Square root of a stationary OU process

Question

We know that an OU process $\sigma_t^2=\int_{-\infty}^t e^{-\delta (t-x)}dV_x$, where $\delta$ is a positive constant and $V_t$ is a subordinator, is stationary, i.e. for all $t$ we have $\mathbb{E}[\sigma_t^2]=\mathbb{E}[\sigma_0^2]$. Can we say the same about $\sigma_t=\sqrt{\int_{-\infty}^t e^{-\delta (t-x)}dV_x}$? Or, more generally, is a continous transfrmation of a stationary process a stationary process as well?