A note on a threshold for temporal regularity of stochastic PDEs

Abstract

We consider solutions to linear parabolic SPDEs of the form du(t) + Au(t)dt = g (t)dβ, 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 ∈^[0,¹2), u ∈ L²⁽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)].