
arXiv: 1905.07545
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.
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
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
| 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). | 2 | |
| 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. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Average | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
