For concreteness take the lattice $Z^3$ -- what's important is random walk is transient.
Consider a model CBRW of branching coalescing random walk.
To be definite (though the point is that the details shouldn't matter) take particles behaving as:
(i) continuous-time simple RW
(ii) splitting at rate \( \beta\), a daughter particle placed at an adjacent site
(iii) two particles coalesce into one if they meet.
Such a process has a (translation-invariant) stationary distribution for which the mean number of particles per site (intensity) equals some \( p(\beta) \). See this Athreya-Swart paper for discussion. There seems no simple exact formula for the complete stationary distribution, but one can calculate \( p(\beta) \sim c \beta\) as \( \beta \to 0 \).
Model 2. This is the constrained Ising model from page 16-17 of The constrained Ising model as an algorithm for storage in dynamic graphs but now with $Z^3$ instead of a finite graph. The model has a parameter \( p \) and in the stationary distribution, site are occupied as an i.i.d. Bernoulli (\( p \)) process. As commented in the linked material, this process evolves in a way that emulates some variant of CBRW.
Problem A. The issue is to show that, for Models 1 and 2 and related models where there is a stationary distribution, as the intensity (which depends on model parameters) tends to 0, the suitable time-rescaled process in which we track the ``lineages" (splits and merges) but not the spatial positions, of the particles, converges in distribution to the following simple limit process.
A limit process. The limit process is constructed w.r.t. a reference particle at time 0. This has a lineage from the infinite past to the infinite future; ``splits" and ``merges" with some other lineage both occur at times of Poisson(1) point processes. A split [merge] creates a new lineage running froward [backward], and recursively such a lineage has its own splits and merges, each creating a new lineage.
In the conjectured convergence result we need to overlook "jitter", the times where a daughter particle soon coalesces with its parent, which we do by asking only for convergence of finite dimensional distributions (in time).
Problem B. As suggested in The constrained Ising model as an algorithm for storage in dynamic graphs, study this model on a graph which is changing in time.
History. This page posted 2012, though the linked "constrained Ising model" material was in the 2006 version.
Update (March 2018). Pillai-Smith (2017) give a detailed treatment of Model 2 on the \(d \ge 3\) dimensional torus in order to study its mixing time. Aghajani et al. study different models of storage in unreliable networks.
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