27 problems

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying fa…

A stochastic mass-action reaction system (SMART) is a continuous-time Markov chain describing molecule counts in a mass-action reaction network, and it is weakly reversible when ev…

Fröhlich's conjecture. The ultraviolet-renormalized Nelson Hamiltonian generates an ergodic semigroup.

Let a quantum system have local channels on its edges, with no dissipators and a single unitary channel on each edge. In this setting, ergodicity refers to the ability of products…

Let $(X_i)$ be the Markovian Langlie design chain described above, with perturbation parameters $(\varepsilon_i)$, and let ergodicity mean the ergodicity established for the chain…

Let MY-MALA denote the Moreau-Yosida Metropolis-adjusted Langevin algorithm, with proximal parameter $\lambda$. A distribution is light-tailed if it belongs to the class of distrib…

Fix $u,v\in\R$. Let $h_0,\tilde h_0\in\operatorname{C}([0,1])$ be arbitrary random functions on the same probability space, and let $\xi$ be white noise on $\mathbb{R}\times[0,1]$.…

Let an extended many-body quantum system satisfy the eigenstate thermalization hypothesis (ETH), and consider quantum trajectories generated by monitoring a sufficiently large set…

Consider a one-dimensional, unimodal target distribution and a Zig-Zag process with zero refresh rate. The process has momentum in the sense that its velocity persists until a swit…

Ergodicity conjecture. The strict inequalities

Consider the generalized join the shortest queue system with retrials and two infinite-capacity orbit queues, whose state is a Markov-modulated two-dimensional random walk. Let the…

Consider the Markov semigroup $\{P_t\}_{t\geq 0}$ corresponding to the weakly damped, stochastically driven KdV equation on $H^2$, with damping parameter $\gamma>0$, forcing…

Weakly reversible positive recurrence conjecture. All weakly reversible mass-action SRNs are positive recurrent.

Differentiability conjecture. The function $\Lambda_{\mathbb{P}_0}$ is differentiable on $(0,\infty)^2$, and consequently

Let $\varrho$ denote the spare-capacity parameter, let $x_i^-\coloneqq\max\{-x_i,0\}$, and let an invariant probability measure refer to an invariant probability measure of the Mar…

Let $\operatorname{GRBM}(R,\mu,\Gamma,U)$ be the generalized reflected Brownian motion defined by the strong solution of the stochastic differential equation in the paper, with tra…

Ergodicity conjecture. Based on numerical experiments, the canonical multi-dimensional Zig-Zag process should be ergodic in general under only mild conditions.

Scaling conjecture. The larger $\lambda$ (and thus the reference drift term), the smaller $\delta$ and the larger $b_1(\delta_{*})$ (and thus a larger window of possibilities for t…

Unconditional convergence conjecture. This limit should exist without imposing any conditions on $\underline{Q}$:

Let Algorithm denote the approximate randomized Hamiltonian Monte Carlo algorithm, and let the potential energy function and the reversible, volume-preserving integrator satisfy su…

Consider a randomly forced fluid system with stochastic forcing acting on a small collection of low-frequency temperature modes, with $N_1=0$ and small $N_2>0$, where $N_1$ and…

Ergodicity conjecture. When $\bar\alpha=0$, ergodicity holds precisely when the dominant regular singularity lies outside the unit circle:

Let ${\cal L}$ be the random walk associated with routing matrices $P$ and $\wt P$, and let conditions … denote the necessary rate and flux conditions introduced in the paper. Ergo…

Finite-moment conjecture. The result known for the operatorname{INGARCH}(1,1) process, namely that all moments of $\lambda_t$ and $N_t$ are finite, should extend to this more gener…

Let $S=\mathbb{R}^2\times\mathbb{S}_1$, let $(X_t)_{t\geq 0}$ be the diffusion on $S$ associated with the melt-spinning SDE, and let $\mu$ be the invariant distribution given by ……