19 problems

Analogous spine-distribution conjecture. An analogous result should hold for every Fleming–Viot process, perhaps under mild technical assumptions.

Consider the skeleton of a Fleming–Viot process, formed by attaching the side branches to its spine, and let $Q$-process and critical binary branching process have their usual mean…

Let $(\vec{X}^N_t)_{0\leq t<\infty}$ be a Fleming–Viot process, and let the ancestries of its particles be traced backwards in time, producing a coalescent process. Kingman coalesc…

Let $\tt FV(\alpha,\theta)$ be the Fleming–Viot diffusion with parameters $\alpha\in(0,1)$ and $\theta>-\alpha$, and let $\tt PDRM(\alpha,\theta)$ denote its stationary Poisson–Dir…

Let $G_t$ be the semigroup associated with the underlying killed McKean–Vlasov dynamics, let $\Psi^N$ be the stationary distribution of the empirical measure-valued Fleming–Viot pr…

Let $U\subseteq\mathbb{R}^d$ be an open domain, and consider the Fleming–Viot particle system with McKean–Vlasov dynamics whose particles interact through a drift depending on thei…

Let a Fleming–Viot diffusion be given, and consider its projection onto the Kingman simplex. Feng–Sun projection conjecture. Petrov's two-parameter Poisson–Dirichlet diffusions sho…

Universality conjecture. Analogous asymptotic spine results should hold for all Fleming–Viot processes, perhaps under mild technical assumptions.

Let $\nu_c$, with $0<c<1-p/q$, be a quasi-stationary distribution of the nearest-neighbor random walk on ${\mathbb N}$ with absorption at the origin, where $q>p$ are the left- and…

Let $p$ and $q$ be the right- and left-jump rates, respectively, with $q>p$, and let $\rho^N$ be the equilibrium density of the Fleming–Viot process driven by the nearest-neighbor…

Let $Z=(Z_t,t\geq0)$ be a Markov driving process on $\Lambda\cup\{0\}$, with $0$ absorbing, and let $\tau$ be its absorption time. For $N\geq2$, let…

Empirical-measure convergence conjecture. As the number $N$ of processes tends to infinity, the empirical measure under the stationary measure converges to the QSD.

Reversibility conjecture. The $\Xi$-Fleming–Viot process is reversible only if it degenerates to the classical Fleming–Viot process.

Consider the Fleming–Viot particle system and the related one-dimensional branching Brownian motion system in which particles branch at a fixed rate and the left-most particle is i…

Fleming–Viot dynamics use particles evolving independently until one attempts to leave the state space, at which point that particle is relocated to the position of another particl…

Burdzy–Holyst–Ingerman–March conjecture. Assuming that $\xi$ has distribution $\lambda^{\text{N}}$, the law of the random measure $m(\cdot,\xi)$ converges to the law concentrated o…

Let $\mathcal V^\Lambda$ be the tree-valued $\Lambda$-resampling dynamics associated with a finite measure $\Lambda\in\mathcal M_f([0,1])$. Continuous-paths conjecture. The followi…

Light-tail Cannings conjecture. Under a suitable time rescaling, $(\mathcal V^N)_{N\in\mathbb N}$ converges weakly to the tree-valued Fleming–Viot dynamics in the Skorohod topology…

Let $cmathcal U^N$ denote an exchangeable tree-valued population dynamics with population size $N$, and let $cmathcal U=(cmathcal U_t)_{t\beq 0}$ be the tree-valued Fleming–Viot dy…