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Strong solutions of stochastic equations with singular time dependent drift
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  • Published: 25 May 2004

Strong solutions of stochastic equations with singular time dependent drift

  • N.V. Krylov1 &
  • M. Röckner2 

Probability Theory and Related Fields volume 131, pages 154–196 (2005)Cite this article

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Abstract.

We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q _L p -integrability of b in ℝ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function ψ blowing up as G∋x→∂G, we prove that the conditions 2D t ψ≤Kψ,2D t ψ+Δψ≤Keɛψ,ɛ ∈ [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.

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Authors and Affiliations

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

    N.V. Krylov

  2. Fakultät für Mathematik, Universität Bielefeld, Bielefeld, D-33501, Germany

    M. Röckner

Authors
  1. N.V. Krylov
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  2. M. Röckner
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Corresponding author

Correspondence to N.V. Krylov.

Additional information

The work of the first author was partially supported by NSF Grant DMS-0140405

Mathematics Subject Classification (2000): 60J60, 31C25

Acknowledgement Financial support by the Humboldt Foundation, the BiBoS-Research Centre and the DFG-Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics” is gratefully acknowledged. The authors are also sincerely grateful to the referees for their helpful suggestions.

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Krylov, N., Röckner, M. Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, 154–196 (2005). https://doi.org/10.1007/s00440-004-0361-z

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  • Received: 12 October 2003

  • Revised: 10 February 2004

  • Published: 25 May 2004

  • Issue Date: February 2005

  • DOI: https://doi.org/10.1007/s00440-004-0361-z

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Key words or phrases:

  • Singular drift
  • Distorted Brownian motion
  • Strong solutions of stochastic equations
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