Construction of pp-energy and associated energy measures on the Sierpi\'{n}ski carpet

We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating the planar Sierpi\'{n}ski carpet for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the Sierpi\'{n}ski carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule.Comment: 80 pages, 9 figures; fixed several typos and some mistakes, edited some proofs. Sections 4 and 5 from the previous version have been merged into one section. Theorem 2.20 and subsection 6.3 are ne

Sierpiński carpet上のp-エネルギーと対応するエネルギー測度の構成

京都大学新制・課程博士博士(情報学)甲第24262号情博第806号京都大学大学院情報学研究科先端数理科学専攻(主査)教授 木上 淳, 教授 磯 祐介, 准教授 白石 大典学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDFA

First-order Sobolev spaces, self-similar energies and energy measures on the Sierpi\'{n}ski carpet

We construct and investigate $(1, p)$-Sobolev space, $p$-energy, and the corresponding $p$-energy measures on the planar Sierpi\'{n}ski carpet for all $p \in (1, \infty)$. Our method is based on the idea of Kusuoka and Zhou [Probab. Theory Related Fields $\textbf{93}$ (1992), no. 2, 169--196], where Brownian motion (the case $p = 2$) on self-similar sets including the planar Sierpi\'{n}ski carpet were constructed. Similar to this earlier work, we use a sequence of discrete graph approximations and the corresponding discrete $p$-energies to define the Sobolev space and $p$-energies. However, we need a new approach to ensure that our $(1, p)$-Sobolev space has a dense set of continuous functions when $p$ is less than the Ahlfors regular conformal dimension. The new ingredients are the use of Loewner type estimates on combinatorial modulus to obtain Poincar\'e inequality and elliptic Harnack inequality on a sequence of approximating graphs. An important feature of our Sobolev space is the self-similarity of our $p$-energy, which allows us to define corresponding $p$-energy measures on the planar Sierpi\'{n}ski carpet. We show that our Sobolev space can also be viewed as a Korevaar-Schoen type space. We apply our results to the attainment problem for Ahlfors regular conformal dimension of the Sierpi\'{n}ski carpet. In particular, we show that if the Ahlfors regular conformal dimension, say $\dim_{\mathrm{ARC}}$, is attained, then any optimal measure which attains $\dim_{\mathrm{ARC}}$ should be comparable with the $\dim_{\mathrm{ARC}}$-energy measure of some function in our $(1, \dim_{\mathrm{ARC}})$-Sobolev space up to a multiplicative constant. In this case, we also prove that the Newton-Sobolev space corresponding to any optimal measure and metric can be identified as our self-similar $(1, \dim_{\mathrm{ARC}})$-Sobolev space.Comment: 162 pages 4 figures; Several updates, sections reorganized, and a new section added; Comments welcom

Photon polarization entanglement induced by biexciton: experimental evidence for violation of Bell's inequality

We have investigated the polarization entanglement between photon pairs generated from a biexciton in a CuCl single crystal via resonant hyper parametric scattering. The pulses of a high repetition pump are seen to provide improved statistical accuracy and the ability to test Bell's inequality. Our results clearly violate the inequality and thus manifest the quantum entanglement and nonlocality of the photon pairs. We also analyzed the quantum state of our photon pairs using quantum state tomography.Comment: 4 pages, 5 figure