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A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains

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  • Kyeong-Hun Kim

    (Korea University)

Abstract

In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain $$\mathcal{O }\subset \mathbb{R }^d$$ admitting the Hardy inequality 0.1 $$\begin{aligned} \int _{\mathcal{O }}|\rho ^{-1}g|^2\,\text{ d}x\le C\int _{\mathcal{O }}|g_x|^2 \text{ d}x, \quad \forall g\in C^{\infty }_0(\mathcal{O }), \end{aligned}$$ where $$\rho (x)=\text{ dist}(x,\partial \mathcal{O }).$$ Existence and uniqueness results are given in weighted Sobolev spaces $$\mathfrak{H }_{p,\theta }^{\gamma }(\mathcal{O },T),$$ where $$p\in [2,\infty ), \gamma \in \mathbb{R }$$ is the number of derivatives of solutions and $$\theta $$ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.

Suggested Citation

  • Kyeong-Hun Kim, 2014. "A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains," Journal of Theoretical Probability, Springer, vol. 27(1), pages 107-136, March.
  • Handle: RePEc:spr:jotpro:v:27:y:2014:i:1:d:10.1007_s10959-012-0459-7
    DOI: 10.1007/s10959-012-0459-7
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    References listed on IDEAS

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    1. Kim, Kyeong-Hun, 2004. "On stochastic partial differential equations with variable coefficients in C1 domains," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 261-283, August.
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