10 problems

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 $D\subset\mathbb{R}^2$ 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 $I=(-1,1)$, 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…

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…

Stable-process unimodality conjecture. The convolution $\mu\ast S_t$ is unimodal for sufficiently large $t>0$.

Let $w_{\alpha}$ denote a periodic symmetric-$\alpha$-stable noise on $\mathbb{T}^d$, with $0<p<\alpha<2$. The endpoint regularity conjecture. … The paper obtains regularity result…

Let $Z$ be a strictly $\alpha$-stable Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and define $T_1=\inf\{t\ge0:A_t>1\}$. The stable Lévy persistence conjecture. If … then ther…