Random walk in a discrete and continuous system with a thin membrane

Random walk in a one-dimensional system with a thin membrane (which is treated as a partially permeable wall with its internal structure being not explicitly involved into our considerations) is discussed for the discrete and continuous time and space variables. The Green's functions of the membrane system for the discrete space variable are obtained using the method of generating function. The Green's functions for the continuous system are obtained from the discrete ones by taking the continuum limit. It is shown that the boundary condition at the membrane, which is commonly used in stationary system (where the flux flowing through the membrane is proportional to the difference of the concentration of the diffusing particle between the membrane surfaces) is appropriate also for the non-stationary system.