Existence of self-similar Dirichlet forms with sub-Gaussian heat kernel estimates on unconstrained Sierpiński carpets in $\mathbb{R}^d$ for $d\geq 4$

Cao and Qiu (arXiv:2305.09292, 2023) constructed a strongly local, regular, self-similar Dirichlet form with a two-sided sub-Gaussian heat kernel estimate on every unconstrained Sierpiński carpet in $\mathbb{R}^3$, and stated that "the situation for higher dimensional (greater than 3) analogs to USC are still not fully understood": their proof runs through Murugan's resolution of the resistance conjecture in the regime $d_H < d_W + 1$, which is automatic in $\mathbb{R}^3$ (where $d_H < 3$ and $d_W \geq 2$) but fails a priori in $\mathbb{R}^d$, $d \geq 4$. This problem asks whether every unconstrained Sierpiński carpet in $\mathbb{R}^d$, $d \geq 4$ (with $d = 4$ the essential case), carries such a Dirichlet form — equivalently, a symmetric diffusion with sub-Gaussian heat kernel estimates — or whether a counterexample exists within the class.

Background

A Sierpiński carpet is built by the following iterated construction: divide the unit cube into $k^d$ congruent closed subcubes, keep $N$ of them with $1 < N < k^d$, and repeat the pattern inside each kept cube. The resulting compact set $K$ is the attractor of the iterated function system (IFS) of the $N$ contraction maps, $K=\bigcup_{i=1}^N \Psi_i(K)$. A central question of analysis on fractals, initiated by the construction of Brownian motion on the Sierpiński gasket and developed by Kigami's theory of Dirichlet forms, is which self-similar sets carry a canonical diffusion. A Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(K,\mu)$ is the abstract analogue of the energy $\int|\nabla f|^2$ that generates a symmetric Markov process (the heat kernel $p_t(x,y)$ is its transition density); it is called strongly local when the energy of a function supported where another is constant vanishes — the analogue of a jump-free diffusion — and regular when its domain contains enough continuous functions. For the classical constrained ("generalized") Sierpiński carpets of Barlow and Bass, where the kept subcubes sit on the $1/k$-grid of the unit cube and the pattern is invariant under the cube's isometry group, Brownian motion was constructed in every dimension $d\geq 2$, the construction is unique, and the heat kernel satisfies the two-sided sub-Gaussian estimate

$c_1\,t^{-d_H/d_W}\exp\Big(-c_2\Big(\frac{d(x,y)^{d_W}}{t}\Big)^{1/(d_W-1)}\Big)\le p_t(x,y)\le c_3\,t^{-d_H/d_W}\exp\Big(-c_4\Big(\frac{d(x,y)^{d_W}}{t}\Big)^{1/(d_W-1)}\Big),\quad 0<t\le 1,$

where $d_H$ is the Hausdorff dimension of $K$ (with respect to the Euclidean metric), $d_W\geq 2$ is the walk dimension (the exponent in the $r^{d_W}$ growth of the mean escape time from a ball of radius $r$; $d_W=2$ is the Gaussian, manifold-like value), and $\mu$ is the normalized Hausdorff measure on $K$.

Kusuoka and Zhou introduced a framework allowing "unconstrained" carpets: the kept subcubes may be translated off the $1/k$-grid, so two neighboring cells can meet in a line segment of irrational length or a rectangle of irrational area. This destroys the exact local reflection symmetry between neighboring cells on which the Barlow–Bass corner-move and Knight-move arguments rest, and makes the existence problem substantially harder. Cao and Qiu settled the planar case (in the recurrent regime, published in Probability Theory and Related Fields, 2024) and then the three-dimensional case: defining an unconstrained Sierpiński carpet in $\mathbb{R}^3$ ($\mathrm{USC}^{(3)}$) as the attractor of similarities $\Psi_i(x)=x/k+c_i$ of the unit cube satisfying non-overlapping (distinct first-level cells intersect in Lebesgue measure zero), face included (the whole boundary $\partial[0,1]^3$ is covered by the union of the first-level cells), strong connectivity (the union of first-level cells is connected with no local cut point, i.e. no point whose removal disconnects a small neighborhood), and symmetry (invariance under every self-isometry of the cube), they proved that every $\mathrm{USC}^{(3)}$ carries a strongly local, regular, self-similar Dirichlet form — self-similar meaning $\mathcal{E}(f,f)=\rho^{-1}\sum_{i=1}^N \mathcal{E}(f\circ\Psi_i,f\circ\Psi_i)$ for some $\rho>0$ — with a sub-Gaussian heat kernel estimate as above, and that in this class one always has $d_H>d_W$ (transience on infinite blow-ups of the carpet, i.e. the diffusion on the increasing union of ever-larger rescaled copies of $K$ escapes to infinity).

The engine of the three-dimensional proof is a theorem of Murugan (arXiv:1809.00767) resolving, in the regime $d_H<d_W+1$, the resistance conjecture of heat-kernel theory — the conjecture that a Poincaré inequality together with a resistance estimate (an upper bound on the effective electrical resistance between a set and the complement of its neighborhood) suffices for two-sided sub-Gaussian heat kernel bounds. Its key observation is that when $d_H<d_W+1$ (with a matching Poincaré inequality) one-dimensional sets are hittable by the diffusion, i.e. reached by it with positive probability. In $\mathbb{R}^3$ this regime is automatic: $d_H<3$ while any strongly local form has $d_W\geq 2$, hence $d_H<3\le d_W+1$. In $\mathbb{R}^d$ with $d\geq 4$ this argument collapses: $d_H$ can be arbitrarily close to $d$, so $d_H-1$ can exceed $2$, and no a priori bound forces $d_W>d_H-1$. Cao and Qiu therefore state that the situation for higher-dimensional (greater than 3) analogues of unconstrained carpets "is still not fully understood." Existence in higher dimensions is also not guaranteed a priori: Cao and Qiu separately constructed (arXiv:2109.12760) a Sierpiński-carpet-like fractal on which no standard self-similar diffusion with sub-Gaussian heat kernel estimate exists.

Problem Statement

Let $d\geq 4$ (the case $d=4$ already contains the essential difficulty: the automatic inequality $d_H<d_W+1$ of $\mathbb{R}^3$ first fails there). Let $K\subset[0,1]^d$ be an unconstrained Sierpiński carpet $\mathrm{USC}^{(d)}$: the attractor $K=\bigcup_{i=1}^{N}\Psi_i(K)$ of similarities $\Psi_i(x)=x/k+c_i$ of the unit cube with $k\geq 3$, an integer $N$ with $N\le k^d-1$ large enough that the face-included condition below is satisfiable, and $\Psi_i([0,1]^d)\subset[0,1]^d$ for each $i$ (the cells are allowed to sit off the $1/k$-grid), where the following axioms hold: (i) non-overlapping — $\Psi_i([0,1]^d)\cap\Psi_j([0,1]^d)$ has Lebesgue measure zero for $i\neq j$; (ii) face included — $\partial[0,1]^d\subset\bigcup_{i=1}^{N}\Psi_i([0,1]^d)$; (iii) strong connectivity — $\bigcup_{i=1}^{N}\Psi_i([0,1]^d)$ is connected and has no local cut point, i.e. no point whose removal disconnects one of its neighborhoods; (iv) symmetry — $\Gamma\big(\bigcup_{i=1}^{N}\Psi_i([0,1]^d)\big)=\bigcup_{i=1}^{N}\Psi_i([0,1]^d)$ for every self-isometry $\Gamma$ of the cube. Let $\mu$ be the normalized Hausdorff measure on $K$. Does every such $K$ admit a strongly local, regular, symmetric, conservative, irreducible Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(K,\mu)$ that is self-similar in the standard sense — $\mathcal{F}\cap C(K)=\{f\in C(K):f\circ\Psi_i\in\mathcal{F}\ \forall\,i\in\{1,\dots,N\}\}$ and $\mathcal{E}(f,f)=\rho^{-1}\sum_{i=1}^{N}\mathcal{E}(f\circ\Psi_i,f\circ\Psi_i)$ for some $\rho>0$ and all $f\in\mathcal{F}\cap C(K)$ — and whose heat kernel $p_t(x,y)$ satisfies, for some constants $c_1,c_2,c_3,c_4>0$ and some exponent $d_W\geq 2$, the two-sided sub-Gaussian estimate

$c_1\,t^{-d_H/d_W}\exp\Big(-c_2\Big(\frac{d(x,y)^{d_W}}{t}\Big)^{1/(d_W-1)}\Big)\le p_t(x,y)\le c_3\,t^{-d_H/d_W}\exp\Big(-c_4\Big(\frac{d(x,y)^{d_W}}{t}\Big)^{1/(d_W-1)}\Big)$

for all $x,y\in K$ and $0<t\le 1$, with $d_H$ the Hausdorff dimension of $K$ and $d(x,y)$ the Euclidean distance? Equivalently: does every $\mathrm{USC}^{(d)}$, $d\geq 4$, carry a symmetric diffusion with sub-Gaussian heat kernel estimates? A construction for the whole class in every dimension $d\geq 4$ — or an explicit $\mathrm{USC}^{(d)}$ in some dimension $d\geq 4$ admitting no such form — would settle the question left open by Cao and Qiu's $\mathbb{R}^3$ paper; dimension $4$ is the critical case in which the mechanism missing beyond $\mathbb{R}^3$ must first be supplied, but a construction in dimension $4$ alone would settle only that dimension.

References

1. Shiping Cao, Hua Qiu, Dirichlet forms on unconstrained Sierpinski carpets in $\mathbb{R}^3$, arXiv:2305.09292 (2023). https://arxiv.org/abs/2305.09292
2. Shiping Cao, Hua Qiu, Dirichlet forms on unconstrained Sierpinski carpets, Probability Theory and Related Fields 189 (2024), 613–657, DOI: 10.1007/s00440-024-01280-6 (preprint arXiv:2104.01529). https://doi.org/10.1007/s00440-024-01280-6
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