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We study the evolution of genealogies of a population of individuals, whosetype frequencies result in an interacting Fleming-Viot process on $\Z$. Weconstruct and analyze the genealogical structure of the population in thisgenealogy-valued Fleming-Viot process as a marked metric measure space, witheach individual carrying its spatial location as a mark. We then show that itstime evolution converges to that of the genealogy of a continuum-sites steppingstone model on $\R$, if space and time are scaled diffusively. We construct thegenealogies of the continuum-sites stepping stone model as functionals of theBrownian web, and furthermore, we show that its evolution solves a martingaleproblem. The generator for the continuum-sites stepping stone model has asingular feature: at each time, the resampling of genealogies only affects aset of individuals of measure $0$. Along the way, we prove some negativecorrelation inequalities for coalescing Brownian motions, as well as extend thetheory of marked metric measure spaces (developed recently by Depperschmidt,Greven and Pfaffelhuber [DGP12]) from the case of probability measures tomeasures that are finite on bounded sets.

Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\Z$