The Annals of Probability

Multiple points of the Brownian sheet in critical dimensions

Robert C. Dalang and Carl Mueller

Full-text: Open access


It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1577-1593.

Received: February 2013
Revised: November 2013
First available in Project Euclid: 3 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 60G15: Gaussian processes 60G60: Random fields

Brownian sheet multiple points Girsanov’s theorem


Dalang, Robert C.; Mueller, Carl. Multiple points of the Brownian sheet in critical dimensions. Ann. Probab. 43 (2015), no. 4, 1577--1593. doi:10.1214/14-AOP912.

Export citation


  • [1] Dalang, R. C. (2003). Level sets and excursions of the Brownian sheet. In Topics in Spatial Stochastic Processes (Martina Franca, 2001). Lecture Notes in Math. 1802 167–208. Springer, Berlin.
  • [2] Dalang, R. C., Khoshnevisan, D., Nualart, E., Wu, D. and Xiao, Y. (2012). Critical Brownian sheet does not have double points. Ann. Probab. 40 1829–1859.
  • [3] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [4] Kendall, W. S. (1980). Contours of Brownian processes with several-dimensional times. Z. Wahrsch. Verw. Gebiete 52 267–276.
  • [5] Khoshnevisan, D. (1997). Some polar sets for the Brownian sheet. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 190–197. Springer, Berlin.
  • [6] Khoshnevisan, D. (2002). Multiparameter Processes: An Introduction to Random Fields. Springer, New York.
  • [7] Khoshnevisan, D. and Shi, Z. (1999). Brownian sheet and capacity. Ann. Probab. 27 1135–1159.
  • [8] Nualart, D. and Pardoux, E. (1994). Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs. Stochastics Stochastics Rep. 48 17–44.
  • [9] Orey, S. and Pruitt, W. E. (1973). Sample functions of the $N$-parameter Wiener process. Ann. Probab. 1 138–163.
  • [10] 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.
  • [11] Walsh, J. B. (1982). Propagation of singularities in the Brownian sheet. Ann. Probab. 10 279–288.
  • [12] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d’Été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.