13 problems

Let $d\geq 3$, let $\mathcal{S}(1)=\{\mathbf{z}\in\mathcal{Z}_d:\|\mathbf{z}\|=1\}$, and define the occupation time of the unit-ball surface centered at $\mathbf{z}$ by … Writing…

Let $R(n)=\{S(0),S(1),\ldots,S(n)\}$ be the range of simple random walk up to time $n$. Erdős–Révész conjecture on repeatedly r-visited random-walk ranges. For every integer…

Erdős–Révész conjecture on favourite-site location after positive occupation. Then

Two-dimensional occupation-time central limit conjecture. If $d=2$ and $\eta_0$ is distributed according to $\nu_p$, then

Riemann-sum error conjecture. The $L_p$-error of $\widehat I_{\mu,n}$ as an estimator of $I_\mu(W)$ is bounded by $c/n^{3/4}$ for some constant $c>0$. This conjecture extends the k…

Let $p$ be the Laplace variable coupled to the occupation time, and let $s$ be the Laplace variable for time. Denote by $\tilde{Q}_p^{(1)}(0,0,s)$ the first perturbative contributi…

Let $X$ be the Lévy process underlying the refracted process $U$, and let $e(q)$ be an independent exponential time with parameter $q>0$. For $b\in\mathbb{R}$ and a complex number…

Occupation-time conjecture. Theorem 4 should remain valid under the weaker assumption that the probing function $f$ is Cesàro $(C,1)$-summable, rather than requiring the existence…

Consider the wider class of critical branching laws with finite variance covered by the cited work of Milos. Let the theorem under discussion denote the paper's occupation-time flu…

Let the limit in assumption (A3) be infinite, so that the clans are infinite and denser than the condition in (A3). Let $F_T=T^{(3-d/\alpha)/2}$ be the norming used in the finite-c…

Let $\varphi\in\mathcal{S}(\mathbb{R}^d)_+$, let $Y_T$ be the rescaled occupation time process, and let $K$ be a compact set in the path space under consideration. Let…

Let a superprocess start from Lebesgue measure, and let $\alpha$ be the stability parameter, $\beta$ the branching parameter, and $d$ the spatial dimension. For a bounded set, cons…

Occupation-time fluctuation conjecture. The conclusion of Theorem 1 should continue to hold under this weaker condition.