25 problems

Let $K$ be the population-size scaling parameter, let $\hat{X}^K$ denote the rescaled epidemic process, and let $\Lambda<0$ be the non-persistence parameter. Consider initial condi…

Let $\mu$ be an initial distribution in the equivalence class indexed by $\sigma$, and let $M^*$ be the normalised transfer operator. A convergence conjecture. If there exists a qu…

Poisson approximation conjecture. We conjecture that $\bar{N}(A)$ is close to the quasi-stationary distribution of the population process, and that the approximation becomes closer…

Let $\tau$ be the absorption or exit time of the random dynamical system, let $\tau>t$ denote survival up to time $t$, and let $\mathrm D\varphi_t$ be the linearisation of the flow…

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…

Quasi-stationary extinction-event conjecture.

Quasi-stationary distribution conjecture. Then:

A Yaglom limit is the limiting conditional distribution of a Markov process given survival, and the initial position is the starting state of that process. Independence conjecture.…

Let $N$ be the population-size parameter, let $s=1$ specify the Verhulst logistic model, and let $R_0$ denote its basic reproduction parameter. For a quasi-stationary distribution…

Consider a regime in which a polymorphic quasi-stationary distribution (QSD) is stable for the dynamics of $\mu A_t$, and consider a finite yet large population-size process corres…

Let $s>0$ and let $r$ be any bounded function. Write $\rho_\alpha(\gamma)$ for the extinction rate associated with the polymorphic quasi-stationary distribution as a function of th…

Let $n$ be the population-size scaling parameter, let $I_h^n(t)$ and $I_v^n(t)$ denote infected hosts and infected vectors, and let $\mathcal{R}_0$ be the basic reproduction number…

Let $n$ be the host population size, let $I_h^n(t)$ denote the number of infected hosts at time $t$, and let $\mathcal{R}_0$ be the basic reproduction number. When…

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…

Consider $N$-BBM and $N$-BRW, with empirical measures viewed under their invariant measure $\lambda^N$. Quasi-stationary distribution conjecture. Both $N$-BBM and $N$-BRW are ergod…

Durrett–Remenik limiting equation conjecture. The limiting equation for an $N$-BBM branching at rate $r$ should be the free-boundary problem above, and the empirical measures in eq…

Fleming–Viot convergence conjecture. Under the stationary distribution, the empirical distribution of this system converges as $N$ tends to infinity to the quasi-stationary distrib…

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

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 $\mu$ be a weak limit point of the quasi-stationary distributions $\mu^\varepsilon$ for the replicator process with $c>0$ sufficiently large. In the rock-paper-scissors case, l…

Let $\mu$ be a weak limit point of the quasi-stationary distributions for the Thompson host–parasitoid process $X_t^\varepsilon$, and suppose that $k\geq 1$. In this regime parasit…

Let $\mu$ be a weak limit point of quasi-stationary distributions for the Leslie–Gower process. In the bistable case, namely when $b_i>1$ for $i=1,2$ and $c_i(b_j-1)>b_i-1$ for…