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Generalized stepping stone model with $Ξ$-resampling mechanism

A generalized stepping stone model with $Ξ$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $Ξ$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $Ξ$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type.