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

Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data