58 problems

Let $\tilde{X}_{t}$ be the continuous version of the density at time $t$, and let $H(x)$ denote its optimal Hölder index at $x$. For $\eta$ in the relevant interval, let $D(\eta)$…

Integrated stable-process tail conjecture. One has

Shi's conjecture. Suppose that $Z$ is symmetric and that $\alpha>1$. Then

Janicki and Woyczynski's conjecture. For every $t>0$, one has

Let $Z=(Z_t)_{t\geq 0}$ be the stable process considered in the paper, let $T$ be the associated stopping time, and let $\alpha$ be its stability index. For every $k>0$, the moment…

Let $D$ be a domain, let $0$ be a point whose accessibility from $D$ is considered, and consider the corresponding conditional symmetric $\alpha$-stable Lévy process at its lifetim…

Tail asymptotic conjecture. The same identity

Let $D\subset\mathbb{R}^2$ be a bounded convex domain symmetric with respect to both coordinate axes. Let $R$ be the smallest rectangle containing $D$ whose sides are parallel to t…

Let $D\subset\mathbb{R}^d$ be a bounded domain, and let $\lambda_1(D)$ and $\lambda_2(D)$ be the first and second eigenvalues for the symmetric $\alpha$-stable process, with…

Let $I=(-1,1)$, and let $\varphi_1^\alpha$ be the ground state eigenfunction for the symmetric stable process of index $0<\alpha<2$ killed upon leaving $I$. Concavity conjecture. T…

Let $\alpha$ be the stable index, let $C_{\alpha}$ be the constant appearing in the stable Lévy generator, and let $f$ belong to a suitable class of nice functions. Consider the…

Let $X^1,\dots,X^N$ be random walks in the domain of attraction of a $\beta$-stable process $U$ with scale function $b(n)$, and suppose that $d/\beta<1$. Let $\mathcal R_n^k$ denot…

Let the geometric $\alpha$-stable process be the symmetric Lévy process with characteristic exponent $\Psi(\xi)=\log(1+|\xi|^\alpha)$, and let $d$ denote the spatial dimension. The…

Diffusive-limit conjecture. As $\eta$ goes to zero, the processes $({\mathbf{X}}^\eta_t({\mathbf{x}}))_{t\ge 0}$ converge in law to the $2$-dimensional process…

Let $\theta$ denote the parameter of the skewed stable Ornstein–Uhlenbeck process, with stability index $\alpha$ and skewness parameter $\beta$. Let $\ell_n(\theta)$ be the true lo…

Let $\tau(\alpha,\rho)$ be the first hitting time of zero for a real strictly $\alpha$-stable process starting from one, where $\alpha\in(1,2]$ and $\rho$ is the positivity paramet…

Fix $\alpha\in(1,2)$. Let $T_n^{(\alpha)}$ be a critical Bienaymé–Galton–Watson tree with offspring distribution $\mu_\alpha$ satisfying $\mu_\alpha([x,\infty))\sim cx^{-\alpha}$ a…

The paper studies random quadrangulations obtained from critical Bienaymé–Galton–Watson trees whose offspring distribution has a stably decaying tail with exponent $\alpha\in(1,2)$…

Let $\gamma_t^x$ denote the field appearing in Theorem, and suppose that $\mu\in K^f{\eta}$ for some $\eta\in(0,\alpha-1)$. In this setting, the field is identified almost surely w…

Let $(X_t)_{t\geq 0}$ be a Lévy process in the domain of attraction of an $\alpha$-stable Lévy process with positivity parameter $\varrho$, and let $T_z$ be the first-passage time…

Let $Z$ be an Ornstein–Uhlenbeck process driven by a stable Lévy process $X$ with index $alpha>1$, so that $X$ has unbounded variation. No-creeping conjecture. The process $Z$ neve…

Roy's conjecture. The process $\mathbf{X}$ is ergodic if and only if the non-singular $G$-system underlying its Rosiński minimal spectral representation is null.

Let $L_{T,n}(y)$ denote the Riemann estimator of the local time at $y\in\mathbb{R}$ for a symmetric stable process, and let $\mathbb{E}[L_T(y)\mid\mathcal{G}_n]$ be the correspondi…

Limit-process conjecture. The limit process of these scaled processes should be the same as the limit process obtained in the setting studied by Pilipenko and Prykhodko.

Let $\tau$ be an integer-valued random variable in the domain of attraction of a $\gamma'$-stable distribution with $1<\gamma'<\gamma$, and let the offspring measure be…