A note on a threshold for temporal regularity of stochastic PDEs

We consider solutions to linear parabolic SPDEs of the form d u ( t ) + A u ( t ) d t = g ( t ) d β , u ( 0 ) = 0 , where A is a positive, invertible, and self-adjoint operator on a Hilbert space X , β is a one-dimensional Brownian motion, and g ( t ) ≡ x ∈ X . We show that, for all α ∈ 0 , 1 2 , u ∈ L 2 Ω ; W α , 2 0 , T ; D ( A 1 / 2 ) if and only if x ∈ D ( A α ) . 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 [ Comptes Rendus. Mathématique 354 (2016)] .