Abstract: We consider random walks on the nonnegative integers in a space-time dependent random environment. We define transition probabilities are given by independent Beta(µ, µ) distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter η. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter η affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on Z.