14 problems

Conjectured supremum-distribution formula. For $t>t_0$ and $x>0$,

Fourier–Hahn–Lévy conjecture. If $f:\mathbb{Z}\to\mathbb{Z}$ is tractable, then the $f$-moment can be estimated to within a constant factor with a Lévy-Tower, either directly or af…

Dichotomy conjecture. Every infinite variation Lévy process is either abrupt or strongly eroded.

Assume that the Lévy process $X$ has finite lifetime $\zeta\in(0,\infty)$. Let … be the last time at which the Euclidean norm is maximal. Set $M:=X_\tau$ if…

Continuity and propagation conjecture. The process $X$ has a continuous transition density function $p(t,x,y)$. Moreover, if

Let $\mu$ be a measure on $(0,1]$ such that … Allow $\mu$ to have atoms and a continuous singular part, and suppose its absolutely continuous part is $g(x)\,dx$, with some…

Bañuelos's conjecture. For every $1<p<\infty$, $T_{m_r}$ is bounded on $L^p(\mathbb{R}^n)$ and

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 $w$ be a periodic noise on $\mathbb{T}^d$, let $\tau\bin\tyu0005\mathbb{R}$, and let $0<p,q\textless\textless=\infty$. The zero–one law. For every such noise and every such Bes…

Let ${\sf HF}$ be a regular Markovian holonomy field, and let an admissible Lévy process determine a Yang–Mills field through the Yang–Mills-fields construction. Yang–Mills charact…

For each $\beta\ge0$, consider a $\beta$-fractionally integrated Lévy process and let $\theta(\beta)$ be its persistence exponent. The Lévy universality conjecture. For every…

For each $\beta\ge0$, let $\theta(\beta)$ denote the persistence exponent of a $\beta$-fractionally integrated Lévy process, when defined. The convexity conjecture. The function ……

Let $Z$ be a Lévy process, let $A_t=\int_0^tZ_s\,\mathrm{d}s$, and for $x>0$ define $T_x=\inf\{t\ge0:A_t>x\}$. The general Lévy persistence conjecture. If … then for every $x>0$, ……

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…