30 problems

Let $\rho_t$ denote a super-Brownian motion and let $L_x$ be its total occupation-time density, defined by … for suitable test functions $\phi$, in dimensions $d\leq 3$. The Hölder…

Let $X$ be $d$-dimensional super-Brownian motion, with total occupation local time $L^x$, range $\mathcal{R}$, and boundary local time measure $\mathcal{L}$ supported on…

Uniform spanning tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{st})}$.

Invasion percolation scaling conjecture. For each $r\geq2$, $\vec t\in\mathbb{R}^{r-1}$ and $\vec k\in\mathbb{R}^{d(r-1)}$, there exist constants $A,V,v$ and $\delta\in(0,1)$ such…

Incipient infinite lattice tree scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that (rhoscal) holds for $\rho^{(\mathrm{lt})}$.

Unoriented percolation scaling conjecture. There exist constants $A,V,v$ and $\delta\in(0,1)$ such that the scaling relation (rhoscal) holds for $\rho^{(\mathrm{pe})}$.

Universal scaling conjecture. ICSBM should arise as the scaling limit of various other critical, infinite, weakly interacting branching-and-motion models above their upper-critical…

Super-Brownian excursion conjecture. High-dimensional critical percolation clusters should have the same scaling limit as critical branching random walk: super-Brownian excursion.

Let $\Sigma\subset \mathbb{R}^d$ be a compact set that is not a removable set for $\Delta$ in $L^2(\mathbb{R}^d)$. A self-adjoint extension of…

Let $\{X_t\}_{t\geq 0}$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $\psi$. Let … and write $\beta:=-\psi'(0)$. For…

Let $d\leq 3$. Let $X_0^N$ converge to an appropriate compactly supported finite measure $X_0$ in $M_F(\mathbb{R}^d)$ as $N\to\infty$, and let $(X_t^N,t\geq 0)$ be the rescaled dis…

Local-time convergence conjecture. As $R\to\infty$, this local-time process will converge to the local time of super-Brownian motion. The source gives no precise mode of convergenc…

Exact Minkowski-content conjecture. Almost everywhere under $\mathbb{N}_0$ or almost surely under $\mathbb{P}_{\delta_0}$,

Boundary-neighborhood approximation conjecture. For some constant $c'>0$,

Let $X$ be $d$-dimensional super-Brownian motion, let $L^x$ be its total occupation local time, and define the random measures $\mathcal{L}^{\lambda}$ on $\mathbb{R}^d$ by … Let…

Let the authors' general conditions be the hypotheses under which rescaled ranges of a single occupancy particle model converge to the range of super-Brownian motion. Range-scaling…

Let $S(X_t)=\overline{\{x:X(t,x)>0\}}$ be the closed support of $X_t$, let $\partial S(X_t)$ denote its boundary, and let $L_t$ be the local-time measure associated with the bounda…

Let $L^x$ denote the local time of super-Brownian motion at $x$, let $\alpha$ be the exponent used in the definition below, and for $\lambda>0$ define a finite measure…

Let ${\cal R}$ be the range of super-Brownian motion, let $F$ be its frontier, and write $\operatorname{dim}$ for Hausdorff dimension. For initial measure $\delta_0$ in spatial dim…

Cox–Klenke conjecture. Under these assumptions, the rescaled measure-valued processes converge to the super-Brownian-motion law $P^{d,\gamma,\sigma^2}$.

Aldous's lattice-tree conjecture. If $d>8$, then $X_N$ should converge to $\mathrm{ISE}$ as $N\to\infty$.

Let $X(t,x)$ be the density of one-dimensional super-Brownian motion, define … and let $BZ_t$ be the boundary of the zero set of $X(t,\cdot)$. Conjecture on the boundary measure. T…

Let $X(t,x)$ be the density of one-dimensional super-Brownian motion, and let $a\downarrow 0$. Conjecture B. … This gives the predicted asymptotic order for the probability that th…

Let $X(t,x)$ be the jointly continuous density of one-dimensional super-Brownian motion, and let … be the boundary of the zero set of its density. Conjecture A. The Hausdorff dimen…

Consider another version of the high-dimensional incipient infinite cluster (IIC), in dimensions above six, and its scaling limit. Hara–Slade conjecture. This scaling limit is inte…