28 problems

Product-form characterization conjecture. The network $(\mathcal{G},\kappa)$ has a product-form stationary distribution if and only if it is complex balanced.

Let $m$ be a sector, let $S(m)$ denote its set of configurations, and let $\mathbb{P}(\sigma_1,\ldots,\sigma_L)$ be the stationary weight of a configuration…

The BAR approach refers to deriving diffusion limits of stationary distributions using the stationary equation and a test function. In the history-dependent M/M/1 queue framework,…

Letac's conjecture. If $\mu$ is not supported on a lattice, then $\pi_{\mu}$ is the unique stationary distribution, and the distributions $\pi_n$ of $x_n$ converge to $\pi_{\mu}$ i…

Let $u$ be the transition rate from the lounge to the queue, let $beta$ satisfy … where $\eta>0$ is constant, and let $A=\lfloor\eta\mu/\alpha\rfloor$. For each $u$, write…

The construction above starts with a finite state space, a prescribed dependence function, and a stationary marginal distribution, and uses an information projection to construct a…

Beta-limit conjecture. As $c\to\infty$, in the two-island case both $X$ and $Y$ converge in distribution to the beta distribution with parameters $(a_1+b_1,a_2+b_2)$; in the seed-b…

Generic 2n-support conjecture. Generically, the support set $\Omega$ needs at least $2n$ elements: one per column to satisfy…

Let the open WASEP have parameters $u$ and $v$, and let $I=\lsem a,b \rsem$ satisfy … For $q$ from the paper's weak-asymmetry assumption with $\varepsilon>0$, consider the low-dens…

Let $P$) be the one-dimensional probabilistic cellular automaton on configurations with state space ${\mathbb{S}}=\{0,1\}$, neighborhood $\{0,1\}$, and transition probabilities … L…

Non-product-form conjecture. Then $\mathcal{G}$ has a non-product-form stationary distribution independently of the reaction rate; equivalently, it belongs to class N.

Let a reflected Brownian motion in a convex polyhedral cone have a stationary distribution, and consider the joint process used to represent sensitivities of its stationary distrib…

The alternative queueing construction uses the same customer discipline for newly arrived and previously present customers: each customer rejects each offered service with probabil…

Mixture classification conjecture. Under these conditions, every stationary gap distribution of an infinite system of competing Brownian particles is equal to $\pi_\rho$ for some p…

Let $\sigma_{\nu}=\sigma/\nu$, and define … Let $y$ be distributed according to the corresponding stationary density proportional to $e^{-\mathcal{E}(y)/(2\sigma_{\nu}^2)}$. Second…

The discrete-state Markov processes studied in the cited work have stationary distributions in product form under certain necessary and sufficient conditions. Continuum-limit conje…

Let the diffusion coefficients satisfy $\sigma_n=1$ for every $n\in\mathbb{Z}$, and assume the conditions of Theorem 2.1 hold. For $(a,b)\in\mathbb{R}^2$, define … and let … For…

Stationary-distribution conjecture. Based on the stationary density of the discrete-time Ornstein–Uhlenbeck approximation, the stationary distribution of $X^*$ can be approximated…

Let $g_w(q_1,\dots,q_n)$ denote the stationary probability density for the continuous TASEP configuration with permutation $w$, and let $\mathbf{k}=(k_1\ge\cdots\ge k_r\ge 1)$ sati…

Let $g_w(q_1,\dots,q_n)$ denote the stationary probability density for the continuous TASEP configuration with permutation $w$, and let $w_0$ be the longest permutation. For…

Let $p(x)$ be a differentiable density with bounded $\nabla \log(p(x))$ on a path-connected compact domain $D\subset\mathbb{R}^d$ with smooth boundary, and let…

Let $\Omega_n$ be the simplex of probability vectors, and consider a second-order Markov chain with stationary vectors satisfying … A $(k-1)$-dimensional face of $\Omega_n$ is the…

Ayyer–Linusson's stationary-state formula conjecture. The displayed form gives the stationary state $|\bar P_m\rangle$ of sector $m$.

Let $\mathcal Q$ be a multiline queue with state space $\Omega^{FM}_m$, and let $\Omega_m$ be the state space of the corresponding inhomogeneous multispecies exclusion process. For…

Let $\hat Z^r(\infty)$ denote the stationary diffusion-scaled total number of customers in the multiclass multiserver queue, and let $\theta$ be the parameter appearing in the expo…