Limit behavior of symmetric random walks with a membrane (Q2849280)

The authors consider the weak convergence of normalized random walks \(\{X(k) : k\in\mathbb Z_+\}\), assuming that its transition probabilities coincide with those of a symmetric random walk with unit steps throughout except for a fixed neighborhood of zero. This neighborhood is called a membrane. The main result generalizes a Harrison and Shepp theorem on the weak convergence to skew Brownian motion in the case where the symmetricity of the random walk fails at a single point. Finally, the authors describe all possible processes that may occur as a limit depending on the properties of a membrane.