A sharp $L_p$-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}(\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.