We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0 < x < 1 with general stochastic boundary conditions at x = 0 and x = 1. Assuming that there is a constant ''wall'' Maxwellian M = (M(i)) compatible with the boundary conditions, and under a technical assumption meaning ''strong thermalization'' at the boundaries, we prove three types of results: I. If no velocity has x-component 0, there are real-valued functions beta1(t) and beta2(t) such that in a measure-theoretic sense f(i)(0, t) --> beta1(t)M(i), f(i)(1, t) --> beta2(t)M(i) as t --> infinity. beta1 and beta2 are closely related and satisfy functional equations which suggest that beta1(t) --> 1 and beta2(t) --> 1 as t --> infinity. II. Under the additional assumption that there is at least one non-trivial collision term containing a product f(k)f(l) with v(k) = v(l), where v(k) denotes the x-component of the velocity associated with f(k), we show that in a measure-theoretic sense beta1(t) and beta2(t) converge to 1 as t --> infinity. This entails L1-convergence of the solution to the unique wall Maxwellian. For this result, v(k) = v(l) = 0 is admissible. III. In the absence of any collision terms, but under the assumption that there is an irrational quotient (v(i) + \v(j)\)/(v(l) + \v(k)\) here v(i), v(l) > 0 and v(j), v(k) < 0), renewal theory entails that the solution converges to the unique wall Maxwellian in L(infinity).
