  We study a Langevin equation describing the stochastic motion of a particle
in one dimension with coordinate $x$, which is simultaneously exposed to a
space-dependent friction coefficient $\gamma(x)$, a confining potential $U(x)$
and non-equilibrium (i.e., active) noise. Specifically, we consider frictions
$\gamma(x)=\gamma_0 + \gamma_1 | x |^p$ and potentials $U(x) \propto | x |^n$
with exponents $p = 1,2$ and $n = 0, 1, 2$. We provide analytical and numerical
results for the particle dynamics for short times and the stationary
probability density functions (PDFs) for long times. The short-time behaviour
displays diffusive and ballistic regimes while the stationary PDFs display
unique characteristic features depending on the exponent values $(p, n)$. The
PDFs interpolate between Laplacian, Gaussian and bimodal distributions, whereby
a change between these different behaviours can be achieved by a tuning of the
friction strengths ratio $\gamma_0/\gamma_1$. Our model is relevant for
molecular motors moving on a one-dimensional track and can also be realized for
confined self-propelled colloidal particles.
