A class of stochastic Besov spaces $B^p L^2(\Omega;\dot H^\alpha(\mathcal{O}))$, $1\le p\le\infty$ and $\alpha\in[-2,2]$, is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation \begin{equation*} {\rm d} u -\Delta u {\rm d} t =f(u) {\rm d} t + {\rm d} W(t) , \end{equation*} under the following conditions for some $\alpha\in(0,1]$: $$ \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{L^2(\Omega;L^2(\mathcal{O}))} \le C t^{\frac{\alpha}{2}} \quad\mbox{and}\quad \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}\le C. $$ The conditions above are shown to be satisfied by both trace-class noises (with $\alpha=1$) and one-dimensional space-time white noises (with $\alpha=\frac12$). The latter would fail to satisfy the conditions with $\alpha=\frac12$ if the stochastic Besov norm $\|\cdot\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$ is replaced by the classical Sobolev norm $\|\cdot\|_{L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$, and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this article, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization, is proved to have order $\alpha$ in both time and space for possibly nonsmooth initial data in $L^4(\Omega;\dot{H}^{\beta}(\mathcal{O}))$ with $\beta>-1$, by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at $t=0$.

翻译：引入了一类随机贝索夫空间 $B^p L^2(\Omega;\dot H^\alpha(\mathcal{O}))$，其中 $1\le p\le\infty$ 且 $\alpha\in[-2,2]$，以表征半线性随机热方程中噪声的正则性：\begin{equation*} {\rm d} u -\Delta u {\rm d} t =f(u) {\rm d} t + {\rm d} W(t) , \end{equation*} 在以下对某个 $\alpha\in(0,1]$ 成立的条件下：$$ \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{L^2(\Omega;L^2(\mathcal{O}))} \le C t^{\frac{\alpha}{2}} \quad\mbox{且}\quad \Big\| \int_0^te^{-(t-s)A}{\rm d} W(s) \Big\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}\le C. $$ 上述条件被证明既适用于迹类噪声（$\alpha=1$），也适用于一维时空白噪声（$\alpha=\frac12$）。若将随机贝索夫范数 $\|\cdot\|_{B^\infty L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$ 替换为经典索博列夫范数 $\|\cdot\|_{L^2(\Omega;\dot H^\alpha(\mathcal{O}))}$，则后者在 $\alpha=\frac12$ 时将不满足条件，这常导致半线性随机热方程数值分析中收敛阶的降低。本文证明，通过利用随机贝索夫空间的实插值性质以及一类局部细化步长以解析 $t=0$ 处解奇异性，采用谱方法进行空间离散化的改进指数欧拉方法，对于可能非光滑的初始数据（属于 $L^4(\Omega;\dot{H}^{\beta}(\mathcal{O}))$ 且 $\beta>-1$），在时间和空间上均能达到 $\alpha$ 阶收敛。