[Paper Review] A review on symmetry properties of birth-death processes

Antonio Di Crescenzo, Barbara Martinucci|arXiv (Cornell University)|Sep 8, 2015

Opinion Dynamics and Social Influence32 references3 citations

TL;DR

This paper reviews symmetry properties in time-homogeneous birth-death processes, establishing necessary and sufficient conditions on transition rates for spatial symmetry that simplify first-passage-time densities and avoiding transition probabilities. The key contribution is a quasi-symmetry relation enabling exact expressions for first-passage times and taboo probabilities in one- and two-dimensional truncated and bilateral processes, including those with catastrophes.

ABSTRACT

In this paper we review some results on time-homogeneous birth-death processes. Specifically, for truncated birth-death processes with two absorbing or two reflecting endpoints, we recall the necessary and sufficient conditions on the transition rates such that the transition probabilities satisfy a spatial symmetry relation. The latter leads to simple expressions for first-passage-time densities and avoiding transition probabilities. This approach is thus thoroughly extended to the case of bilateral birth-death processes, even in the presence of catastrophes, and to the case of a two-dimensional birth-death process with constant rates.

Motivation & Objective

• To identify necessary and sufficient conditions on birth and death rates for spatial symmetry in truncated birth-death processes with absorbing or reflecting boundaries.
• To extend symmetry-based methods to bilateral birth-death processes, including those with total catastrophes that reset the process to state 0.
• To generalize symmetry properties to two-dimensional birth-death processes with constant transition rates, focusing on symmetry across the line $x_2 = x_1 + r$.
• To derive closed-form expressions for first-passage-time densities and taboo probabilities using the symmetry framework.
• To establish quasi-symmetry relations for transition probabilities and first-passage densities under specific rate conditions.

Proposed method

• Derives forward Kolmogorov equations for transition probabilities in truncated birth-death processes on state space $\{0,1,\ldots,N\}$ with absorbing or reflecting endpoints.
• Introduces a quasi-symmetry condition via the ratio $x_n = \frac{\mu_1\cdots\mu_n}{\lambda_{N-1}\cdots\lambda_{N-n}}$, ensuring time-invariant path probability ratios.
• Applies the method of images and symmetry transformations to bilateral processes, particularly under total catastrophes that force passage through state 0.
• Extends symmetry to two-dimensional processes by defining a reflection across the line $x_2 = x_1 + r$, leading to a transformation $\mathbf{k} \to (k_2 - r, k_1 + r)$.
• Uses the symmetry relation $P(n_2 - r, n_1 + r, t \mid \mathbf{k}) = \xi^{n_2 - k_2 - n_1 + k_1} P(\mathbf{n}, t \mid \mathbf{k})$ to relate symmetric path probabilities.
• Derives first-passage densities via the identity $g(x,x+r,t|\mathbf{k}) = \frac{|k_2 - k_1 - r|}{t} P(x,x+r,t|\mathbf{k})$, linking them directly to transition probabilities.

Experimental results

Research questions

• RQ1 Under what conditions on birth and death rates do truncated birth-death processes exhibit spatial symmetry about the midpoint $N/2$?
• RQ2 How can symmetry be extended to bilateral birth-death processes with catastrophes, particularly total catastrophes that reset the process to state 0?
• RQ3 What symmetry properties emerge in two-dimensional birth-death processes with constant rates, and how do they relate to first-passage through the line $x_2 = x_1 + r$?
• RQ4 Can first-passage-time densities and taboo probabilities be expressed in closed form using symmetry-based transformations?
• RQ5 What is the relationship between the first-passage probability and the symmetry parameter $\xi$ under the quasi-symmetry condition?

Key findings

• The necessary and sufficient condition for spatial symmetry in truncated birth-death processes is $\frac{\mu_n}{\lambda_{N-n}} = \frac{\mu_{n-1}}{\lambda_{N-n+1}}$ for all $n=1,\ldots,N-1$, ensuring the ratio of symmetric path probabilities is time-independent.
• For bilateral processes with total catastrophes, the symmetry property allows exact expressions for first-passage densities through state 0 and corresponding avoiding transition probabilities.
• In two-dimensional birth-death processes with constant rates, the symmetry across the line $x_2 = x_1 + r$ leads to the identity $P(n_2 - r, n_1 + r, t \mid \mathbf{k}) = \xi^{n_2 - k_2 - n_1 + k_1} P(\mathbf{n}, t \mid \mathbf{k})$, where $\xi = \frac{\lambda_1 + \mu_2}{\mu_1 + \lambda_2}$.
• The first-passage density to the line $x_2 = x_1 + r$ is given by $g(x,x+r,t|\mathbf{k}) = \frac{|k_2 - k_1 - r|}{t} P(x,x+r,t|\mathbf{k})$, linking it directly to the transition probability.
• The taboo probability $P^{\langle r\rangle}(\mathbf{n},t|\mathbf{k})$ is expressed as $P(\mathbf{n},t|\mathbf{k}) - \xi^{n_1 + r - n_2} P(n_2 - r, n_1 + r, t|\mathbf{k})$, valid under the symmetry condition.
• The ultimate first-passage probability $\pi_r(\mathbf{k})$ satisfies $\pi_r(k_2 - r, k_1 + r) = \xi^{k_1 + r - k_2} \pi_r(\mathbf{k})$, and is equal to 1 if $\lambda_1 + \mu_2 \geq \mu_1 + \lambda_2$ and $k_2 < k_1 + r$, or if $\lambda_1 + \mu_2 \leq \mu_1 + \lambda_2$ and $k_2 > k_1 + r$, otherwise $\xi^{k_2 - k_1 - r}$.