Structural Properties of Conditioned Random Walks on Integer Lattices with Random Local Constraints

TL;DR

This paper studies a multidimensional random walk with random local time constraints, introducing a core process to analyze its behavior and generalize previous one-dimensional results as the high level tends to infinity.

Contribution

It introduces a regenerative core process to analyze conditioned random walks with local constraints, extending known one-dimensional results to higher dimensions.

Findings

Representation formulas for the walk's distribution via the core process

Limiting behavior as the high level tends to infinity

Generalization of 1D results to multidimensional settings

Abstract

We consider a random walk on a multidimensional integer lattice with random bounds on local times, conditioned on the event that it hits a high level before its death. We introduce an auxiliary "core" process that has a regenerative structure and plays a key role in our analysis. We obtain a number of representations for the distribution of the random walk in terms of the similar distribution of the "core" process. Based on that, we prove a number of limiting results by letting the high level to tend to infinity. In particular, we generalise results for a simple symmetric one-dimensional random walk obtained earlier in the paper by Benjamini and Berestycki (2010).