In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.
Let $B$ be a Brownian motion, and let $\mathcal C_{\mathrm p}$ be the space of all continuous periodic functions $f\:\mathbb{R}\rightarrow \mathbb{R}$ with period 1. It is shown that the set of all $f\in \mathcal C_{\mathrm p}$ such that the stochastic convolution $X_{f,B}(t)= \int _0^tf(t-s)\mathrm{d}B(s)$, $t\in [0,1]$ does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.