27 problems

Bouleau–Hirsch energy image density conjecture. Every Dirichlet structure satisfies the energy image density property.

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

A simple nested fractal is a fractal in the class described in the paper, equipped with its Brownian motion and associated Dirichlet form. For a fractal $F$ with Hausdorff measure…

Equivalence conjecture. For all finitely ramified fractals, the condition

Let $F$ be a finitely ramified fractal considered in the paper, let $\nu$ be its energy measure, and let $Z(x)$ be the matrix defining the weak gradient metric. The generalized Rie…

Bouleau–Hirsch energy image density conjecture. The invertibility of the generalized Malliavin matrix of $f$ should imply that the law of $f$ is absolutely continuous with respect…

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 $\beta>1$,…

$L^p$-boundedness conjecture. The function $\widetilde{H}$ should resolve the issue of $L^p$-unboundedness: unlike $\widetilde{G}$ for $p>2$ and $H$ for $1<p<2$, it is expected to…

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 $mathcal M_{z,z';\theta}$ be the boundary z-measure on the Thoma simplex $Omega$, whose points are written $(\alpha;\beta)$ with nonincreasing nonnegative coordinate sequences…

Boundary-support null-set conjecture. The set

Let the Sierpiński gasket carry Brownian motion, and let the domain of its Dirichlet form consist of the functions associated with that diffusion. A function is Lipschitz when it i…

A regular Dirichlet form decomposes into a strongly local part, a jumping part, and a killing part; its associated Hunt process correspondingly has diffusion, jump, and killing com…

In the setting of Example 3, let $(\mcE,\mcF)$ be the Dirichlet form on the configuration space, and let $\pi_d$ and $W_2$ denote the projection measure and Wasserstein distance in…

Let $T_t^{[m]}$ be the semigroup under consideration, and let its associated Dirichlet form be a lower Dirichlet form in the sense of the cited reference. Assume that tagged partic…

Let $m_i$ be positive continuous functions on $\mathbb{R}$ such that $\lim_{|x|\to\infty}m_i(x)=\infty$ for $i=1,2$. Let $(\underline{\mathcal{E}},\underline{\operatorname{dom}})$…

Morrey embedding conjecture. For the Sierpinski carpet,

Morrey embedding conjecture. On nested fractals, $\delta_{\mathcal{E}}=1$ and, for every $p>1$, there exists $C>0$ such that

Let $\delta_\mathcal{E}$ denote the infimum of the exponents $p$ for which the Sobolev embedding associated with the Dirichlet form $\mathcal{E}$ holds. Let $d_W\approx 2.097$ be t…

Let $M$ be a compact Riemannian manifold, and let $u:S^1\rightarrow M$ solve the stochastic heat equation … Suppose that the Markov processes constructed by Dirichlet form in the c…

The paper considers the Markov process constructed in Section 2 from the Dirichlet form $\mathscr E$, and the solution to equation constructed by Hairer. Equality-in-law conjecture…

Let $(\mathcal{E}^{\mathrm{upper}},\mathcal{D}^{\mathrm{upper}})$ and $(\mathcal{E}^{\mathrm{lower}},\mathcal{D}^{\mathrm{lower}})$ be the upper and lower Dirichlet forms associate…

Let $\mu$ be a determinantal random point field with a continuous kernel, and let $(\mathcal{E},\di)$ be the associated pre-Dirichlet form on $\Lm$. The notation $\Lm$ denotes the…