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^pL²(Ω;H^α(O)), 1 ⩽ p ⩽ ∞ and α ∈ [−2, 2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation (Formula presented.) under the following conditions for some α ∈ (0,1] (Formula presented.) The conditions above are shown to be satisfied by both trace-class noises (with α =1) and one-dimensional space-time white noises (with α=12). The latter would fail to satisfy the conditions with α=12 if the stochastic Besov norm ‖⋅‖_{B∞L2(Ω;H˙α(O))} is replaced by the classical Sobolev norm ‖⋅‖_{L2(Ω;H˙α(O))}, and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation. In this paper, the convergence of a modified exponential Euler method, with a spectral method for spatial discretization, is proved to have order αin both the time and space for possibly nonsmooth initial data in L⁴(Ω;H˙^β(O)) with β > −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.