Conditioned one-way simple random walk and representation theory

TL;DR

This paper explores conditioned one-way simple random walks in the positive orthant and connects their behavior to the representation theory of Lie algebras, using crystal bases and combinatorial transformations.

Contribution

It introduces a generalized Pitman transformation for these walks based on crystal bases, linking stochastic processes with algebraic combinatorics.

Findings

Describes the law of conditioned random walks in various semi-groups.

Develops a generalized Pitman transformation using crystal bases.

Connects stochastic processes with Lie algebra representation theory.

Abstract

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].