8 problems

Let $(X,d,m,\mathcal E,\mathcal F)$ be an MMD space, and let $d_m$ denote the martingale dimension of its associated diffusion. Suppose that $(X,d,m,\mathcal E,\mathcal F)$ satisfi…

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…

Energy measure singularity dichotomy. Then $\Gamma(f,f) \perp m$ for all $f \in \mathcal{F}$. The conjecture predicts singularity of every energy measure under the stated structura…

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

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 $d=4$, 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…