The Dimension of the Frontier of Planar Brownian Motion

Gregory F. Lawler (Duke University)

Abstract


Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals $2(1 - \alpha)$ where $\alpha$ is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on $\alpha$ due to Werner, the Hausdorff dimension is greater than $1.015$.

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Pages: 29-47

Publication Date: March 10, 1996

DOI: 10.1214/ECP.v1-975

References

  1. Bishop, C., Jones, P., Pemantle, R., and Peres, Y. (1995). The dimension of the Brownian frontier is greater than 1, preprint.
  2. Burdzy, K. and Lawler, G. (1990). Non-intersection exponents for random walk and Brownian motion. Part II: Estimates and applications to a random fractal. Ann. Probab. 18 981--1009. Math. Review 91g:60097
  3. Burdzy, K., and San Martin, J. (1989). Curvature of the convex hull of planar Brownian motion near its minimum point. Stoch. Proc. Their Appl. 33 89-103. exponent for random walk intersections. J. Stat. Phys. 56 1--12. Math. Review 91e:60233
  4. Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. Wiley Math. Review 92j:28008
  5. Kaufman, R. (1969). Une propriete metrique du mouvement brownien. C. R. Acad. Sci., Paris 268 727-728. Math. Review 39:#2219
  6. Lawler, G. (1996). Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1, paper no. 2, pp. 1-20.
  7. Mandelbrot, B. (1983). The Fractal Geometry of Nature. . W. H. Freeman. Math. Review 84h:00021
  8. Puckette, E. and Werner, W. (1995). Simulations and conjectures for disconnection exponents, preprint.
  9. Werner, W. (1995). An upper bound to the disconnection exponent for two-dimensional Brownian motion. Bernoulli 1, 371-380.
  10. Werner, W. (1996). Bounds for disconnection exponents. Electron. Comm. Probab. 1, paper no. 4, pp. 19-28.
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