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Monotone duality of interacting particle systems

Jan Niklas Latz, Jan M. Swart

TL;DR

This work extends duality theory for monotone interacting particle systems by allowing the dual to start from infinite initial states and by introducing an upper invariant law for the dual, framing the dual as an independent interacting system in its own right. It develops a graphical Poisson construction and a backward stochastic flow to define a monotone dual process on the compact space ${\cal H}(\Lambda)$, proves the dual is a Feller process with caglad paths, and establishes upper invariant laws and ergodicity results that connect forward survival with dual stability. The cooperative contact process is used as a concrete instantiation, yielding phase-structure insights, duality-driven expressions for survival probabilities, and continuity results for the forward process with respect to parameters $\alpha$ and $\delta$. The results enable robust analysis of long-time behavior on general grids, including Cayley graphs, and offer a framework for exploring phase transitions, stability, and convergence in more complex, non-additive monotone systems.

Abstract

The duality theory for monotone interacting particle systems was initiated by Gray (1986) and further developed by Sturm and Swart (2018). It contains the better known additive duality as a special case but differs in the sense that the dual process contains not only single particles but also pairs, triples, and general $n$-tuples of particles, which correspond to the fact that in the forward process sometimes several particles are needed to create one particle at a later time. In earlier work, the dual process was constructed for finite initial states only, but, assuming that the empty state is a trap for the forward process, we show that the dual process can be started in infinite initial states and has an upper invariant law. It can therefore be viewed as some sort of interacting particle system in its own right. For the monotone dual of a cooperative contact process, we show that the upper invariant law is the long-time limit started from any nontrivial homogeneous invariant law. We use this to prove continuity of the survival probability of the forward process as a function of its parameters.

Monotone duality of interacting particle systems

TL;DR

This work extends duality theory for monotone interacting particle systems by allowing the dual to start from infinite initial states and by introducing an upper invariant law for the dual, framing the dual as an independent interacting system in its own right. It develops a graphical Poisson construction and a backward stochastic flow to define a monotone dual process on the compact space , proves the dual is a Feller process with caglad paths, and establishes upper invariant laws and ergodicity results that connect forward survival with dual stability. The cooperative contact process is used as a concrete instantiation, yielding phase-structure insights, duality-driven expressions for survival probabilities, and continuity results for the forward process with respect to parameters and . The results enable robust analysis of long-time behavior on general grids, including Cayley graphs, and offer a framework for exploring phase transitions, stability, and convergence in more complex, non-additive monotone systems.

Abstract

The duality theory for monotone interacting particle systems was initiated by Gray (1986) and further developed by Sturm and Swart (2018). It contains the better known additive duality as a special case but differs in the sense that the dual process contains not only single particles but also pairs, triples, and general -tuples of particles, which correspond to the fact that in the forward process sometimes several particles are needed to create one particle at a later time. In earlier work, the dual process was constructed for finite initial states only, but, assuming that the empty state is a trap for the forward process, we show that the dual process can be started in infinite initial states and has an upper invariant law. It can therefore be viewed as some sort of interacting particle system in its own right. For the monotone dual of a cooperative contact process, we show that the upper invariant law is the long-time limit started from any nontrivial homogeneous invariant law. We use this to prove continuity of the survival probability of the forward process as a function of its parameters.
Paper Structure (17 sections, 41 theorems, 148 equations, 1 figure)

This paper contains 17 sections, 41 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Monotone particle systems
  3. Graphical representations
  4. Duality
  5. The dual process
  6. Survival and stability
  7. Ergodicity of the dual process
  8. Discussion and open problems
  9. Proofs
  10. The dual space
  11. The dual process
  12. The upper invariant law
  13. Homogeneous invariant laws
  14. The cooperative contact process
  15. Appendix
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that the rates $(r_m)_{m\in{\cal G}}$ satisfy Then almost surely, for each $s\in{\mathbb R}$ and $x\in S^\Lambda$, there exists a unique cadlag function $X^{s,x}:[s,\infty)\to S^\Lambda$ such that $X^{s,x}_s=x$ and Setting ${\mathbf X}_{s,t}(x):=X^{s,x}_t$$(s\leq t,\ x\in S^\Lambda)$ defines a collection of random continuous maps $({\mathbf X}_{s,t})_{s\leq t}$ from $S^\Lambda$ into itsel

Figures (1)

  • Figure 1: Conjectured phase diagram for the cooperative contact process on ${\mathbb Z}^2$. The density of the upper invariant law $\rho(\alpha,\delta)$ is positive for $\delta<\delta_{\rm c}(\alpha)$ and the survival probability $\theta(\alpha,\delta)$ is positive for $\delta<\delta'_{\rm c}(\alpha)$. Numerically, one sees that $\rho(\alpha,\delta)>0$ also for $\delta=\delta_{\rm c}(\alpha)$ in the regime where $\delta'_{\rm c}(\alpha)<\delta_{\rm c}(\alpha)$ while the functions $\rho$ and $\theta$ are continuous everywhere else.

Theorems & Definitions (41)

  • Theorem 1: Poisson construction of particle systems
  • Proposition 2: Finite perturbations
  • Lemma 3: Finite systems
  • Lemma 4: Preserved subspaces
  • Proposition 5: Encoding monotone lower semi-continuous functions
  • Proposition 6: Dual topology
  • Theorem 7: Dual process
  • Lemma 8: Monotonicity of the dual process
  • Proposition 9: Order on the dual state space
  • Theorem 10: Upper invariant law
  • ...and 31 more