A Note on a threshold for temporal regularity of stochastic PDEs

We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} \beta, \qquad u(0)=0, \] 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 $\alpha\in [0,\frac{1}{2}),$ \[ u\in L^2(\Omega;W^{\alpha,2}(0,T;\mathsf{D}(A^{1/2}))) \quad \text{ if and only if }\quad x\in \mathsf{D}(A^{\alpha}). \] 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].