, Berlin, 1975

[4] H. Watanabe, “The local time of self-intersections of brownian motions as generalized brownian functionals”, Lett. in Math. Phys., 23 (1991), 1–9

[5] P. Imkeller, V. Perez-Abreu, J. Vives, “Chaos expansions of double intersection local time of brownian motion in ${\mathbb R}^d$ and renormalization”, Stoch.Proc.and Appl., 56 (1995), 1–34

[6] P. Malliavin, Stochastic Analysis, Springer, Berlin, 1997

[7] J. Rosen, “A renormalized local time for multiple intersections of planar brownian motion”, Seminaire de Probabilities XX, 20, 1986, 515–531

[8] S. Albeverio, Y. Hu, X. Y. Zhou, “A remark on non-smoothness of the self-intersection local time of planar brownian motion”, Stat. and Prob. Lett., 32 (1997), 57–65

[9] S. Orey, “Gaussian sample functions and the Hausdorff dimension of level crossings”, Wahrscheinlichkeitstheorie verw. Geb., 15 (1970), 249–256

[10] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Springer, Berlin, 1984

[11] Y. Hu, D. Nualart, “Renormalized self-intersection local time for fractional brownian motion”, Annals of Prob., 33:3 (2005), 948–983

[12] Y. Hu, D. Nualart, “Regularity of renormalized self-intersection local time for fractional Brownian motion”, J. of Commun. in Information and Systems (CIS), 7:1 (2007), 21–30