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