A Note on a threshold for temporal regularity of stochastic PDEs

We consider solutions to linear parabolic SPDEs of the form $$\mathrm{d}u(t)+Au(t)\mathrm{d}t=g(t)\mathrm{d}\beta ,u(0)=0,$$ where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $\beta$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $\alpha \in [0,\frac{1}{2}),$ $$u\in L^2(\mathrm{\Omega };W^{\alpha ,2}(0,T;\mathsf{D}(A^{1\mathrm{/}2})))\text{ if and only if }x\in \mathsf{D}(A^{\alpha })\mathrm{.}$$ In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient $g$ to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [C. R. Math. Acad. Sci. Paris 354 (2016), 33-37].