24 problems

Energy measure singularity dichotomy conjecture. The energy measures associated with the Dirichlet form are singular with respect to the reference measure $m$.

The paper considers characterizations of heat kernel estimates and Harnack inequalities for diffusions, pure jump processes, and processes with both diffusions and jumps, where cut…

Grigor’yan–Hu–Lau resistance conjecture. The form $(\mathcal{E},\mathcal{F})$ satisfies $\textup{HKE}(\beta)$ if and only if it satisfies both $\textup{PI}(\beta)$ and…

Let $d$ be a metric, let $\mu$ be a measure, and let $p_t(x,y)$ be the heat kernel associated with a strongly local Dirichlet form on the metric measure space. For some $\beta>1$,…

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space. Suppose it satisfies sub-Gaussian heat kernel estimates with volume growth exponent $\alpha$ and walk dimension $\beta$, and le…

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, meaning a metric measure Dirichlet space equipped with the associated diffusion. Suppose it satisfies sub-Gaussian heat kernel…

Let $b1 b2$ and $d_m$ satisfy … and … Here $b1$ is the volume growth exponent, $b2$ is the escape time exponent, and $d_m$ is the martingale dimension, also called the martingale i…

Resistance conjecture. Nice two-sided heat kernel estimates are equivalent to the conjunction of volume doubling, $\operatorname{PI}_{2}(\Psi)$), and…

BCK2's conjecture. A similar combination of the various exponents should appear in sub-Gaussian annealed heat-kernel bounds for the random walk on the two-dimensional uniform spann…

Log-correction conjecture. Some logarithmic correction is necessary in the upper bound; equivalently, the upper bound $c_4t^{-d/2}$ is not sharp. The conjecture concerns the critic…

Heat-kernel comparability conjecture. The heat kernel of $X$ is comparable to that of $Z$. The conjecture proposes robustness of heat-kernel estimates under comparable perturbation…

Generalized Sierpinski carpet criticality conjecture. For generalized Sierpinski carpets and similar fractals,

Sierpinski carpet weak Bakry–Émery conjecture. The Sierpinski carpet satisfies $wBE(\kappa)$ for some

Weak Bakry–Émery conjecture. On a non-compact metric graph with finitely many edges, the weak Bakry–Émery curvature condition holds for every $t>0$.

Let $(M,d, u)$ be the metric-measure space and let $V$ and $V{epsilon}(t)$ be as defined above. Assume that $textup{Vol}(V; alpha1, alpha2)$ and…

Let $\{\boldsymbol{G}_t\}$ be a family of conductance graphs such that $t\mapsto\pi_t(x)$ is non-decreasing, the graphs are uniformly elliptic, have uniform volume growth $v(r)$ wi…

Let $I$ be the unit interval regarded as a p.c.f. fractal, with metric $d$, measure $\b5$, heat kernel $p(t,x,y)$, spectral dimension $d_S$, and volume function…

Gaussian estimates and Riesz transform conjecture. The heat kernel of the Hodge Laplacian should have Gaussian estimates, and the Riesz transform should be bounded on $L^p$ for eve…

Let $k_n(x,y)$ be the transition kernel of the random walk on $\Gamma$, let $V$ be the volume function, and let $\zeta(n)$ be the scale appearing in the heat-kernel estimates. Writ…

Volume-doubling generalisation conjecture. The condition $(\mathrm{bacra})$ should imply $(DU\!E^v)$ even when $v$ depends on $x$.

Let $N$ be a nilpotent group, and let $P^\sigma(t,0)$ denote the Poisson kernel associated with the setting above. The preceding theorem gives an estimate for $P^\sigma(t,0)(x)$ on…

Let $D$ be a half-space-like $C^{1,1}$ domain, and suppose that the Dirichlet heat kernel estimates described above are considered for rotationally symmetric Lévy processes in…

Let $d=4$, and let $\mathbb P$ be any i.i.d. environment law satisfying $\mathbb P(0<\omega_b\leq 1)=1$. Write $\cmss P_\omega^{2n}(0,0)$ for the quenched return probability at tim…

Endpoint $L^1$ gradient-bound conjecture. The same inequality should hold for $p=1$, namely, there should exist a constant $C_1>1$ such that for every smooth…