The transport of fine aerosol particles diluted in a carrier fluid in turbulent channel flow have been theoretically analysed in the framework of a Eulerian description. The two-component (particle laden) fluid is described as a binary gas mixture with disparate molar masses leading to inertial non-equilibrium. To account for slight inertia effects, the governing equations are expanded in terms of the particle Stokes number (particle relaxation time to flow characteristic time ≡ Stk<<1) and the first correction is retained. Furthermore, the particles are weakly diffusive such that the particle Schmidt number Sc is a large parameter and the rate of Brownian deposition of particles along the walls is slow compared to the particle transport rate in the wall-normal direction. This approximation allows decoupling both variations and solving the Reynolds-averaged transport equation of the particle mass fraction across the channel width. Through most of the channel cross-section the particle concentration distributes according to the counterbalance between inertial (turbophoretical) drift and turbulent dispersion. Such a solution is not valid in the regions adjacent to the walls where thin Brownian boundary layers develop with a relative thickness of order Sc^(-1/4), compared to the viscous sublayer thickness. The Sc-asymptotic matching of the profiles on both regions provides with an analytical expression for the particle deposition flux. The concentration profile and the deposition flux depend on the statistics of the turbulent flow through several coefficients. The DNS computation of the carrier flow allows to assig precise values to these coefficients and fully quantify the theoretical predictions in terms of the parameters of the problem.