Abstract
We consider a d-dimensional random field that solves a possibly non-linear system of stochastic partial differential equations, such as stochastic heat or wave equations. We present results, obtained in joint works with Davar Khoshnevisan and Eulalia Nualart, and with Marta Sanz-Solé, on upper and lower bounds on the probabilities that the random field visits a deterministic subset of \(\mathbb {R}^d\), in terms, respectively, of Hausdorff measure and Newtonian capacity of the subset. These bounds determine the critical dimension above which points are polar, but do not, in general, determine whether points are polar in the critical dimension. For linear SPDEs, we discuss, based on joint work with Carl Mueller and Yimin Xiao, how the issue of polarity of points can be resolved in the critical dimension.
The research of the author is partly supported by the Swiss National Foundation for Scientific Research.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Balan, R.: Linear SPDEs driven by stationary random distributions. J. Fourier Anal. Appl. 18(6), 1113–1145 (2012)
Biermé, H., Lacaux, C., Xiao, Y.: Hitting probabilities and the Hausdorff dimension of the inverse images of anisotropic Gaussian random fields. Bull. Lond. Math. Soc. 41, 253–273 (2009)
Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Pure and Applied Mathematics, vol. 29. Academic Press, New York (1968)
Chen, Z.L., Zhou, Q.: Hitting probabilities and the Hausdorff dimension of the inverse images of a class of anisotropic random fields. Acta Math. Sin. 31, 1895–1922 (2015)
Clarke de la Cerda, J., Tudor, C.A.: Hitting times for the stochastic wave equation with fractional colored noise. Rev. Mat. Iberoam. 30, 685–709 (2014)
Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. Latin Am. J. Probab. Stat. (ALEA) 3, 231–271 (2007)
Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems for non-linear stochastic heat equations with multiplicative noise. Probab. Theory Related Fields 144, 371–427 (2009)
Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension \(k\ge 1\). J. SPDE’s Anal. Comput. 1–1, 94–151 (2013)
Dalang, R.C., Mueller, C.: Multiple points of the Brownian sheet in critical dimensions. Ann. Probab. 43(4), 1577–1593 (2015)
Dalang, R.C., Mueller, C., Xiao, Y.: Polarity, of points for Gaussian random fields. Ann. Probab. 45(6B), 4700–4751 (2017)
Dalang, R.C., Nualart, E.: Potential theory for hyperbolic SPDEs. Ann. Probab. 32, 2099–2148 (2004)
Dalang, R.C., Sanz-Solé, M.: Hitting probabilities for non-linear systems of stochastic waves. Mem. Am. Math. Soc. 237(1120), 1–75 (2015)
Doob, J.L.: Classical potential theory and its probabilistic counterpart. Grundlehren der Mathematischen Wissenschaften, vol. 262. Springer, New York (1984)
Khoshnevisan, D.: Multiparameter Processes. An Introduction to Random Fields. Springer Monographs in Mathematics. Springer, New York (2002)
Khoshnevisan, D.: Intersections of Brownian motions. Exp. Math. 21(3), 97–114 (2003)
Khoshnevisan, D., Shi, Z.: Brownian sheet and capacity. Ann. Probab. 27(3), 1135–1159 (1999)
Kohatsu-Higa, A.: Lower bound estimates for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126, 421–457 (2003)
Lévy, P.: Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris (1948)
Mueller, C., Tribe, R.: Hitting properties of a random string. Electron. J. Probab. 7(10) (2002)
Nualart, E.: On the density of systems of non-linear spatially homogeneous SPDEs. Stochastics 85(1), 48–70 (2013)
Nualart, E., Viens, F.: The fractional stochastic heat equation on the circle: time regularity and potential theory. Stoch. Process. Appl. 119(5), 1505–1540 (2009)
Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press, New York (1978)
Sanz-Solé, M., Sarrà, M.: Hölder continuity for the stochastic heat equation with spatially correlated noise, Seminar on Stochastic Analysis, Random Fields ans Applications, III (Ascona, 1999). Progr. Prob. 52, 259–268 (2002)
Sanz-Solé, M., Viles, N.: Systems of stochastic Poisson equations: hitting probabilities. Stochastic Process. Appl. (2016). (to appear). arXiv:1612.04567
Talagrand, M.: Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23(2), 767–775 (1995)
Talagrand, M.: Multiple points of trajectories of multiparameter fractional Brownian motion. Probab. Theory Related Fields 112–4, 545–563 (1998)
Xiao, Y.: Sample path properties of anisotropic Gaussian random fields. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1962, pp. 145–212. Springer, New York (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Dalang, R.C. (2018). Hitting Probabilities for Systems of Stochastic PDEs: An Overview. In: Eberle, A., Grothaus, M., Hoh, W., Kassmann, M., Stannat, W., Trutnau, G. (eds) Stochastic Partial Differential Equations and Related Fields. SPDERF 2016. Springer Proceedings in Mathematics & Statistics, vol 229. Springer, Cham. https://doi.org/10.1007/978-3-319-74929-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-74929-7_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74928-0
Online ISBN: 978-3-319-74929-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)