  We describe all countable particle systems on $\mathbb{R}$ which have the
following three properties: independence, Gaussianity and stationarity. More
precisely, we consider particles on the real line starting at the points of a
Poisson point process with intensity measure $\mathfrak{m}$ and moving
independently of each other according to the law of some Gaussian process
$\xi$. We classify all pairs $(\mathfrak{m},\xi)$ generating a stationary
particle system, obtaining three families of examples. In the first, trivial
family, the measure $\mathfrak{m}$ is arbitrary, whereas the process $\xi$ is
stationary. In the second family, the measure $\mathfrak{m}$ is a multiple of
the Lebesgue measure, and $\xi$ is essentially a Gaussian stationary increment
process with linear drift. In the third, most interesting family, the measure
$\mathfrak{m}$ has a density of the form $\alpha e^{-\lambda x}$, where $\alpha
>0$, $\lambda\in\mathbb{R}$, whereas the process $\xi$ is of the form
$\xi(t)=W(t)-\lambda\sigma ^2(t)/2+c$, where $W$ is a zero-mean Gaussian
process with stationary increments, $\sigma ^2(t)=\operatorname {Var}W(t)$, and
$c\in\mathbb{R}$.
