Osada's jump conjecture: the continuity threshold for Hausdorff-measure time changes of exotic diffusions on Sierpinski carpets

For every exponent α above the Hausdorff dimension d_H of the Sierpinski carpet, Osada constructed a self-similar diffusion X^α on the carpet that is reversible with respect to a singular self-similar measure μ^α of scaling exponent α — an "exotic" Brownian motion, distinct from the Barlow–Bass Brownian motion — and studied the time change Y^α of X^α by the carpet's Hausdorff measure v. He proved that Y^α is a v-reversible diffusion whenever d_H < α < β + d_w, where β is the minimal Hausdorff dimension of the carpet's boundary faces and d_w is the walk dimension of X^α (its space–time scaling exponent $t\sim r^{d_w}$), and conjectured an abrupt change at the threshold: for β + d_w ≤ α < d_H + d_w the process Y^α should fail to have continuous sample paths and should exhibit jumps. No later work has established or refuted this. The problem is to determine the sharp continuity threshold of this singular time change and to describe the jump regime.

Background

The Sierpinski carpet. Starting from the unit cube [0,1]^d, delete the open middle cube of side 1/3 and iterate on the remaining N = 3^d − 1 subcubes. The limit set F is the standard Sierpinski carpet; it is the basic example of an infinitely ramified fractal (it cannot be disconnected by removing finitely many small pieces) and has Hausdorff dimension $d_H=\log(3^d-1)/\log 3$ (≈ 1.893 for d = 2). Let $\hat F$ be the unbounded carpet, the union of the integer translates of F, and let $v$ denote (a constant multiple of) the $d_H$-dimensional Hausdorff measure of $\hat F$.

Brownian motion on the carpet. Barlow and Bass constructed a Brownian motion on $\hat F$ as the scaling limit of reflected Brownian motions on the pre-fractal domains and proved sub-Gaussian heat kernel bounds with walk dimension $d_w^{BB}>2$ — the walk dimension is the exponent $d_w$ in the space–time scaling $t\sim r^{d_w}$ of the diffusion, equivalently in the on-diagonal heat-kernel decay $p_t(x,x)\asymp t^{-d_H/d_w}$ ($d_w=2$ for ordinary Brownian motion, and $d_w>2$ reflects the sub-diffusive slowing caused by the fractal's bottlenecks); the process is v-reversible. Barlow, Bass, Kumagai and Teplyaev later proved that, up to scalar multiples, this is the only local regular Dirichlet form on a generalized Sierpinski carpet that is invariant under the full group of local isometries of the carpet, so the law of the symmetric "standard" Brownian motion on the carpet is unique.

Dirichlet forms and time changes. A symmetric Hunt process — a strong Markov process with right-continuous sample paths that is well behaved under taking limits of stopping times — on a locally compact state space corresponds to a regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(M,\mu)$; the form is local precisely when X is a diffusion (has continuous paths). Under the Revuz correspondence (the standard dictionary between positive measures and additive functionals of X), a smooth measure $\nu$ — in Dirichlet-form theory, a measure charging no set of zero capacity and having finite energy integrals — determines a positive continuous additive functional $A_t$ of X, and the time change $Y_t=X_{\tau_t}$, $\tau_t=\inf\{s:A_s>t\}$, is a $\nu$-symmetric Hunt process living on the quasi-support of $\nu$ — the smallest quasi-closed set carrying $\nu$ up to sets of zero capacity, where capacity is the capacity associated with $(\mathcal{E},\mathcal{F})$. Whether Y is itself a diffusion is a delicate question: it holds exactly when the quasi-support of $\nu$ coincides with its ordinary support, i.e. when X hits every piece of the support of $\nu$ immediately in the right sense; when it fails, Y jumps across the complementary gaps, which is one of the standard mechanisms producing jump processes from diffusions. In classical potential theory on $\mathbb{R}^d$, two-sided Green function bounds feed a Wiener-type criterion that decides this question; such machinery is exactly what is missing for the singular processes below.

Osada's exotic diffusions. Osada constructed a different family of diffusions on the carpet. For each $\alpha>d_H$ and each admissible weight m he builds a self-similar Radon measure $\mu^{[\alpha,m]}$ on $\hat F$ with scaling exponent α, that is $\mu(3U)=3^{\alpha}\mu(U)$, which is singular with respect to v: the measure spreads mass over the carpet together with its dense cell-boundary skeleton (the union, over all scales $3^{-n}$, of the $(d-1)$-dimensional grid pieces of the carpet), with density proportional to $m\,d\lambda$ (Lebesgue measure) on the level-0 skeleton pieces. The canonical Euclidean square field $\mathbb{D}[f,g]=\sum_i \partial_i f\,\partial_i g$, integrated against $\mu^{\alpha}$ with a Neumann-type boundary condition along the skeleton, defines a local regular Dirichlet form on $L^2(\hat F,\mu^{\alpha})$. The associated diffusion $X^{\alpha}$ is self-similar, invariant under cell translations, reversible with respect to $\mu^{\alpha}$, satisfies Gaussian-type heat kernel upper bounds with scaling $t^{-\alpha/2}$, and is different from the Barlow–Bass Brownian motion. Its walk dimension is $d_w=2$ for the concrete constructions on the carpet and on $\mathbb{R}^d$; in the axiomatic framework of the 2006 source paper $d_w$ denotes the walk dimension entering the Green-function bound $G_r(x,y)\le c\,\max\{|x-y|^{-\alpha+d_w},1\}$, which is hypothesis (A.2) there (together with (A.1), a lower bound on transition densities). These exotic Brownian motions are invariant only under part of the local isometries, so they lie outside the uniqueness theorem quoted above; their existence shows that uniqueness fails in the larger class of merely cell-translation-invariant diffusions.

The singular time change and the conjecture. For the form of $X^{\alpha}$, the Hausdorff measure v is a smooth (finite-energy) Revuz measure exactly when $\alpha<d_H+d_w$, and in that window the time change $Y^{\alpha}$ of $X^{\alpha}$ by v is a well-defined v-reversible Hunt process. Whether $Y^{\alpha}$ is a diffusion was already an open question in Osada's 2001 paper. The 2006 source paper settles it below a threshold: if $d_H<\alpha<\beta+d_w$, where $\beta=\min_i \beta_i$ and the $\beta_i$ are the Hausdorff dimensions of the boundary faces of the fractal ($\beta=d-1$ for the standard carpet, whose faces are flat), then $Y^{\alpha}$ is a diffusion. The proof replaces the unavailable Wiener test with a quasi-support criterion derived from the one-sided bounds (A.1)–(A.2) and the self-similar structure. For the standard planar carpet this reads: continuity of $Y^{\alpha}$ for $d_H<\alpha<3$, while for $3\le\alpha<d_H+2\approx 3.89$ the conjecture predicts jumps. The upper endpoint is forced: for $\alpha\ge d_H+d_w$ the time change degenerates because v is no longer smooth for the form of $X^{\alpha}$. What is missing is any argument, in either direction, for the window $\beta+d_w\le\alpha<d_H+d_w$: the Green bound (A.2) alone gives no lower control of the hitting behavior needed to force continuity, and no construction of jumps is known either.

Problem Statement

Let $\hat F\subset\mathbb{R}^d$ be the unbounded standard Sierpinski carpet with Hausdorff dimension $d_H=\log(3^d-1)/\log 3$, let $v$ be its normalized $d_H$-dimensional Hausdorff measure, and let $X^{\alpha}$ be Osada's self-similar diffusion on $\hat F$ reversible with respect to the singular self-similar measure $\mu^{[\alpha,m]}$ of scaling exponent $\alpha>d_H$ (Dirichlet form $\mathcal{E}^{\alpha}(f,g)=\int_{\hat F}\mathbb{D}[f,g]\,d\mu^{[\alpha,m]}$ with the Neumann-type condition along the cell-boundary skeleton; definitions as in Osada 2001, 2006). Let $d_w$ be the walk dimension of $X^{\alpha}$, i.e. the exponent in the Green-function bound $G_r(x,y)\le c\max\{|x-y|^{-\alpha+d_w},1\}$ ($d_w=2$ for the concrete constructions on the carpet and on $\mathbb{R}^d$), and let $\beta=\min_i\beta_i$ where the $\beta_i$ are the Hausdorff dimensions of the boundary faces of the fractal ($\beta=d-1$ for the standard carpet). It is known that the time change $Y^{\alpha}$ of $X^{\alpha}$ by $v$ exists as a $v$-reversible Hunt process for $d_H<\alpha<d_H+d_w$, and that $Y^{\alpha}$ is a diffusion whenever $d_H<\alpha<\beta+d_w$.

Prove or refute Osada's conjecture: for every admissible $\alpha$ with $\beta+d_w\le\alpha<d_H+d_w$, the process $Y^{\alpha}$ fails to have continuous sample paths and is a jump process.

In full, the scientific objective is:

(i) Determine the sharp continuity threshold $\alpha_c$ of the $v$-time change within the admissible window $(d_H,\,d_H+d_w)$; the conjecture is that $\alpha_c=\beta+d_w$, so that continuity holds exactly for $\alpha<\beta+d_w$ and fails for $\alpha\ge\beta+d_w$.

(ii) If the conjecture holds, characterize the jump regime: identify the quasi-support of $v$ with respect to $X^{\alpha}$, describe the Dirichlet space of the (then non-local) form of $Y^{\alpha}$, and, if possible, the structure of its jump measure (e.g. whether the jumps are comparable to a Lévy-type kernel and how the face dimensions $\beta_i$ enter).

The content of part (i) is the continuity statement of the conjecture above; the characterisation of the jump regime in part (ii) is a further objective.

The conjecture is stated both for the standard carpet with $d_w=2$ — where the conjectured jump window is $d+1\le\alpha<d_H+2$, that is $3\le\alpha<3.89$ for the planar carpet — and for the general self-similar framework of Osada (2006); in the latter setting the resolution is under hypotheses (A.1)–(A.2) of that paper and must not rely on unproven inputs. A refutation consists of an admissible $\alpha$ in the window $[\beta+d_w,\,d_H+d_w)$, with all standing hypotheses verified, for which $Y^{\alpha}$ is nonetheless continuous, or a proof that no jumps occur anywhere in the window.

References

1. Hirofumi Osada, "Singular time changes of diffusions on Sierpinski carpets", Stochastic Processes and their Applications 116(4):675–689, 2006. DOI: 10.1016/j.spa.2005.11.004, https://doi.org/10.1016/j.spa.2005.11.004 (source paper)
2. Hirofumi Osada, "A family of diffusion processes on Sierpinski carpets", Probability Theory and Related Fields 119(2):275–310, 2001. DOI: 10.1007/PL00008761, https://doi.org/10.1007/PL00008761 (construction of the μ^[α,m] measures and the X^α diffusions)
3. Martin T. Barlow, Richard F. Bass, Takashi Kumagai, Alexander Teplyaev, "Uniqueness of Brownian motion on Sierpiński carpets", Journal of the European Mathematical Society 12(3):655–701, 2010. DOI: 10.4171/JEMS/211, https://doi.org/10.4171/JEMS/211 (preprint: arXiv:0812.1802, https://arxiv.org/abs/0812.1802, where the paper appears under the title "Brownian motion on the Sierpinski carpet")
4. Martin T. Barlow, Richard F. Bass, "Brownian Motion and Harmonic Analysis on Sierpinski Carpets", Canadian Journal of Mathematics 51(4):673–744, 1999. DOI: 10.4153/CJM-1999-031-4, https://doi.org/10.4153/CJM-1999-031-4
5. Hirofumi Osada, "Exotic Brownian motions", Kyushu Journal of Mathematics 61(1):233–257, 2007. DOI: 10.2206/KYUSHUJM.61.233, https://doi.org/10.2206/KYUSHUJM.61.233
6. Hirofumi Osada, "Harnack inequalities for exotic Brownian motions", Kyushu Journal of Mathematics 56(2):363–380, 2002. DOI: 10.2206/kyushujm.56.363, https://doi.org/10.2206/kyushujm.56.363
7. Jun Kigami, "Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance", Memoirs of the American Mathematical Society 259(1250), 2019. DOI: 10.1090/memo/1250, https://doi.org/10.1090/memo/1250
8. Shiping Cao, Zhen-Qing Chen, "Boundary trace theorems for symmetric reflected diffusions", 2024. arXiv:2410.19201, https://arxiv.org/abs/2410.19201 (adjacent methodology for jump kernels of boundary trace processes, not on fractals)