15 problems

Super-Brownian scaling conjecture. The critical-dimensional model should have the same super-Brownian scaling limit as the high-dimensional model once appropriate logarithmic corre…

Nonexistence conjecture. For $\beta<1$, the renormalized SILT $\tilde{\gamma}_X$ does not exist when

Skoulakis–Adler conjecture. The conditional log-Laplace transform of $X_t$ should be the unique solution to a nonlinear stochastic partial differential equation.

Let $X_t$ be the limiting measure-valued process characterized above, and let $d$ denote the spatial dimension. A Girsanov theorem relates the law of $X_t$ to that of a standard su…

Let the incipient infinite cluster be obtained as the scaling limit of increasingly large finite critical percolation clusters, with lattice spacing scaled by $n^{-1/4}$ in dimensi…

Mytnik–Xiong's persistence conjecture. The process $X$ will converge weakly, as $t\to\infty$, to some non-trivial random measure $X_\infty$ on ${\mathbb R}^d$.

Central limit conjecture. Theorem 2 and Corollary 2.2 should both hold in dimensions $d=3,4$ after replacing $T^{-1/2}$ by $T^{-3/4}$ when $d=3$ and by $T^{-1/2}(log T)^{-1}$ when…

A critical interacting particle system in a random environment is a particle system at criticality whose large-scale behavior is affected by environmental randomness. Universality…

Density conjecture. A density for $X_t$ exists only when $d=1$.

Let $X'_t$ be the super Ornstein–Uhlenbeck process whose initial measure $X'_0$ is fixed and compactly supported, and consider the survival set. Let $\beta$ and $\gamma$ be the mod…

Let $D_U(\eta)$ denote the Hausdorff dimension of the set of points in an open set $U\subset\mathsf{R}$ where the optimal Hölder index of $X_t$ equals $\eta$, and let…

Superprocess extension conjecture. To extend the application of superprocesses to more general nonlinear equations, one should move from processes on measures to processes on gener…

Let $X_t(x)$ denote the density solving the stochastic partial differential equation associated with the one-dimensional superprocess, with spatially correlated noise determined by…

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…

Consider the $(d,\alpha,\beta,\gamma)$-branching particle system, where $d$ is the spatial dimension, $\alpha$ is the stability parameter, $\beta$ is the branching parameter, and…