The Annals of Probability

Polarity of points for Gaussian random fields

Robert C. Dalang, Carl Mueller, and Yimin Xiao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space–time white noise, or colored noise in spatial dimensions $k\geq1$. Our approach builds on a delicate covering argument developed by M. Talagrand [Ann. Probab. 23 (1995) 767–775; Probab. Theory Related Fields 112 (1998) 545–563] for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic PDEs.

Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4700-4751.

Dates
Received: May 2015
Revised: November 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069271

Digital Object Identifier
doi:10.1214/17-AOP1176

Mathematical Reviews number (MathSciNet)
MR3737922

Zentralblatt MATH identifier
06838131

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

Keywords
Hitting probabilities polarity of points critical dimension harmonizable representation stochastic partial differential equations

Citation

Dalang, Robert C.; Mueller, Carl; Xiao, Yimin. Polarity of points for Gaussian random fields. Ann. Probab. 45 (2017), no. 6B, 4700--4751. doi:10.1214/17-AOP1176. https://projecteuclid.org/euclid.aop/1513069271


Export citation

References

  • [1] Balan, R. M. (2012). Linear SPDEs driven by stationary random distributions. J. Fourier Anal. Appl. 18 1113–1145.
  • [2] 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.
  • [3] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York.
  • [4] Cairoli, R. and Walsh, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134 111–183.
  • [5] Cambanis, S. and Liu, B. (1970). On harmonizable stochastic processes. Inf. Control 17 183–202.
  • [6] Clarke de la Cerda, J. and Tudor, C. A. (2014). Hitting times for the stochastic wave equation with fractional colored noise. Rev. Mat. Iberoam. 30 685–709.
  • [7] Cohen, S. and Istas, J. (2013). Fractional Fields and Applications. Mathématiques & Applications (Berlin) 73. Springer, Heidelberg.
  • [8] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 no. 6, 29 pp.
  • [9] 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.
  • [10] Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2009). Hitting probabilities for systems for non-linear stochastic heat equations with multiplicative noise. Probab. Theory Related Fields 144 371–427.
  • [11] Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2013). Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $k\geq1$. Stoch. Partial Differ. Equ., Anal. Computat. 1 94–151.
  • [12] 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.
  • [13] Dalang, R. C. and Mueller, C. (2015). Multiple points of the Brownian sheet in critical dimensions. Ann. Probab. 43 1577–1593.
  • [14] Dalang, R. C. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099–2148.
  • [15] Dalang, R. C. and Sanz-Solé, M. (2010). Criteria for hitting probabilities with applications to systems of stochastic wave equations. Bernoulli 16 1343–1368.
  • [16] Dalang, R. C. and Sanz-Solé, M. (2015). Hitting probabilities for nonlinear systems of stochastic waves. Mem. Amer. Math. Soc. 237 v+75.
  • [17] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • [18] Itô, K. (1954). Stationary random distributions. Mem. Coll. Sci., Univ. Kyoto, Ser. 1: Math 28 291–326.
  • [19] Khoshnevisan, D. (2002). Multiparameter Processes. An Introduction to Random Fields. Springer, New York.
  • [20] Khoshnevisan, D. (2003). Intersections of Brownian motions. Expo. Math. 21 97–114.
  • [21] Khoshnevisan, D. and Shi, Z. (1999). Brownian sheet and capacity. Ann. Probab. 27 1135–1159.
  • [22] Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 165–294. Springer, Berlin.
  • [23] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) 23. Springer, Berlin.
  • [24] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
  • [25] Mueller, C. and Tribe, R. (2002). Hitting properties of a random string. Electron. J. Probab. 7. 29 pp.
  • [26] Mueller, C. and Wu, Z. (2009). A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of talagrand. Electron. Commun. Probab. 14 55–65.
  • [27] Nualart, E. and Viens, F. (2009). The fractional stochastic heat equation on the circle: Time regularity and potential theory. Stochastic Process. Appl. 119 1505–1540.
  • [28] Sanz-Solé, M. (2005). Malliavin Calculus. With Applications to Stochastic Partial Differential Equations. EPFL Press, Lausanne.
  • [29] Talagrand, M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23 767–775.
  • [30] Talagrand, M. (1998). Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theory Related Fields 112 545–563.
  • [31] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In Ecole D’Eté de Probabilités de Saint-Flour XIV. Lect. Notes in Math. 1180 265–439. Springer, Berlin.
  • [32] Wu, D. (2011). On the solution process for a stochastic fractional partial differential equation driven by space–time white noise. Statist. Probab. Lett. 81 1161–1172.
  • [33] Xiao, Y. (1997). Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Related Fields 109 129–157.
  • [34] Xiao, Y. (2009). Sample path properties of anisotropic Gaussian random fields. In A Minicourse on Stochastic Partial Differential Equations (D. Khoshnevisan and F. Rassoul-Agha, eds.). Lecture Notes in Math. 1962 145–212. Springer, Berlin.
  • [35] Yaglom, A. M. (1957). Some classes of random fields in $n$-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 273–320.