New Research


2013/10/01

Stochastic analysis on fractals

Professor Masanori Hino

Research Group of Stochastic Analysis consists of Prof. Masanori Hino and Associate Prof. Watanabe Yusuke. The following is an introduction to Professor Hino's research.

 

Stochastic analysis on fractal sets as a means of building mathematical models of natural phenomena, such as heat transfer in heterogeneous materials, has been attracting the attention of physicists and mathematicians for the past 30 years. Due to the nature of their structure, fractals are expected to show various properties that differ from those of smooth spaces. However, it is not at all clear how to mathematically define this difference or prove its existence, and many problems in this area remain unsolved. Many techniques that are useful for smooth spaces do not extend to fractals, and so research often has to start by developing techniques for analyzing these sets.

 

As one example of a typical research topic, let us explain briefly the problem of noise generated by stochastic processes over fractals. Although white noise is associated with Brownian motion in Euclidean space, noise attributable to natural stochastic processes over fractals is likely to be qualitatively different. However, there is not yet a decisive theory describing how the concepts of noise and qualitative difference can be formalized into a mathematically analyzable form. One of our research goals is to use such fundamental research to reveal the mathematical structures that emerge from randomness and the spatial structures that emerge from stochastic processes.

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