The Mathematical Work of Norbert Wiener (1894‐1964)

Kybernetes ◽  
1994 ◽  
Vol 23 (6/7) ◽  
pp. 34-45 ◽  
Author(s):  
S.D. Chatterji
Keyword(s):  
Brownian Motion ◽  
Harmonic Analysis ◽  
Potential Theory ◽  
Significant Contribution ◽  
Early Research ◽  
Integration Theory ◽  
Mathematical Work ◽  
Norbert Wiener

Describes some parts of Wiener’s significant contribution to mathematics in as simple and non‐technical a language as possible. Looks at Wiener’s early research and how he applied integration theory to potential theory, but not without first explaining the background to integration theory. Then describes one of Wiener’s most important works – that on Brownian motion, and how other theories such as harmonic analysis flowed from his study of Brownian motion. Concludes with a brief chronology of Wiener’s life.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals Potential theory of subordinate killed Brownian motion

2018 ◽  
Vol 371 (6) ◽  
pp. 3917-3969 ◽  
Author(s):  
Panki Kim ◽  
Renming Song ◽  
Zoran Vondraček
Keyword(s):  
Brownian Motion ◽  
Potential Theory

The measure theory of random fractals

1986 ◽  
Vol 100 (3) ◽  
pp. 383-406 ◽  
Author(s):  
S. James Taylor
Keyword(s):  
Brownian Motion ◽  
Potential Theory ◽  
Measure Theory ◽  
Motion Path ◽  
Natural Question ◽  
Random Fractals ◽  
Limited Space ◽  
Definition Of ◽  
Good Problem

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

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