  Biased lattice random walks (BLRW) are used to model random motion with drift
in a variety of empirical situations in engineering and natural systems such as
phototaxis, chemotaxis or gravitaxis. When motion is also affected by the
presence of external borders resulting from natural barriers or experimental
apparatuses, modelling biased random movement in confinement becomes necessary.
To study these scenarios, confined BLRW models have been employed but so far
only through computational techniques due to the lack of an analytic framework.
Here, we lay the groundwork for such an analytical approach by deriving the
Green's functions, or propagators, for the confined BLRW in arbitrary
dimensions and arbitrary boundary conditions. By using these propagators we
construct explicitly the time dependent first-passage probability in one
dimension for reflecting and periodic domains, while in higher dimensions we
are able to find its generating function. The latter is used to find the mean
first-passage passage time for a $d$-dimensional box, $d$-dimensional torus or
a combination of both. We show the appearance of surprising characteristics
such as the presence of saddles in the spatio-temporal dynamics of the
propagator with reflecting boundaries, bimodal features in the first-passage
probability in periodic domains and the minimisation of the mean first-return
time for a bias of intermediate strength in rectangular domains. Furthermore,
we quantify how in a multi-target environment with the presence of a bias
shorter mean first-passage times can be achieved by placing fewer targets close
to boundaries in contrast to many targets away from them.
