24 problems

Erdős–Taylor conjecture. For $d=2$, the maximal local time satisfies

Exact Hausdorff-measure conjecture. There exist positive finite constants $C_1$ and $C_2$ such that

Let a simple random walk on $\mathbb{Z}$ have local times, and call the sites attaining the maximal local time its favorite sites. The set of favorite sites is tight if it remains…

Let $(X_i)$ be a sequence of independent identically distributed integer-valued random variables with … where $C$ is a normalizing constant. Define the random walk…

Let $V(n)$ denote a favourite site of simple random walk at time $n$, and let $\nu>1$ be the constant from the Brownian favourite-site upper-tail conjecture. For any non-decreasing…

Brownian favourite-site upper-tail conjecture. There exists a constant $\nu>1$ such that

Let $D$ be an admissible domain containing the origin, let $D_N$ be its scaled-up lattice approximation, let $\tau_\varrho$ be the hitting time of the boundary vertex $\varrho$, an…

Let $\check{\mathbb{F}}_c$ be the class of spaces under consideration and let the stochastic processes associated with its elements be equipped with the identity referred to as the…

Jego's conjecture. Under the setup of the source,

A Gaussian stochastic field is a stochastic field whose finite-dimensional distributions are Gaussian. Its boundary behaviour means the behaviour of the field when restricted to th…

Let $L_{T,n}(y)$ denote the Riemann estimator of the local time at $y\in\mathbb{R}$ for a symmetric stable process, and let $\mathbb{E}[L_T(y)\mid\mathcal{G}_n]$ be the correspondi…

Let $\xi$, $T_{\xi}$, and $|T_{\xi}|$ be as in the conditioned Galton–Watson tree theorem, and write $\sigma^2=\operatorname{Var}\xi$. Let $H_n^{\xi}$ and $C_n^{\xi}$ denote respec…

Uniform convergence conjecture. Uniform convergence over compact subsets of $[0,\infty)$ in the general case $H<\frac{1}{2\rho+1}$ should eventually be proved by obtaining suitable…

Heuristic convergence conjecture. Under suitable hypotheses, the sequence $n^{a(\rho+1)-1}G_{t,\lambda,a}^{(n,\rho)}[g]$ converges to a scalar multiple of $L_t^{(\rho)}(\lambda)$.

Tóth's conjecture. The number of times at which the walk is at one of exactly three favorite sites is finite almost surely:

Let $B=\{B_t\}_{t\geq 0}$ be a one-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let … where…

Let $B=(B_t)_{t\geq 0}$ be Brownian motion, let $L_t^x$ denote its local time at $x\in\mathbb{R}$, and define the increment … Let $Z\sim\mathcal{N}(0,1)$ be independent of…

Let $X$ be Brownian motion, let $L_t^x$ denote its local time, and let … be the set of favorite points at time $t$. Define … The preceding result establishes … for every $\gamma>3$…

Hu–Nualart–Xu's extension conjecture. The central limit theorem

Let $\{L^z_t\}$ be the jointly continuous local times of a one-dimensional Brownian motion, let $L^*_t=\sup_{z\in\mathbb R}L^z_t$, and let $V_t$ be any point $z$ such that…

Let $\hat\alpha_t'(y)$ denote the derivative with respect to the spatial variable of the derivative self-intersection local time of fractional Brownian motion at time $t$, and let…

A branching random walk is a particle system in which particles move according to a random walk and reproduce according to a critical branching mechanism; its local time process re…

Critical roughness conjecture. The local time may be a rough path of roughness $p>3$.

Let $(X_n)_{n\geq 0}$ be the random walk on the line considered in the paper, with transition parameters $p_i$, and let $\xi(i,\infty)$ denote the local time at site $i$. Assume th…