8 problems

Let $f$ be a Gaussian stationary process, and consider its persistence probability over an interval $[0,T]$. Slepian's conjecture. Under mild decay conditions, the persistence prob…

Let $f_\rho$ be a Gaussian stationary process with spectral measure $rho\in\mathcal{L}\cap\mathcal{M}$, and let $a<b$ with $a\ne-b$. A two-sided barrier event is the event that…

Let $ell\in\mathbb{R}$ and let $\rho,nu\in\mathcal{L}$ be spectral measures. Write $psi_\rho^\ell$ for the ball exponent and $theta_\rho^\ell$ for the persistence exponent. Theorem…

Let $\rho$ be a spectral measure in $\rho\in\mathcal{M}$, and let $\theta_\rho^\ell$ denote its persistence exponent at level $ell$. Positivity conjecture. For every $ell>0$, the p…

Let $B^H$ be a standard fractional Brownian motion with Hurst parameter $H\in(0,1)$, and define its one-sided integrated version by … Write $\theta_I(H):=\theta(I^H)$ for the persi…

The persistence exponent is related to the largest eigenvalue of an integral operator, and the paper establishes Theorem, Theorem, and equations of type for the setting under consi…

Strict-convexity conjecture. The function

Let $(X_i)_{i\geq 0}$ be a stationary Gaussian sequence with mean $0$, variance $1$, and nonnegative correlations satisfying, for some $H\in[0,1)$, $K>0$, and slowly varying functi…