Diffusions with sub-Gaussian heat kernel estimates on the Sierpiński carpet for general self-similar cell weights

Every known construction of a diffusion (equivalently, a regular symmetric Dirichlet form) on the Sierpiński carpet with two-sided sub-Gaussian heat kernel estimates rests on symmetry: the classical Barlow–Bass and Kusuoka–Zhou constructions use the canonical (uniform) measure and local symmetry, and Cao–Qiu–Wang (arXiv:2511.22156, 2025) recently treated the first non-canonical case, a self-similar measure whose four corner level-1 cells carry weight 1 and whose four edge cells carry weight ρ > 0, a pattern invariant under the full dihedral symmetry of the square but not under local symmetries. That work explicitly poses, and leaves open, the general question: whether one can assign different weights to the level-1 cells of the Sierpiński carpet and define other diffusion processes on the carpet that also satisfy sub-Gaussian heat kernel estimates with respect to the Euclidean metric and the associated self-similar measure, as is possible on many p.c.f. self-similar sets. The question remains open: no published or preprinted work treats weight patterns without global symmetry, no classification of admissible weights exists, and uniqueness of the scaling limit is open even for the symmetric pattern already constructed.

Background

The standard Sierpiński carpet is the plane fractal $F=\bigcap_{n\ge 0}F_n$ obtained from the unit square by repeatedly removing the open middle ninth of every remaining square; at level 1 the carpet consists of eight closed subsquares $F_1,\dots,F_8$ of side $1/3$, each a scaled copy of $F$. A self-similar measure on $F$ assigns each level-1 cell a positive weight $\rho_i$ and is normalized so that $\mu(A)=\big(\sum_{j=1}^{8}\rho_j\big)^{-1}\sum_{i=1}^{8}\rho_i\,\mu(\Phi_i^{-1}A)$; the canonical (Lebesgue-like) measure is the special case $\rho_1=\cdots=\rho_8$. A Dirichlet form is an analytic encoding of a symmetric Markov process: every regular Dirichlet form on $L^2(F,\mu)$ corresponds to a $\mu$-symmetric diffusion (Hunt process), so constructing a diffusion on $F$ and constructing such a form are equivalent tasks. The heat kernel $p_t(x,y)$ of the diffusion is said to satisfy a two-sided sub-Gaussian estimate (with respect to the Euclidean metric) when $C_1\mu(B(x,t^{1/\beta}))^{-1}\exp\{-C_2(|x-y|^\beta/t)^{1/(\beta-1)}\}\le p_t(x,y)\le C_3\mu(B(x,t^{1/\beta}))^{-1}\exp\{-C_4(|x-y|^\beta/t)^{1/(\beta-1)}\}$ for positive constants $C_1,\dots,C_4$; the exponent $\beta>2$ is the walk dimension, governing how the mean exit time from a ball of radius $r$ grows ($\asymp r^\beta$). The carpet is infinitely ramified — it has no local cut points, so no cell can be detached by removing finitely many points — which makes its analysis far harder than for post-critically finite (p.c.f.) self-similar sets (finitely ramified sets such as the Sierpiński gasket, where Kigami's theory gives self-similar Dirichlet forms for flexible cell weights, Sabot characterized the admissible weights on the gasket, and Hambly–Kumagai derived the corresponding transition density estimates). For carpets, the classical route (Kusuoka–Zhou; Barlow–Bass) builds the form as the scaling limit of weighted energies $\int|\nabla f|^2\,d\mu_0$ on the pre-carpet (the increasing union of the approximating sets), and its key steps — the Knight-move lower bound on hitting distributions (a uniform lower bound for the probability that the approximating walk, started near one side of a block of cells, exits through a prescribed far-away, non-adjacent side, the two sides lying in knight-move position as in chess), the elliptic Harnack inequality (harmonic functions on balls attain comparable maxima and minima), and exponential resistance growth $R_n\asymp\lambda^n$ of the effective resistance between opposite sides of the level-$n$ pre-carpet (the resistance of the domain viewed as an electrical network, the reciprocal of the energy-minimizing conductance between the two sides) — all relied on symmetry assumptions. Here one must distinguish the global symmetry of the square (the dihedral group $D_4$, the eight-element symmetry group of the square, acting on the whole carpet) from local symmetry (invariance under the symmetry group of every sub-piece of the carpet); all pre-2025 carpet constructions required local symmetry, and the uniqueness theorem of Barlow–Bass–Kumagai–Teplyaev — that, up to scalar multiples, only one local regular Dirichlet form on a generalized carpet is invariant under local symmetries — is also confined to that class. A parallel line studies time changes (running the canonical carpet diffusion on a different clock, so that it spends time in each region according to a new measure): Gu–Lau–Qiu–Ruan, building on the carpet-case preprint of Gu–Qiu–Ruan, proved Kigami's conjecture for weight functions generated by symmetric cell weights — time changes of the standard carpet diffusion by self-similar measures satisfy sub-Gaussian estimates — but with respect to geodesic metrics adapted to the weights (distances built from the weight functions, measured along shortest paths) rather than the Euclidean one, and the process there is a time change of the canonical diffusion rather than a new self-similar energy. In this landscape, Cao–Qiu–Wang (arXiv:2511.22156, 2025) constructed the first genuinely non-canonical carpet diffusion: for the weight pattern $\rho_{2i}=\rho$, $\rho_{2i-1}=1$ (corner cells weight 1, edge cells weight $\rho$, for any fixed $\rho>0$), which has full $D_4$ global symmetry but breaks local symmetry, they proved the elliptic Harnack inequality, resistance growth with exponent $\lambda$, mean exit time scaling, and the two-sided sub-Gaussian heat kernel estimate with respect to the Euclidean metric, with walk dimension $\beta=\big(\log(4+4\rho)+\log\lambda\big)/\log 3$. Their limit process is obtained only along a subsequence of graph approximations, and no uniqueness is claimed. The paper frames the general question — assigning different weights to cells and defining diffusion processes on the carpet that also satisfy sub-Gaussian heat kernel estimates — as a natural question and states that it addresses only a simple case of it.

Problem Statement

Let $F$ be the standard Sierpiński carpet with level-1 similarity maps $\Phi_1,\dots,\Phi_8$ (indexed counterclockwise, $\Phi_1$ fixing the origin), and let $(\rho_1,\dots,\rho_8)\in(0,\infty)^8$ be an arbitrary weight vector, inducing the self-similar measure $\mu$ with weights $\rho_i$; in the Kusuoka–Zhou framework the same weights induce the weighted energy $\widetilde{\mathcal{E}}_0(f)=\int_{\widetilde{F}_0}|\nabla f|^2\,d\mu_0$ on the pre-carpet $\widetilde{F}_0$, where $\mu_0$ assigns each level-$n$ cell $F_w$ the weight $\rho_{w_1}\cdots\rho_{w_n}$. The problem, posed as open by Cao–Qiu–Wang (arXiv:2511.22156), is whether one can assign different weights to the cells and define other diffusion processes on the Sierpiński carpet that also satisfy sub-Gaussian heat kernel estimates, as is possible on many p.c.f. self-similar sets: given a weight vector $(\rho_1,\dots,\rho_8)$, determine whether there exists a non-degenerate regular symmetric Dirichlet form on $L^2(F,\mu)$, equivalently a $\mu$-symmetric diffusion on $F$ (and on the unbounded carpet), whose transition density satisfies the two-sided sub-Gaussian heat kernel estimate $C_1\mu(B(x,t^{1/\beta}))^{-1}\exp(-C_2((|x-y|^\beta/t)^{1/(\beta-1)}))\le p_t(x,y)\le C_3\mu(B(x,t^{1/\beta}))^{-1}\exp(-C_4((|x-y|^\beta/t)^{1/(\beta-1)}))$ with respect to the Euclidean metric, for suitable constants and a walk dimension $\beta$ determined by the pattern. The question is to determine, for every weight vector $(\rho_1,\dots,\rho_8)\in(0,\infty)^8$, whether such a form exists — equivalently, to characterize the admissible weights. The case $\rho_{2i}=\rho$, $\rho_{2i-1}=1$ ($1\le i\le 4$; all corner cells weight 1, all edge cells weight $\rho$, for any fixed $\rho>0$), which retains full global $D_4$ symmetry, is settled as regards existence by the source paper and is therefore excluded. Weight patterns without global symmetry, such as patterns with unequal corner weights, are entirely open.

References

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