Random walks in octants, and related structures

A diffusion walk in Z2 is a (random) walk with unit step vectors →, ↑, ←, and ↓. Particles from different sources with opposite charges cancel each other when they meet in the lattice. This cancellation principle is applied to enumerate diffusion walks in shifted half-planes, quadrants, and octants (a three-dimensional version is also considered). Summing over time we calculate expected numbers of visits and first passage probabilities. Comparing those quantities to analytically obtained expressions leads to interesting identities, many of them involving integrals over products of Chebyshev polynomials of the first and second kind. We also explore what the expected number of visits means when the diffusion in an octant is bijectively mapped onto other combinatorial structures, like pairs of non-intersecting Dyck paths, vicious walkers, bicolored Motzkin paths, staircase polygons in the second octant, and {→↑}-paths confined to the third hexadecant enumerated by left turns.