A note on a threshold for temporal regularity of stochastic PDEs

We consider solutions to linear parabolic SPDEs of the form \[ \dd u(t) + A u(t)\,\dd t = g(t)\, \dd \W, u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $\W$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $\alpha\in [0,\frac{1}{2})$, \[ u\in L^2(\Omega;W^{\alpha,2}(0,T;\Do(A^{1/2}))) \text{ if and only if } x\in \Do(A^{\alpha}). \] 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. Hofmanov\'a in [C. R. Math. Acad. Sci. Paris 354 (2016), 33–37].