Abstract. In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. Homogenized equations for the covariance and variance of the solution of stochastic elliptic PDEs are derived. In addition to the homogenized equations, a scaling law for the covariance and variance as the cell size tends to zero is given. For the homogenized problems, existence and uniqueness results and a priori bounds are given and further properties are proven. The multiscale problem stems from the modeling of the electrostatics in nanoscale field-effect sensors, where the fluctuations arise from randomly distributed charge concentrations in the cells of a boundary layer. Finally, numerical results and a spectral approximation are presented. 1. Introduction. The motivation for the present study of stochastic elliptic pde stems from the desire to model field-effect nano-sensors and hence to understand their physics. Elliptic equations, such as the Poisson equation and the linearized Poisson-Boltzmann equation, are the basic equations for their electrostatics, and the stochastic equations considered here make it possible to study fluctuations and noise in nanostructures. Furthermore, a multiscale problem is inherent in these nanoscale structures and it is solved by homogenization.First, we introduce the physical problem. Recently, nanoscale field-effect biosensors [19][20][21]24] and gas sensors [16,22] have been demonstrated experimentally. A schematic diagram of such a sensor structure is shown in Fig. 1. The length scale of the biomolecules is in the Angstrom or nanometer range, whereas the length of the nanowire is in the micrometer range. This gives rise to a multiscale problem, since it is not possible to resolve both the boundary layer and the whole simulation domain using a single numerical grid.This simulation problem also gives rise to a stochastic problem, since binding and unbinding events (in the case of biosensors) or chemical reactions (in the case of gas sensors) occur in the boundary layer. Additionally, the movement of the molecules in the boundary layer can be modeled by calculating their electrostatic free energy and by using a Boltzmann distribution [11]. These effects imply that the charge