8 problems

Let $\mathcal{P}$ be the operator associated with the one-parameter family of analytic Markov maps, acting on the Hilbert space $\mathcal{H}$, and let $r\in[0,1]$ denote the parame…

Let $U\subset\mathbb{R}^d$ be a bounded domain satisfying the hypotheses of either Theorem or Theorem, and let $u(t,x)$ be the corresponding solution to the stochastic heat equatio…

Endpoint fourth-moment conjecture. The conjecture is

Fix $n\geq 1$. For each $p$, let $\kappa_p^{(n)}$ denote the critical diffusion constant associated with $p$-intermittency. Distinct critical values conjecture. For all fixed…

Fix $n\in\mathbb{N}$. Let $d$ be the spatial dimension, let $\kappa\geq 0$ be the diffusion constant of the catalyst, and let $\kappa_p^{(n)}$ denote the critical values governing…

Let $p_t(x,y)$ be the transition probability at time $t$ for the rate-one random walk with kernel $p(,)$, and let $\blambda_p(\kappa)$ denote the associated Lyapunov exponent. Suppo…

Voter-model Lyapunov-exponent conjecture. On $[0,\infty)$, $\kappa\mapsto\lambda_p(\kappa)$ is strictly decreasing and convex, with

Threshold conjecture. In $d\geq4$, there exists a strictly increasing sequence $0<\kappa_2<\kappa_3<\ldots$ such that, for $p=2,3,\ldots$, the system is $p$-intermittent if and onl…