  The Fleming-Viot measure-valued diffusion is a Markov process describing the
evolution of (allelic) types under mutation, selection and random reproduction.
We enrich this process by genealogical relations of individuals so that the
random type distribution as well as the genealogical distances in the
population evolve stochastically. The state space of this tree-valued
enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS)
consists of marked ultrametric measure spaces, equipped with the marked
Gromov-weak topology and a suitable notion of polynomials as a separating
algebra of test functions. The construction and study of the TFVMS is based on
a well-posed martingale problem. For existence, we use approximating finite
population models, the tree-valued Moran models, while uniqueness follows from
duality to a function-valued process. Path properties of the resulting process
carry over from the neutral case due to absolute continuity, given by a new
Girsanov-type theorem on marked metric measure spaces. To study the long-time
behavior of the process, we use a duality based on ideas from Dawson and Greven
[On the effects of migration in spatial Fleming-Viot models with selection and
mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if
the Fleming-Viot measure-valued diffusion is ergodic. As a further application,
we consider the case of two allelic types and additive selection. For small
selection strength, we give an expansion of the Laplace transform of
genealogical distances in equilibrium, which is a first step in showing that
distances are shorter in the selective case.
