2011/04/06 by Donald A. Dawson, Dawson, Donald A., Andreas Greven +1
Mathematics · #60J60 #60J68 #60J70 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60J60 #msc:60J68 #msc:60J70
paper · pdf · doi:10.48550/arxiv.1104.1099
69 pages
arxiv created 2011/04/06 · openalex publication_date 2011/04/06 · arxiv updated 2011/04/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Consider a system X = ((xξ(t)), ξ∈ ΩN)t ≥ 0 of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space (\CP(\I))ΩN, where \I is the type space, ΩN the geographic space is assumed to be a countable group and \CP denotes the probability measures. We establish various duality relations for this process. These dualities are function-valued processes which are driven by a coalescing-branching random walk, that is, an evolving particle system which in addition exhibits certain changes in the function-valued part at jump times driven by mutation. In the case of a finite type space \I we construct a set-valued dual process, which is a Markov jump process, which is very suitable to prove ergodic theorems which we do here. The set-valued duality contains as special case a duality relation for any finite state Markov chain. In the finitely many types case there is also a further tableau-valued dual which can be used to study the invasion of fitter types after rare mutation. This is carried out in \citeDGsel and \citeDGInvasion.