Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise
Download PDF
Download PDF
  • Published: 08 April 2008

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

  • Robert C. Dalang1,
  • Davar Khoshnevisan2 &
  • Eulalia Nualart3 

Probability Theory and Related Fields volume 144, pages 371–427 (2009)Cite this article

Abstract

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y),u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process \({\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}\), in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes \({t \mapsto u(t,x)}\) and \({x \mapsto u(t,x)}\). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bally, V., Millet, A., Sanz-Solé, M.: Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23, 178–222 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bally, V., Pardoux, E.: Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9, 27–64 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dalang, R.C., Nualart, E.: Potential theory for hyperbolic SPDEs. Ann. Probab. 32, 2099–2148 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Dalang, R.C., Khoshnevisan, D., Nualart, E.: Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA 3, 231–271 (2007)

    MathSciNet  Google Scholar 

  5. Kahane, J.-P.: Some random series of functions. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  6. Khoshnevisan, D.: Multiparameter Processes. An Introduction to Random Fields. Springer, Heidelberg (2002)

  7. Kohatsu-Higa, A.: Lower bound estimates for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126, 421–457 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Morien, P.-L.: The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation. Bernouilli 5, 275–298 (1998)

    Article  MathSciNet  Google Scholar 

  9. Mueller, C., Tribe, R.: Hitting properties of the random string. Elect. J. Probab. 7, 1–29 (2002)

    MathSciNet  Google Scholar 

  10. Nualart, D., Pardoux, E.: Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs. Stoch. Stoch. Reports 48, 17–44 (1994)

    MATH  MathSciNet  Google Scholar 

  11. Nualart, D.: The Malliavin Calculus a nd Related Topics. Springer, Heidelberg (1995)

    Google Scholar 

  12. Nualart, D.: Analysis on Wiener space and anticipating stochastic calculus. Ecole d’Eté de Probabilités de Saint-Flour XXV. Lect. Notes in Math. vol. 1690, pp. 123–227. Springer, Heidelberg (1998)

  13. Sanz-Solé, M.: Malliavin calculus with applications to stochastic partial differential equations. EPFL Press, Lausanne (2005)

    MATH  Google Scholar 

  14. Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. Ecole d’Eté de Probabilités de Saint-Flour XIV. Lect. Notes in Math., vol. 1180, pp. 266–437. Springer, Heidelberg (1986)

  15. Watanabe, S.: Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Institute of Fundamental Research. Lectures on Math. and Physics, vol. 73. Springer, Berlin (1984)

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    Robert C. Dalang

  2. Department of Mathematics, The University of Utah, 155 S.1400 E, Salt Lake City, UT, 84112-0090, USA

    Davar Khoshnevisan

  3. Institut Galilée, Université Paris 13, 93430, Villetaneuse, France

    Eulalia Nualart

Authors
  1. Robert C. Dalang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Davar Khoshnevisan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eulalia Nualart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert C. Dalang.

Additional information

R.C. Dalang is supported in part by the Swiss National Foundation for Scientific Research.

D. Khoshnevisan’s research is supported in part by a grant from the US National Science Foundation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Dalang, R.C., Khoshnevisan, D. & Nualart, E. Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise. Probab. Theory Relat. Fields 144, 371–427 (2009). https://doi.org/10.1007/s00440-008-0150-1

Download citation

  • Received: 30 March 2007

  • Revised: 28 February 2008

  • Published: 08 April 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0150-1

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Hitting probabilities
  • Stochastic heat equation
  • Space-time white noise
  • Malliavin calculus

Mathematics Subject Classification (2000)

  • Primary: 60H15
  • 60J45
  • Secondary: 60H07
  • 60G60
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature