Abstract
Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the $\Lambda$-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding $\Lambda$-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support.
Highlights
The Fleming-Viot process is a probability-measure-valued stochastic process for population genetics
The compact support property for solutions to SPDEs can be found in Mueller and Perkins [12]
Blath [3] showed that the Λ-Fleming-Viot process with underlying Brownian motion does not have a compact support if the Λ-coalescent does not come down from infinity
Summary
The Fleming-Viot process is a probability-measure-valued stochastic process for population genetics. The earliest work on the compact support property for classical Fleming-Viot process is due to Dawson and Hochberg [4] It was shown in [4] that at any fixed time T > 0 the classical Fleming-Viot process with underlying Brownian motion has a compact support and the support has a Hausdorff dimension not greater than two. Blath [3] showed that the Λ-Fleming-Viot process with underlying Brownian motion does not have a compact support if the Λ-coalescent does not come down from infinity. We find both upper and lower bounds on the Hausdorff dimension for the compact support
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