A sharp $L_p$-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients (1905.07545v1)

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}(\omega,t)u_{x^ix^j}+f)dt + (\sigma^{ik}(\omega,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 $(\omega,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $\sigma^{ik}$ merely satisfy \begin{align} \label{abs only} \left(\alpha^{ij}(\omega,t)\right)_{d\times d}:= \left(a^{ij}(\omega,t)-\frac{1}{2}\sum_{k=1}^{\infty} \sigma^{ik}(\omega,t)\sigma^{jk}(\omega,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}^\gamma_p(\tau,\delta)} &\leq N(d,p) \bigg( |u_0|{\mathbb{B}p^{\gamma+2 \left(1-1/ p \right)}} + | f|{\mathbb{H}^\gamma_p( \tau,\delta^{1-p} )} \label{abs est} &\qquad \qquad+|g_x|^p_{\mathbb{H}^\gamma_p( \tau, |\sigma|^p \delta^{1-p},l_2)}+ | g_x|_{\mathbb{H}^\gamma_p( \tau,\delta^{1-p/2},l_2)} \bigg), \end{align} where $p\geq 2$, $\gamma\in \mathbf{R}$, $\tau$ is an arbitrary stopping time, $\delta(\omega, t)$ is the smallest eigenvalue of $\alpha^{ij}(\omega, t)$, $\mathbb{H}_p^\gamma(\tau, \delta)$ is a weighted stochastic Sobolev space, and $\mathbb{B}_p^{\gamma+2 \left(1-1/ p \right)}$ is a stochastic Besov space.