Spectral properties of the Brownian self-transport operator

The problem of the Laplacian transfer across an irregular resistive interface (a membrane or an electrode) is investigated with use of the Brownian self-transport operator. This operator describes the transfer probability between two points of a surface, through Brownian motion in the medium neighbouring the surface. This operator governs the flux across a semi-permeable membrane as diffusing particles repetitively hit the surface until they are finally absorbed. In this paper, we first give a theoretical study of the properties of this operator for a planar membrane. It is found that the net effect of a decrease of the surface permeability is to induce a broadening of the region where a particle, first hitting the surface on one point, is finally absorbed. This result constitutes the first analytical justification of the Land Surveyor Approximation, a formerly developed method used to compute the overall impedance of a semi-permeable membrane. In a second step, we study numerically the properties of the Brownian self-transport operator for selected irregular shapes.