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Random walks with drift in the positive quadrant

Tuan Anh Nguyen, Vitali Wachtel

TL;DR

This work analyzes two-dimensional random walks in the positive quadrant with drift along an axis, constructing positive harmonic functions for walks killed on exiting the quadrant and obtaining precise tail asymptotics for the exit time together with integral and local limit theorems for walks conditioned to stay inside the quadrant. By linking discrete random walks to a Brownian motion analogue in a half-space, the authors derive explicit densities and renewal-structure-based formulas that yield the asymptotics $\mathbb{P}(T_x>n) \sim \varkappa W(x)n^{-1/2}$ and a conditional limit in terms of a Gaussian density $p$. They develop sharp local limit theorems on the lattice, including refined regimes where the endpoint lies near the boundary, and apply these results to singular lattice walks, obtaining exact asymptotics for the number of walks ending on a boundary line and, via exponential tilting, enumerate constrained quadrant paths. The framework advances understanding of constrained walks in cones with drift along the boundary and provides concrete tools for both probabilistic analysis and combinatorial enumeration.

Abstract

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 $\{(k,1),\ k\ge1\}$.

Random walks with drift in the positive quadrant

TL;DR

This work analyzes two-dimensional random walks in the positive quadrant with drift along an axis, constructing positive harmonic functions for walks killed on exiting the quadrant and obtaining precise tail asymptotics for the exit time together with integral and local limit theorems for walks conditioned to stay inside the quadrant. By linking discrete random walks to a Brownian motion analogue in a half-space, the authors derive explicit densities and renewal-structure-based formulas that yield the asymptotics and a conditional limit in terms of a Gaussian density . They develop sharp local limit theorems on the lattice, including refined regimes where the endpoint lies near the boundary, and apply these results to singular lattice walks, obtaining exact asymptotics for the number of walks ending on a boundary line and, via exponential tilting, enumerate constrained quadrant paths. The framework advances understanding of constrained walks in cones with drift along the boundary and provides concrete tools for both probabilistic analysis and combinatorial enumeration.

Abstract

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 and determine asymptotics for the number of walks of length which end on the line .
Paper Structure (12 sections, 12 theorems, 178 equations)

This paper contains 12 sections, 12 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Two-dimensional Brownian motion killed at leaving a half-space
  3. Random walks in the upper half-plane.
  4. Tail of $T_x$ and the conditional limit theorem
  5. Tail asymptotics for $T_x$.
  6. Conditional limit theorem.
  7. Properties of the function $W$.
  8. Local limit theorem for the walk conditioned to stay in the positive quadrant
  9. Proof of Theorem \ref{['t11']}
  10. Proof of Theorem \ref{['t12']}
  11. Proof of Corollary \ref{['t15']}
  12. Singular walks ending at one of the axes

Key Result

Theorem 1.1

Assume that eq:1st.moment and eq:2nd.moment hold. Then, for every fixed $x\in{\mathbbm{R}}\times(0,\infty)$, where and This function is harmonic for $S(n)$ killed at leaving the quadrant, that is, Furthermore, for every fixed $x\in{\mathbbm{R}}_+^2$ and all $x\in{\mathbbm{R}}_+^2$ one has where $\Delta= [0,1)^d$ and the function $p(y)$ is defined in eq:p-def.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1: cf. HRT2023
  • proof : Proof of Lemma \ref{['c01']}
  • Corollary 2.2
  • proof : Proof of Corollary \ref{['c10']}
  • Proposition 3.1
  • Lemma 3.2
  • ...and 8 more