Generalized Birth-Death Process on Finite Lattice

TL;DR

This paper introduces a generalized birth-death process on finite lattices, providing conditions for transition matrix commutation and efficient computation of transition probabilities in multi-dimensional settings.

Contribution

It extends the theory of birth-death processes to finite multi-dimensional lattices and derives conditions for matrix commutation and minimal constraints on transition probabilities.

Findings

Derived necessary and sufficient conditions for matrix commutation.

Established minimal constraints for transition probabilities.

Extended results to q-dimensional finite grids.

Abstract

We consider a generalized birth-death process (GBDP) whose state space is a finite subset of a $q$-dimensional lattice. It is assumed that there can be a jump of finite step size in all possible directions such that the probability of simultaneous transition in more than one direction is zero. Such processes are of interest as their transition probability matrix is diagonalizable under suitable conditions. Thus, their $k$-step transition probabilities can be efficiently obtained. For GBDP on two-dimensional finite grid, we obtain a sufficient and necessary condition for the vertical and horizontal transition probability matrices to commute. Later, we extend these results to the case of $q$-dimensional finite grid. We obtain the minimum number of constraints required on transition probabilities that ensure the commutation of transition probability matrices for each possible direction.