Brownian Motion and Potential Theory

1996 ◽  
pp. 301-388
Author(s):  
Michael E. Taylor
Keyword(s):  
Brownian Motion ◽  
Potential Theory

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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