English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Fleming-Viot (FV) process is a measure-valued diffusion that models theevolution of type frequencies in a countable population which evolves underresampling (genetic drift), mutation, and selection. In the classic FV modelthe fitness (strength) of types is given by a measurable function. In thispaper, we introduce and study the Fleming-Viot process in random environment(FVRE), when by random environment we mean the fitness of types is a stochasticprocess with c\`adl\`ag paths. We identify FVRE as the unique solution to a socalled quenched martingale problem and derive some of its properties viamartingale and duality methods. We develop the duality methods for generaltime-inhomogeneous and quenched martingale problems. In fact, some importantaspects of the duality relations only appears for time-inhomogeneous (andquenched) martingale problems. For example, we see that duals evolve backwardin time with respect to the main Markov process whose evolution is forward intime. Using a family of function-valued dual processes for FVRE, we prove that,as the number of individuals $N$ tends to $\infty$, the measure-valued Moranprocess $\mu_N^{e_N}$ (with fitness process $e_N$) converges weakly inSkorokhod topology of c\`adl\`ag functions to the FVRE process $\mu^e$ (withfitness process $e$), if $e_N \rightarrow e$ a.s. in Skorokhod topology ofc\`adl\`ag functions. We also study the long-time behaviour of FVRE process$(\mu_t^e)_{t\geq 0}$ joint with its fitness process $e=(e_t)_{t\geq 0}$ andprove that the joint FV-environment process $(\mu_t^e,e_t)_{t\geq 0}$ isergodic under the assumption of weak ergodicity of $e$.

An Ergodic Theorem for Fleming-Viot Models in Random Environments