Mathematical analysis of differential operators derived from singular geometric structures and their probabilistic counterparts

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17554
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: KAJINO Naotaka 神戸大学, 理学研究科, 准教授 (90700352)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: フラクタル解析 / ラプラシアン / 固有値漸近挙動 / アポロニウスの詰め込み / 円詰込フラクタル / クライン群 / リウヴィルブラウン運動 / 劣ガウス型熱核評価

Outline of Final Research Achievements

As the main results of this research project, the principal investigator has studied certain important examples of circle packing fractals, such as the Apollonian gasket, realized as the minimum closed invariant sets of discrete groups of complex linear fractional transformations. Specifically, he has constructed a geometrically natural energy form and a Laplacian (a differential operator in space variable which is central to the formulation of heat and wave equations) on those fractals, found out an explicit expression of them and proved an asymptotic formula for the eigenvalues of this Laplacian.
He has also conducted joint research on the Liouville Brownian motion (LBM), which is a stochastic process running in a certain random geometry and has been extensively studied recently in probability theory, and a sufficient condition for estimates from above has been obtained in a general framework as an intermediate step for proving estimates of the transition density function of the LBM.

Free Research Field

フラクタルおよび測度距離空間上の解析学

Academic Significance and Societal Importance of the Research Achievements

与えられたフラクタルの上に幾何的に自然なラプラシアンを構成する問題は，30年の歴史を持つフラクタル上の解析学の最も基本的な問題であるにも拘らず，一部の理想的な自己相似性を有するフラクタル以外に対してはほとんど研究されてこなかった．アポロニウスの詰め込みの場合を端緒として一般の円詰込フラクタルに対して幾何的に自然なLaplacianの構成法を見出せたことは大きな進展であり，今後はより広範なフラクタルへの拡張が期待される．
Liouville Brown運動の確率密度関数の評価を得る研究は，既存の研究では追求されていない詳細な劣ガウス型評価の証明を目指すものであり，世界的に見てもその独自性は高い．