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Journal of Differential Equations
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A sharp L-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

A sharp \(L_p\)-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients
Authors: Kim, Ildoo; Kim, Kyeong-Hun;

A sharp L-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

Abstract

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(��,t)u_{x^ix^j}+f)dt + (��^{ik}(��,t)u_{x^i}+g^k)dw^k_t, \quad u(0,x)=u_0, \end{align} where $\{w^k_t:k=1,2,\cdots\}$ is a sequence of independent Brownian motions. The coefficients are merely measurable in $(��,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $��^{ik}$ merely satisfy \begin{align} \label{abs only} \left(��^{ij}(��,t)\right)_{d\times d}:= \left(a^{ij}(��,t)-\frac{1}{2}\sum_{k=1}^{\infty} ��^{ik}(��,t)��^{jk}(��,t)\right) \geq 0. \end{align} In this article, we prove that there exists a unique solution $u$ to \eqref{abs eqn}, and \begin{align} \notag \|u_{xx}\|_{\mathbb{H}^��_p(��,��)} &\leq N(d,p) \bigg( \|u_0\|_{\mathbb{B}_p^{��+2 \left(1-1/ p \right)}} + \| f\|_{\mathbb{H}^��_p( ��,��^{1-p} )} \label{abs est} &\qquad \qquad+\|g_x\|^p_{\mathbb{H}^��_p( ��, |��|^p ��^{1-p},l_2)}+ \| g_x\|_{\mathbb{H}^��_p( ��,��^{1-p/2},l_2)} \bigg), \end{align} where $p\geq 2$, $��\in \mathbf{R}$, $��$ is an arbitrary stopping time, $��(��, t)$ is the smallest eigenvalue of $��^{ij}(��, t)$, $\mathbb{H}_p^��(��, ��)$ is a weighted stochastic Sobolev space, and $\mathbb{B}_p^{��+2 \left(1-1/ p \right)}$ is a stochastic Besov space.

Keywords

degenerate stochastic partial differential equations, unbounded coefficients, Stochastic partial differential equations (aspects of stochastic analysis), Smoothness and regularity of solutions to PDEs, Probability (math.PR), FOS: Mathematics, PDEs with randomness, stochastic partial differential equations, maximal \(L_p\)-regularity theory, Mathematics - Probability

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
2
Average
Average
Average
Green