Joint continuity of the local times of fractional brownian sheets
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, p. 727-748
Désignons par ${B}^{H}=\left\{{B}^{H}\left(t\right),t\in {ℝ}_{+}^{N}\right\}$ le $\left(N,d\right)$-drap Brownien fractionnaire de paramètre $H=\left({H}_{1},...,{H}_{N}\right)\in {\left(0,1\right)}^{N}$ défini par ${B}^{H}\left(t\right)=\left({B}_{1}^{H}\left(t\right),...,{B}_{d}^{H}\left(t\right)\right)\left(t\in {ℝ}_{+}^{N}\right)$, où ${B}_{1}^{H},...,{B}_{d}^{H}$ sont des copies indépendantes du drap Brownien fractionnaire à valeurs réelles ${B}_{0}^{H}$. Nous montrons que le temps local de ${B}^{H}$ est bicontinu lorsque $d<{\sum }_{\ell =1}^{N}{H}_{\ell }^{-1}$. Cela résout une conjecture de Xiao et Zhang (Probab. Theory Related Fields 124 (2002)). Nous obtenons aussi des résultats fins concernant la régularité Hölderienne, locale et globale, du temps local. Ces résultats nous permettent d’étudier certaines propriétés analytiques et géométriques des trajectoires de ${B}^{H}$.
Let ${B}^{H}=\left\{{B}^{H}\left(t\right),t\in {ℝ}_{+}^{N}\right\}$ be an $\left(N,d\right)$-fractional brownian sheet with index $H=\left({H}_{1},...,{H}_{N}\right)\in {\left(0,1\right)}^{N}$ defined by ${B}^{H}\left(t\right)=\left({B}_{1}^{H}\left(t\right),...,{B}_{d}^{H}\left(t\right)\right)\left(t\in {ℝ}_{+}^{N}\right)$, where ${B}_{1}^{H},...,{B}_{d}^{H}$ are independent copies of a real-valued fractional brownian sheet ${B}_{0}^{H}$. We prove that if $d<{\sum }_{\ell =1}^{N}{H}_{\ell }^{-1}$, then the local times of ${B}^{H}$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global Hölder conditions for the local times of ${B}^{H}$. These results are applied to study analytic and geometric properties of the sample paths of ${B}^{H}$.
DOI : https://doi.org/10.1214/07-AIHP131
Classification:  60G15,  60G17
@article{AIHPB_2008__44_4_727_0,
author = {Ayache, Antoine and Wu, Dongsheng and Xiao, Yimin},
title = {Joint continuity of the local times of fractional brownian sheets},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {44},
number = {4},
year = {2008},
pages = {727-748},
doi = {10.1214/07-AIHP131},
zbl = {1180.60032},
mrnumber = {2446295},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2008__44_4_727_0}
}

Ayache, Antoine; Wu, Dongsheng; Xiao, Yimin. Joint continuity of the local times of fractional brownian sheets. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 727-748. doi : 10.1214/07-AIHP131. http://www.numdam.org/item/AIHPB_2008__44_4_727_0/

 R. J. Adler. The Geometry of Random Fields. Wiley, New York, 1981. | MR 611857 | Zbl 0478.60059

 A. Ayache, S. Leger and M. Pontier. Drap Brownien fractionnaire. Potential Anal. 17 (2002) 31-43. | MR 1906407 | Zbl 1006.60029

 A. Ayache and Y. Xiao. Asymptotic properties and Hausdorff dimension of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407-439. | MR 2169474 | Zbl 1088.60033

 D. A. Benson, M. M. Meerschaert and B. Baeumer. Aquifer operator-scaling and the effect on solute mixing and dispersion. Water Resour. Res. 42 (2006) W01415.

 S. M. Berman. Local times and sample function properties of stationary Gaussian processes. Trans. Amer. Math. Soc. 137 (1969) 277-299. | MR 239652 | Zbl 0184.40801

 S. M. Berman. Gaussian sample function: uniform dimension and Hölder conditions nowhere. Nagoya Math. J. 46 (1972) 63-86. | MR 307320 | Zbl 0246.60038

 S. M. Berman. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973) 69-94. | MR 317397 | Zbl 0264.60024

 A. Bonami and A. Estrade. Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003) 215-236. | MR 1988750 | Zbl 1034.60038

 J. Cuzick and J. Dupreez. Joint continuity of Gaussian local times. Ann. Probab. 10 (1982) 810-817. | MR 659550 | Zbl 0492.60032

 M. Dozzi. Occupation density and sample path properties of N-parameter processes. Topics in Spatial Stochastic Processes (Martina Franca, 2001) 127-166. Lecture Notes in Math. 1802. Springer, Berlin, 2002. | MR 1975519 | Zbl 1042.60031

 T. Dunker. Estimates for the small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 13 (2000) 357-382. | MR 1777539 | Zbl 0971.60041

 W. Ehm. Sample function properties of multi-parameter stable processes. Z. Wahrsch. Verw Gebiete 56 (1981) 195-228. | MR 618272 | Zbl 0471.60046

 D. Geman and J. Horowitz. Occupation densities. Ann. Probab. 8 (1980) 1-67. | MR 556414 | Zbl 0499.60081

 D. Geman, J. Horowitz and J. Rosen. A local time analysis of intersections of Brownian paths in the plane. Ann. Probab. 12 (1984) 86-107. | MR 723731 | Zbl 0536.60046

 G. H. Hardy. Inequalities. Cambridge Univ. Press, 1934. | JFM 60.0169.01 | Zbl 0010.10703

 H. Kesten. An iterated logarithm law for local time. Duke Math. J. 32 (1965) 447-456. | MR 178494 | Zbl 0132.12701

 D. Khoshnevisan. Multiparameter Processes: An Introduction to Random Fields. Springer, New York, 2002. | MR 1914748 | Zbl 1005.60005

 D. Khoshnevisan, D. Wu and Y. Xiao. Sectorial local non-determinism and the geometry of the Brownian sheet. Electron. J. Probab. 11 (2006) 817-843. | MR 2261054 | Zbl 1111.60020

 D. Khoshnevisan and Y. Xiao. Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 (2007) 3125-3151. | MR 2299449 | Zbl 1124.60037

 D. Khoshnevisan, Y. Xiao and Y. Zhong. Local times of additive Lévy processes. Stoch. Process. Appl. 104 (2003) 193-216. | MR 1961619 | Zbl 1075.60520

 D. M. Mason and Z. Shi. Small deviations for some multi-parameter Gaussian processes. J. Theoret. Probab. 14 (2001) 213-239. | MR 1822902 | Zbl 0982.60024

 T. S. Mountford. A relation between Hausdorff dimension and a condition on time sets for the image by the Brownian sheet to possess interior-points. Bull. London Math. Soc. 21 (1989) 179-185. | MR 976063 | Zbl 0668.60044

 T. S. Mountford and D. Baraka. A law of the iterated logarithm for fractional Brownian motions. Preprint, 2005. | Zbl 1157.60030

 B. Øksendal and T. Zhang. Multiparameter fractional Brownian motion and quasi-linear stochastic partial differential equations. Stochastics Stochastics Rep. 71 (2000) 141-163. | MR 1922562 | Zbl 0986.60056

 L. D. Pitt. Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978) 309-330. | MR 471055 | Zbl 0382.60055

 C. A. Rogers and S. J. Taylor. Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8 (1961) 1-31. | MR 130336 | Zbl 0145.28701

 J. Rosen. Self-intersections of random fields. Ann. Probab. 12 (1984) 108-119. | MR 723732 | Zbl 0536.60066

 M. Talagrand. Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23 (1995) 767-775. | MR 1334170 | Zbl 0830.60034

 D. Wu and Y. Xiao. Geometric properties of fractional Brownian sheets. J. Fourier Anal. Appl. 13 (2007) 1-37. | MR 2296726 | Zbl 1127.60032

 Y. Xiao. Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 (1997) 129-157. | MR 1469923 | Zbl 0882.60035

 Y. Xiao. Properties of local nondeterminism of Gaussian and stable random fields and their applications. Ann. Fac. Sci. Toulouse Math. XV (2006) 157-193. | Numdam | MR 2225751 | Zbl 1128.60041

 Y. Xiao. Sample path properties of anisotropic Gaussian random fields. Submitted, 2007. | Zbl 1167.60011

 Y. Xiao and T. Zhang. Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 (2002) 204-226. | MR 1936017 | Zbl 1009.60024