Abstract
We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. ) and for the self-duality of Feller's branching diffusion with logistic growth (cf. ). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above.
Highlights
Two processes (Xt)t≥0 and (Yt)t≥0 with state spaces E1 and E2, respectively, are called dual with respect to the duality function H if H : E1 × E2 → Ê is a measurable and bounded function and if Ex[H(Xt, y)] = Ey[H(x, Yt)] holds for all x ∈ E1, y ∈ E2 and all t ≥ 0
The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation
We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above
Summary
Denote by Xt ∈ Æ0 the number of particles at time t ≥ 0 of the branching-coalescing particle process defined by the initial value X0 = n and the following dynamics: Each particle splits into two particles at rate b, each particle dies at rate d and each ordered pair of particles coalesces into one particle at rate c All these events occur independently of each other. = 1, we deduce from (1.6) (and from the convergence properties of (XtN )t≥0 and of (YtN )t≥0) the moment duality of a branching-coalescing particle process with a resampling-selection model It remains to prove that the birth and death events become asymptotically independent as N → ∞ It is known, see e.g. Section 2 in [3], that the dual process of the Moran model (MtN )t≥0, N ≥ 1, is a coalescing random walk. The dynamics of the process (1 − YtN )t≥0 is obtained from the dynamics of (YtN )t≥0 by interchanging the roles of the types 0 and 1
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