25 problems

Long-term behaviour conjecture. 1) If $\alpha<0$ and $\alpha+\beta\nu(G)<0$, then the process $X$ is positive recurrent with the stationary distribution $\mu$ defined in Theorem. 2…

Let $\Omega_1$ be the space of decreasing nonnegative sequences with sum $1$, and let $\widehat{\mu}_1$ denote the Poisson-Dirichlet distribution on $\Omega_1$ with parameter…

Let $G=(V,E)$ be a finite graph with positive edge rates $r_{xy}$, and let $L$ be the generator of the symmetric simple exclusion process on the $n$-particle configuration space…

Let $\Lambda$ be a finite graph, and let $L$ be the generator of the symmetric simple exclusion process with positive jump rates $J_{xy}$ on its edges. Let $H_\Lambda^{(n)}$ be the…

Flip-process distribution conjecture. The following are equivalent:

The preceding theorem concerns a collection of generators for controlled Markovian quantum dynamics, with the relevant notation and generator class defined there. Converse characte…

Nonlinear Perron–Frobenius conjecture. There exists a value $E^\star(\boldsymbol{\gamma})$ such that: for $E>E^\star(\boldsymbol{\gamma})$, all orbits tend towards the system bound…

Conditional Poincaré inequality inheritance conjecture. Suppose the standard form of the Poincaré inequality holds for $X$ with constant $c_0$, and suppose ${\cal Z}$ is obtained u…

Near second-order Markov conjecture. The MA filtering of the first-order Markov process generates nearly a second-order Markov process.

Let $0 < \gamma \leq \alpha < 1$. Consider the unlabelled $(\alpha,\gamma)$-chain on rooted multifurcating trees with $n$ leaves, obtained by selecting and deleting a uniform leaf…

Extended-spectrum interweaving conjecture. There exists an interweaving relation from $(\exp(tL))_{t\geq 0}$ to $(\exp(t\widetilde L))_{t\geq 0}$ if and only if the extended spectr…

An MMPP is a Markovian modulated Poisson process; write $c^2$ for the squared coefficient of variation of its interarrival time. Squared coefficient-of-variation conjecture. For an…

Let $D_{\infty}$ denote the infinite Ehrenfest process and let $L$ be the Ornstein–Uhlenbeck operator. An intertwining existence conjecture. There exists a non-trivial intertwining…

Let $\pi$ be the invariant measure, let $f\in L_0^2(\pi)$, and let $\phi\in L_0^2(\pi)$ be the corresponding solution of the overdamped Poisson equation. Let $\alpha_f$ be the matr…

For $i \in \{1,\ldots,n\}$, let $(\mathcal{L}_{i},\mathcal{D}(\mathcal{L}_i))$ be generators of ergodic Feller semigroups. Let $A_k^i$, $B^i$, and $\mathcal{J}$ arise from decompos…

Let $\mathcal{G}$ be the set of coupling operators and let $\mathcal{G}^0$ be the subset for which the corresponding Feller coupling has the required invariant measure. Convexity c…

Let $(\bar{E},\bar{\tau})$ be the product state space, let $(\mathcal{L}_i,\mathcal{D}(\mathcal{L}_i))$ be the generators of the marginal Feller semigroups, and let $\mathcal{G}$ d…

Let $H_r$ be the state space of height configurations and let $\widetilde X^{(z,z')}_r$ be the Markov process on $H_r$ whose deterministic component evolves each height according t…

Freidlin–Wentzell/semi-classical limit. For $\lambda\ll 1$,

Let $M_L$ be the Markov matrix of the asymmetric annihilation process on a system of size $L$. The characteristic polynomial has only $2L$ distinct eigenvalues out of a possible…

In the setting of the preceding proposition, consider a countable class of maps and the BIC penalty. BIC consistency conjecture. We almost surely have Property 2 from the propositi…

Let $(A(t),B(t))$ be the fast continuous-time embedding of the simple harmonic urn, let … and write $\mathbb{E}_n$ for expectation conditional on $A(0)=n$ and $B(0)=0$. Let…

Let $M_L$ be the Markov matrix of the asymmetric annihilation model, with rows and columns indexed by $i=0,\ldots,2^L-1$. For an integer $m$, define $n_1(m)$ to be $(-1)$ raised to…

Characteristic-polynomial factorization conjecture. The characteristic polynomial and its successive ratios are

Conditional distribution conjecture. The conditional distribution of $Y_t$ given $\mathcal{F}_{s,u}$ is