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

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 $\alpha\in(0,1]$: $$ \Big| \int0^{te{-(t-s)A}{\rm} d} W(s) \Big|{L^{2(\Omega;L2(\mathcal{O}))}} \le C t^{{\frac{\alpha}{2}}} \quad\mbox{and}\quad \Big| \int0^{te{-(t-s)A}{\rm} d} W(s) \Big|{B^\infty L^{2(\Omega;\dot} H^{\alpha(\mathcal{O}))}\le} C. $$ 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 $t=0$.