Effective dimension of points visited by Brownian motion

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• 作者：Bjø ; rn Kjos-Hanssen ; Anil Nerode
• 关键词：Brownian motion ; Algorithmic randomness ; Effective randomness
• 刊名：Theoretical Computer Science
• 出版年：2009
• 出版时间：17 February 2009
• 年：2009
• 卷：410
• 期：4-5
• 页码：347-354
• 全文大小：561 K

文摘

We consider the individual points on a Martin-Löf random path of Brownian motion. We show that (1) Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension <1. The proof of (1) shows that, for almost all times t, the path f is Martin-Löf random relative to t and so the effective dimension of (t,f(t)) is 2.