8 problems

Let $(X_i)$ be centered i.i.d. increments with law $\mu$ and finite variance, let $S_n$ and $A_n$ be the associated random walk and integrated walk, and write…

Weighted-sum persistence conjecture. If $\mu$ has finite variance and $\sigma_n\asymp n^p$ for some $p>0$, then

Khokhlov-Molchan conjecture. One has

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 $B^H$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, let $A_t^H=\int_0^tB_s^H\,\mathrm{d}s$, and define $T_1^H=\inf\{t\ge0:A_t^H>1\}$. The fractional Brownian…

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…