Erratum to: An \(L_{p}\) -theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

Journal of Theoretical Probability, Sep 2015

The purpose of this erratum is to correct the assumptions in Theorem 2.10 of [2] (Kim in J Theor Probab 22:220–238, 2009).

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Erratum to: An \(L_{p}\) -theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

J Theor Probab Erratum to: An Lp-theory of Stochastic PDEs of Divergence Form on Lipschitz Domains Kyeong-Hun Kim 0 B Kyeong-Hun Kim 0 Department of Mathematics, Korea University , 1 Anam-dong Sungbuk-gu, Seoul 136-701 , South Korea The purpose of this erratum is to correct the assumptions in Theorem 2.10 of [2] (Kim in J Theor Probab 22:220-238, 2009). Theorem 0; 1 Let p ∈ [2; ∞) and Assumptions 2; 1; 2; 3; 2; 4; 2; 7 and 2; 8 be satisfied; There exists κ = κ (δ0; p; d; K ) ∈ (0; 1) such that if - d − 2 + p − κ < θ < d − 2 + p + κ then for any f ∈ ψ −1H −p,1θ (O, τ ), f i ∈ L p,θ (O, τ ), g ∈ L p,θ (O, τ ) and u0 ∈ U p1,θ (O), Eq. (1.1) with initial data u0 has a unique solution u ∈ H p,θ (O, τ ), and for 1 this solution In Theorem 2.10 of [2], in place of (0.1), the weaker condition is assumed. The error of the proof of [2, Theorem 2.10] occurred because it relied on a result proved in [3, Theorem 2.1], which is related to non-divergence type SPDE. The result of [3, Theorem 2.1] is proved for the range of θ satisfying (0.3), but it turns out that [3, Theorem 2.1] is false unless much stronger restriction on θ is assumed (see [1] for details). Theorem 2.1 of [3] is corrected in [1, Theorem 2.12] for θ satisfying (0.1). Thus the proof of Theorem 2.10 of [2] goes throughout without any change if condition (0.1) is assumed. Acknowledgements The author thank Prof. N.V. Krylov for finding the error mentioned above. 1. Kim , K. : A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains . J. Theor. Probab . 27 , 107 - 136 ( 2014 ) 2. Kim , K. : An L p-theory of stochastic PDEs of divergence form on Lipschitz domains . J. Theor. Probab . 22 , 220 - 238 ( 2009 ) 3. Kim , K. : An L p-theory of SPDEs on Lipschitz domains . Potential Anal . 29 , 303 - 326 ( 2008 )


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Kyeong-Hun Kim. Erratum to: An \(L_{p}\) -theory of Stochastic PDEs of Divergence Form on Lipschitz Domains, Journal of Theoretical Probability, 2017, 395-396, DOI: 10.1007/s10959-015-0637-5