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Invariant measure for the contact process with modified border in the non-attractive region

Célio Terra

TL;DR

The paper analyzes a one-dimensional contact process with modified border on $\mathbb{Z}$, where external boundary infections occur at rate $\lambda_e$ and interior infections at rate $\lambda_i$. It proves the existence of an invariant measure $\tilde{\mu}$ for the process seen from the right edge in the non-attractive regime $\lambda_i=\lambda_c$, $\lambda_e>\lambda_c$, and shows that $\Psi \xi^A_t$ converges to $\tilde{\mu}$ for any infinite initial condition, with $\xi^{\tilde{\mu}}_t$ distributed according to $\tilde{\mu}$ and a random asymptotic edge velocity $V$ arising from stationary increments of the right edge. The analysis extends the edge-based convergence framework to the critical-border setting and establishes a full invariant-measure description along with an edge-velocity law. Additionally, it proves non-survival along the critical curve within the attractive region, thereby completing the phase diagram for the modified-border model and highlighting edge-driven long-term behavior as the system approaches criticality.

Abstract

We investigate a modified one-dimensional contact process with varying infection rates. Specifically, the infection spreads at rate $λ_e$ along the boundaries of the infected region and at rate $λ_i$ elsewhere. We establish the existence of an invariant measure when $λ_i = λ_c$ and $λ_e > λ_c$, where $λ_c$ is the critical parameter of the standard contact process. Moreover, we show that, when viewed from the rightmost infected site, the process converges weakly to this invariant measure. Finally, we prove that along the critical curve within the attractive region of the phase space, the infection almost surely dies out.

Invariant measure for the contact process with modified border in the non-attractive region

TL;DR

The paper analyzes a one-dimensional contact process with modified border on , where external boundary infections occur at rate and interior infections at rate . It proves the existence of an invariant measure for the process seen from the right edge in the non-attractive regime , , and shows that converges to for any infinite initial condition, with distributed according to and a random asymptotic edge velocity arising from stationary increments of the right edge. The analysis extends the edge-based convergence framework to the critical-border setting and establishes a full invariant-measure description along with an edge-velocity law. Additionally, it proves non-survival along the critical curve within the attractive region, thereby completing the phase diagram for the modified-border model and highlighting edge-driven long-term behavior as the system approaches criticality.

Abstract

We investigate a modified one-dimensional contact process with varying infection rates. Specifically, the infection spreads at rate along the boundaries of the infected region and at rate elsewhere. We establish the existence of an invariant measure when and , where is the critical parameter of the standard contact process. Moreover, we show that, when viewed from the rightmost infected site, the process converges weakly to this invariant measure. Finally, we prove that along the critical curve within the attractive region of the phase space, the infection almost surely dies out.

Paper Structure

This paper contains 4 sections, 8 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Convergence to invariant measure
  4. Non-survival at criticality

Key Result

Theorem 1.1

Suppose $\lambda_e \le \lambda_i$ and $\theta(\lambda_i, \lambda_e)>0$ or $\lambda_e=\lambda_i = \lambda_c$. There is a measure $\mu$ such that for every initial condition $A \subseteq \mathbb{Z}$ with $|A|=+\infty$ and $\sup A < +\infty$, $\Psi \xi^A_t \rightarrow \mu$ weakly. Moreover, $\Psi \xi^{

Figures (1)

  • Figure 1: Phase space for the contact process with modified border

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • proof : Proof of Theorem \ref{['thm_measure']}.
  • Proposition 3.1
  • Proposition 3.2
  • proof : Proof of Proposition \ref{['prop_uniquenessmeasure']}.
  • proof : Proof of Proposition \ref{['prop_fullmeasure']}.
  • proof : Proof of Theorem \ref{['thm_criticaldies']}.
  • ...and 4 more