Strong solutions of stochastic equations with singular time dependent drift

N. V. Krylov, M. Röckner

Research output: Contribution to journalArticlepeer-review

222 Scopus citations

Abstract

We prove existence and uniqueness of strong solutions to stochastic equations in domains G ⊂ ℝd with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L q_Lp-integrability of b in ℝ x 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 2Dtψ ≤ Kψ, 2Dtψ + Δψ ≤ Keεψ, ε ∈ [0, 2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.

Original languageEnglish (US)
Pages (from-to)154-196
Number of pages43
JournalProbability Theory and Related Fields
Volume131
Issue number2
DOIs
StatePublished - Feb 2005

Keywords

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

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