Numerical approximation of the stochastic heat equation with a distributional reaction term

Abstract We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of Hölder continuous processes. For a suitable choice of sequence $(b^{k})_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the stochastic partial differential equation with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^{k}$, converges to $0$ in $L^{m}(\varOmega )$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, we recover the (almost) optimal rate of convergence $\big(\frac{1}{2}-\varepsilon \big)$-in space and $\big(\frac{1}{4}-\varepsilon \big)$-in time. The proofs are based on new regularizing properties of the Ornstein–Uhlenbeck process, its discrete version and their combined effect.