Random walks with drift in the positive quadrant

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive harmonic functions for such walks and find tail asymptotics for the exit time from the positive quadrant. Moreover, we prove integral and local limit theorems. Finally, we apply our local limit theorems to singular lattice walks with steps ${(1,-1),(1,1),(-1,1)}$ and determine asymptotics for the number of walks of length $n$ which end on the line ${(k,1),\ k\ge1}$.

💡 Research Summary

This paper studies two‑dimensional random walks with a non‑zero drift that is aligned with one of the coordinate axes, specifically μ=(μ₁,0) with μ₁>0, and finite second moments. The walk is conditioned to stay inside the positive quadrant K=ℝ₊². The authors first introduce the one‑dimensional ladder height χ⁻ of the vertical component S₂(n) and its renewal function V(u). V is a positive harmonic function for the one‑dimensional walk killed when it leaves the upper half‑plane. Using V they construct a two‑dimensional positive harmonic function
W(x)=V(x₂)−E