# Graphical representation of certain moment dualities and application to population models with balancing selection

@article{Jansen2012GraphicalRO,
title={Graphical representation of certain moment dualities and application to population models with balancing selection},
author={Sabine Jansen and Noemi Kurt},
journal={Electronic Communications in Probability},
year={2012},
volume={18},
pages={1-15}
}
• Published 25 July 2012
• Mathematics
• Electronic Communications in Probability
We investigate dual mechanisms for interacting particle systems. Generalizing an approach of Alkemper and Hutzenthaler in the case of coalescing duals, we show that a simple linear transformation leads to a moment duality of suitably rescaled processes. More precisely, we show how dualities of interacting particle systems of the form $H(A,B)=q^{|A\cap B|}, A,B\subset\{0,1\}^N, q\in[-1,1),$ are rescaled to yield moment dualities of rescaled processes. We discuss in particular the case \$q=-1…

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## References

SHOWING 1-10 OF 21 REFERENCES
GRAPHICAL REPRESENTATION OF SOME DUALI- TY RELATIONS IN STOCHASTIC POPULATION MO- DELS
• Mathematics
• 2007
We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. [1]) and for the self-duality of Feller’s branching diffusion
Coexistence in locally regulated competing populations and survival of branching annihilating random walk
• Mathematics
• 2007
We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic
Dual Families of Interacting Particle Systems on Graphs
A simple condition for IPS (Interacting Particle Systems) with nearest neighbor interactions to be self-dual is given. It follows that any IPS with the contact transition and no spontaneous birth is
Duals and thinnings of some relatives of the contact process
This paper considers contact processes with additional voter model dynamics. For such models, results of Lloyd and Sudbury can be applied to find a self-duality, as well as dualities and thinning
Branching-coalescing particle systems
• Mathematics
• 2005
We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting
Quantum Operators in Classical Probability Theory: II. The Concept of Duality in Interacting Particle Systems
• Mathematics
• 1995
Duality has proved to be a powerful tool in the theory of interacting particle systems. The approach in this paper is algebraic rather than via Harris diagrams. A form of duality is found which
A countable representation of the Fleming-Viot measure-valued diffusion
• Mathematics
• 1996
The Fleming-Viot measure-valued diffusion arises as the infinite population limit of various discrete genetic models with general type space. The paper gives a countable construction of the process
Markov Processes: Characterization and Convergence
• Mathematics
• 2005
Introduction. 1. Operator Semigroups. 2. Stochastic Processes and Martingales. 3. Convergence of Probability Measures. 4. Generators and Markov Processes. 5. Stochastic Integral Equations. 6. Random
Interacting Particle Systems
The Construction, and Other General Results.- Some Basic Tools.- Spin Systems.- Stochastic Ising Models.- The Voter Model.- The Contact Process.- Nearest-Particle Systems.- The Exclusion Process.-