  Because of the spatially long-ranged nature of spontaneous fluctuations in
thermal non-equilibrium systems, they are affected by boundary conditions for
the fluctuating hydrodynamic variables. In this paper we consider a liquid
mixture between two rigid and impervious plates with a stationary concentration
gradient resulting from a temperature gradient through the Soret effect. For
liquid mixtures with a large Lewis number, we are able to obtain explicit
analytical expressions for the intensity of the non-equilibrium concentration
fluctuations as a function of the frequency $\omega$ and the wave number $q$ of
the fluctuations. In addition we elucidate the spatial dependence of the
intensity of the non-equilibrium fluctuations responsible for a non-equilibrium
Casimir effect.
