71 problems

A Hunt process is a stochastic process with the strong Markov and regularity properties assumed in probabilistic potential theory. A set is semi-polar if it is contained in a count…

Peres's cutoff conjecture. A process exhibits cutoff if and only if it satisfies the product condition

Let $D$ be a domain, let $(X_t)$ be a Markov process of jump type, and let the representation formula referred to in the source be denoted by … generalizes to a wide class of Marko…

Let $\{{\cal G}^{f}; f \in {\cal S}\}$ be a collection of generators, with each ${\cal G}^{f}$ associated to a semigroup ${\cal T}^{f}$. In the setting where the semigroups…

Consider the diffusion operator on $[0,infty)$ … and define … The uniqueness criterion is $\int_0^\infty\mu[0,x]e^{-C(x)}=\infty$. Strong ergodicity criterion. The criterion for st…

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…

A q-graded Markovian matrix is intended to generalize the bi-graded Markovian matrices discussed above, although the source does not define the grading or the required Markovian pr…

Markovian value-function conjecture. In view of the Markov structure, a solution $(V_t)_{t\geqslant 0}$ to this recursion must have the form

Let $G$ be an abelian group, let $V\subseteq G$ be a symmetric set of generators, and let $\Delta$ be the corresponding symmetric-walk Laplacian. Let $\mathcal{L}$ be the generator…

Let $G$ be an abelian group and let a lifted random walk on $G$ be obtained by augmenting the position with a direction variable, producing a second-order Markov chain. Write…

Consider the lifted random walk on the discrete torus $\mathbb{Z}_n^d$ with fixed dimension $d$ and direction-change probability $p\in(0,1)$. Its relaxation time is denoted by…

Let $X$ be the Markov process on $[0,1]$ and let $Y$ be the Markov process on $\mathbb{Z}_{\geq 0}$ with generator … Let $B([0,1]\times\mathbb{Z}_{\geq 0})$ be the Banach space of…

Let $f_{\mathtt{ISE}}$ be the random differentiable density of the Integrated SuperBrownian Excursion arising as the scaling limit of the vertical profile of suitable conditioned r…

Let $E$ be the state space, let $T>0$, and let $bar{ u}{tau,zeta}$ denote the law associated with the mean-field process started at time $tauin[0,T]$ from $zetainmathcal{P}^q(E)$.…

Let $E$ be the state space of the Markov processes considered in sections 3–5, and let $\overline E$ be a Ray–Knight compactification of $E$. The boundary of interest is the set…

The result concerns robust mean estimation for data that are both Markovian and adversarially contaminated. In particular, the estimator texttt{RUMEM} achieves, under the stated Hu…

Semi-Markov property conjecture. The process $X$ is semi-Markov, even in the general case in which $E$ is a locally compact separable Hausdorff space.

Donsker–Varadhan representation conjecture. The rate function $I_{\mathscr D}(\beta)$ in the large deviation principle should satisfy

Convergence conjecture. The state $\phi$ converges in probability to the set of Fock states, which correspond to the computational basis in $\mathcal{S}_N$.

The diffusion Monte Carlo (DMC) method uses a free-evolution transition together with an absorbing well centered at the origin; uniform convergence means convergence whose bounds a…

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…

Spectral-extent diagonal-maximum conjecture. The concentration of $\max_{1\le l \le D}J_{ll}$ around $\tilde{\gamma}$, known in the symmetric case under mild conditions on the weig…

Generalization conjecture. The results could be greatly generalized in this direction for degree distributions with fast decaying tails.

Pure-jump limiting semigroup conjecture. In general, the limiting semigroup $e^{t\mathcal{L}_\infty}$ will be associated with a pure-jump continuous-time Markov process living in t…