  Consider a real Gaussian stationary process $f_\rho$, indexed on either
$\mathbb{R}$ or $\mathbb{Z}$ and admitting a spectral measure $\rho$. We study
$\theta_{\rho}^\ell=-\lim\limits_{T\to\infty}\frac{1}{T}
\log\mathbb{P}\left(\inf_{t\in[0,T]}f_{\rho}(t)>\ell\right)$, the persistence
exponent of $f_\rho$. We show that, if $\rho$ has a positive density at the
origin, then the persistence exponent exists; moreover, if $\rho$ has an
absolutely continuous component, then $\theta_{\rho}^\ell>0$ if and only if
this spectral density at the origin is finite. We further establish continuity
of $\theta_{\rho}^\ell$ in $\ell$, in $\rho$ (under a suitable metric) and, if
$\rho$ is compactly supported, also in dense sampling. Analogous continuity
properties are shown for $\psi_{\rho}^\ell=-\lim\limits_{T\to\infty}\frac{1}{T}
\log\mathbb{P}\left(\inf_{t\in[0,T]}|f_{\rho}(t)|\le \ell\right)$, the ball
exponent of $f_\rho$, and it is shown to be positive if and only if $\rho$ has
an absolutely continuous component.
