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A generalized stepping stone model with $\Xi$-resampling mechanism is a twodimensional probability-measure-valued stochastic process whose moment dual issimilar to that of the classical stepping stone model except that Kingman'scoalescent is replaced by $\Xi$-coalescent. We prove the existence of such aprocess by specifying its moments using the dual function-valued$\Xi$-coalescent process with geographical labels and migration, and thenverifying a multidimensional Hausdorff moment problem. We also characterize thestationary distribution of the generalized stepping stone model and show thatit is not reversible if the mutation operator is of uniform jump-type.

Generalized stepping stone model with $\Xi$-resampling mechanism