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A square root law for diffusing particles

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1153)

Abstract

In an earlier paper, we proved that a plot of log n independent Brownian motions in dimension d=1, for times in [0,n], is nearly certain to give the appearance of a shaded region with square root boundaries, when subjected to the rescaling of the functional iterated logarithm law. Here we prove that for every finite dimension d the same conclusion holds if just one point is plotted from each of log n Brownian paths having variance parameter equal to 1, provided these points are selected uniformly in the time interval [0,n] and independently of the paths.

Keywords

  • Brownian Motion
  • Variance Parameter
  • Shade Region
  • Finite Dimension
  • Iterate Logarithm

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References

  1. LePage, R. and Schreiber, B. M. An iterated logarithm law for families of Brownian paths. To appear, Z. W. Verw. Geb. (1984).

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  2. Richardson, L. F. Some measurements of atmospheric turbulance. Phil. Trans. Roy. Soc. Lond, A. Vol. 221 (1921), p. 1.

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  3. Taylor, G. I. Diffusion by continuous movements. Proc. London Math. Soc. 20 (1921), 196–211.

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© 1985 Springer-Verlag

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LePage, R., Schreiber, B.M. (1985). A square root law for diffusing particles. In: Beck, A., Dudley, R., Hahn, M., Kuelbs, J., Marcus, M. (eds) Probability in Banach Spaces V. Lecture Notes in Mathematics, vol 1153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074958

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  • DOI: https://doi.org/10.1007/BFb0074958

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15704-5

  • Online ISBN: 978-3-540-39645-1

  • eBook Packages: Springer Book Archive