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One more square root law for Brownian motion and its application to SPDEs
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  • Published: 04 November 2003

One more square root law for Brownian motion and its application to SPDEs

  • N.V. Krylov1 

Probability Theory and Related Fields volume 127, pages 496–512 (2003)Cite this article

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Abstract.

It is proved that there is a function p(c)≥0 such that p(c)>0 if c is large enough, and (a.s.) for any t∈[0,1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0,2− k]×[0,c2− k /2] for all k belonging to a set with lower density ≥p(c). This law of square root helps show that solutions of one-dimensional SPDEs are Hölder continuous up to the boundary.

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References

  1. Davis, B.: On Brownian slow points, Z. Wahrscheinlichkeitstheory verw. Gebiete. 64, 359–367 (1983)

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  2. Krylov, N.V.: Nonlinear elliptic and parabolic equations of second order. Nauka, Moscow, 1985 in Russian; English translation by Reidel, Dordrecht, 1987

  3. Krylov, N.V.: Introduction to the theory of diffusion processes. Amer. Math. Soc., Providence, RI, 1995

  4. Krylov, N.V.: Brownian trajectory is a regular lateral boundary for the heat equation. SIAM J. Math. Anal. 34 (5), 1167–1182 (2003)

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Author information

Authors and Affiliations

  1. University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA

    N.V. Krylov

Authors
  1. N.V. Krylov
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Corresponding author

Correspondence to N.V. Krylov.

Additional information

The work was partially supported by NSF Grant DMS-0140405

Mathematics Subject Classification (2000): 60G17, 35K05, 60H15

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Cite this article

Krylov, N. One more square root law for Brownian motion and its application to SPDEs. Probab. Theory Relat. Fields 127, 496–512 (2003). https://doi.org/10.1007/s00440-003-0301-3

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  • Received: 05 April 2003

  • Revised: 15 September 2003

  • Published: 04 November 2003

  • Issue Date: December 2003

  • DOI: https://doi.org/10.1007/s00440-003-0301-3

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Keywords

  • Square root law
  • Brownian motion
  • Stochastic partial differential equations
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