  We consider a model quantum field theory with a scalar quantum field in de
Sitter space-time in a Bohmian version with a field ontology, i.e., an actual
field configuration $\varphi({\bf x},t)$ guided by a wave function on the space
of field configurations. We analyze the asymptotics at late times
($t\to\infty$) and provide reason to believe that for more or less any wave
function and initial field configuration, every Fourier coefficient
$\varphi_{\bf k}(t)$ of the field is asymptotically of the form $c_{\bf
k}\sqrt{1+{\bf k}^2 \exp(-2Ht)/H^2}$, where the limiting coefficients $c_{\bf
k}=\varphi_{\bf k}(\infty)$ are independent of $t$ and $H$ is the Hubble
constant quantifying the expansion rate of de Sitter space-time. In particular,
every field mode $\varphi_{\bf k}$ possesses a limit as $t\to\infty$ and thus
"freezes." This result is relevant to the question whether Boltzmann brains
form in the late universe according to this theory, and supports that they do
not.
