Some intersection properties of random walk paths

  title={Some intersection properties of random walk paths},
  author={Paul L. Erdos and Stephen Taylor},
  journal={Acta Mathematica Academiae Scientiarum Hungarica},
We complete the solution of this problem in Section 3 . Clearly, there is no problem for d-== I or 2 . The solution takes a different form in the cases d= 3, d-4, and d = 5. For example, if d = 4, an interesting consequence of the result is that, with probability 1, there are infinitely many n for which 17+ (0, n) and /1 4(2n, x ) have a point in common . This in turn implies that any two independent random walks in 4-space have infinitely many points in common . This at first surprised us… 

Some problems concerning the structure of random walk paths

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The Hausdorff α-dimensional measure of Brownian paths in n-space

  • Stephen Taylor
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1953
Dvoretsky, Erdös and Kakutani (3), showed that Brownian paths in 4 dimensions have zero 2-dimensional capacity, and their method gives the same result for Brownian paths in n-space whenever n ≥ 3.

Double points of Brownian paths in n-space

  • Acta Sci. Math . Szeged
  • 1950