The Annals of Probability

Critical Brownian sheet does not have double points

Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart, Dongsheng Wu, and Yimin Xiao

Full-text: Open access


We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^{d}$ has double points if and only if $d<4N$. In particular, in the critical case where $d=4N$, the Brownian sheet does not have double points. This answers an old problem in the folklore of the subject. We also discuss some of the geometric consequences of the mentioned decoupling, and establish a partial result concerning $k$-multiple points in the critical case $k(d-2N)=d$.

Article information

Ann. Probab., Volume 40, Number 4 (2012), 1829-1859.

First available in Project Euclid: 4 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60G15: Gaussian processes

Brownian sheet multiple points capacity Hausdorff dimension


Dalang, Robert C.; Khoshnevisan, Davar; Nualart, Eulalia; Wu, Dongsheng; Xiao, Yimin. Critical Brownian sheet does not have double points. Ann. Probab. 40 (2012), no. 4, 1829--1859. doi:10.1214/11-AOP665.

Export citation


  • [1] Biermé, H., Lacaux, C. and Xiao, Y. (2009). Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull. Lond. Math. Soc. 41 253–273.
  • [2] Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134 111–183.
  • [3] Čencov, N. N. (1956). Wiener random fields depending on several parameters. Dokl. Akad. Nauk SSSR (N.S.) 106 607–609.
  • [4] Chen, X. (1994). Hausdorff dimension of multiple points of the $(N,d)$ Wiener process. Indiana Univ. Math. J. 43 55–60.
  • [5] Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2007). Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3 231–271.
  • [6] Dalang, R. C. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099–2148.
  • [7] de la Peña, V. H. and Giné, E. (1999). Decoupling: From Dependence to Independence. Randomly Stopped Processes. $U$-statistics and Processes. Martingales and Beyond. Springer, New York.
  • [8] Dvoretzky, A., Erdös, P. and Kakutani, S. (1950). Double points of paths of Brownian motion in $n$-space. Acta Sci. Math. Szeged 12 75–81.
  • [9] Dvoretzky, A., Erdös, P. and Kakutani, S. (1954). Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3 364–371.
  • [10] Dvoretzky, A., Erdős, P., Kakutani, S. and Taylor, S. J. (1957). Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 53 856–862.
  • [11] Kakutani, S. (1944). On Brownian motions in $n$-space. Proc. Imp. Acad. Tokyo 20 648–652.
  • [12] Khoshnevisan, D. (1997). Some polar sets for the Brownian sheet. In Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 190–197. Springer, Berlin.
  • [13] Khoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. Springer, New York.
  • [14] Khoshnevisan, D. and Shi, Z. (1999). Brownian sheet and capacity. Ann. Probab. 27 1135–1159.
  • [15] Khoshnevisan, D., Wu, D. and Xiao, Y. (2006). Sectorial local non-determinism and the geometry of the Brownian sheet. Electron. J. Probab. 11 817–843 (electronic).
  • [16] Khoshnevisan, D. and Xiao, Y. (2007). Images of the Brownian sheet. Trans. Amer. Math. Soc. 359 3125–3151 (electronic).
  • [17] Lévy, P. (1940). Le mouvement brownien plan. Amer. J. Math. 62 487–550.
  • [18] Mountford, T. S. (1989). Uniform dimension results for the Brownian sheet. Ann. Probab. 17 1454–1462.
  • [19] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [20] Orey, S. and Pruitt, W. E. (1973). Sample functions of the $N$-parameter Wiener process. Ann. Probab. 1 138–163.
  • [21] Peres, Y. (1996). Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177 417–434.
  • [22] Peres, Y. (1999). Probability on trees: An introductory climb. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 193–280. Springer, Berlin.
  • [23] Rosen, J. (1984). Self-intersections of random fields. Ann. Probab. 12 108–119.
  • [24] Ville, J. (1942). Sur un problème de géométrie suggéré par l’étude du mouvement brownien. C. R. Acad. Sci. Paris 215 51–52.
  • [25] Wiener, N. (1923). Differerential space. J. Math. Phys. 2 131–174.
  • [26] Wiener, N. (1938). The homogeneous chaos. Amer. J. Math. 60 897–936.
  • [27] Xiao, Y. (1999). Hitting probabilities and polar sets for fractional Brownian motion. Stochastics Stochastics Rep. 66 121–151.
  • [28] Xiao, Y. (2009). Sample path properties of anisotropic Gaussian random fields. In A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math. 1962 (D. Khoshnevisan and F. Rassoul-Agha, eds.) 145–212. Springer, Berlin.