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Infinite Analysis Lab (Naotaka Kajino)
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A probability theory lab that analyzes the Laplacian and stochastic processes on fractals.
AboutAbout the lab
Unraveling stochastic processes on fractals.
The Kajino Lab studies the Laplacian and stochastic processes on fractals, in the field of probability theory.
Fractal is a general term for spaces whose geometric properties are entirely different from those of smooth spaces such as Euclidean space or Riemannian manifolds.
On such spaces, the lab aims to construct a geometrically natural Laplacian and the corresponding stochastic process, and to analyze them in detail.
In recent years, the lab has also been working on constructing and analyzing nonlinear energy functionals of Lp type on fractals.
Key terms
- Fractal
- A shape in which similar structures appear repeatedly in fine detail when magnified. The Sierpiński gasket is a typical example.
- Laplacian
- A differential operator that measures how bumpy the values of a function are. It appears in the equation describing how heat spreads.
- Heat kernel
- A function describing how much of the heat emitted from a single point has spread to each location at each time. It corresponds to the transition probability density of a stochastic process.
ResearchResearch themes
The lab aims to construct a geometrically natural Laplacian and the corresponding stochastic process on fractals, which are spaces whose geometric properties are entirely different from those of smooth spaces, and to analyze them in detail. The main part of the research is understanding the Laplacian and sub-Gaussian heat kernel estimates on Euclidean self-similar fractals, and in recent years random fractals and self-conformal fractals have also become objects of study.
*Illustrative imageSub-Gaussian heat kernel estimates and energy measures
The lab studies energy measures and harmonic measures, which are difficult to analyze, in connection with the sub-Gaussian heat kernel estimates that typically hold on Euclidean self-similar fractals.
*Illustrative imageQuasiconformal structures and higher-dimensional Sierpiński gaskets
On the other hand, the lab has also shown that for the d-dimensional Sierpiński gasket with d≥3, a replacement of the same kind cannot make it satisfy Gaussian heat kernel estimates.
*Illustrative imageLiouville quantum gravity and Liouville Brownian motion
In joint research, the lab investigates Liouville Brownian motion, the natural diffusion process under Liouville quantum gravity, which is a random geometry on surfaces. The lab has proved the continuity of the heat kernel and an upper bound for it.
*Illustrative imageLaplacians on self-conformal fractals
The lab is also constructing and analyzing geometrically natural Laplacians on self-conformal fractals such as limit sets of Kleinian groups and Julia sets of complex dynamical systems.
MembersFaculty
- Associate ProfessorNaotaka Kajino
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Pseudo Kleinian OpenCL K 002” PantheraLeo1359531 / CC BY-SA 4.0 (cropped)
- Sub-Gaussian heat kernel estimates and energy measures: “Apollonian gasket symmetrical 3” Cmglee / CC BY-SA 4.0 (cropped)
- Quasiconformal structures and higher-dimensional Sierpiński gaskets: “Sierpinski Triangle mod 3” Kabrick1 / CC BY 4.0 (cropped)
- Liouville quantum gravity and Liouville Brownian motion: “Discrete Gaussian free field on 60 x 60 square grid” Samuelswatson / Public domain (cropped)
- Laplacians on self-conformal fractals: “Julia set (highres 01)” Solkoll / Public domain (cropped)
- Related lab (無限解析研究部門
（CROYDON, David） ): “Schramm-Loewner Evolution” Tesseract16 / CC0 (cropped) - Related lab (無限解析研究部門
（星 裕一郎） ): “Kleinian group limit set on sphere” Adam majewski / CC BY-SA 4.0 (cropped) - Related lab (無限解析研究部門
（望月 新一） ): “Kyoto University RIMS” KATSURA Roen / CC BY 3.0 (cropped) - Related lab (無限解析研究部門
（玉川 安騎男） ): “Klein-map” PSL27 / CC BY-SA 4.0 (cropped)
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