Skip to main content
×
×
Home

Beurling's projection theorem via one-dimensional Brownian motion

  • Wendelin Werner (a1)
Abstract

We prove some elementary intuitive estimates on moving boundaries hitting times by one-dimensional Brownian motion (in ℝ and on the circle). These results give an alternative approach to Beurling's radial projection theorem on harmonic measure in a disc.

Copyright
References
Hide All
[1]Ahlfors, L. V.. Conformal invariants, topics in geometric function theory (McGraw-Hill, 1973).
[2]Beurling, A.. Etudes sur un problème de majoration (Thèse; Uppsala, 1933).
[3]Burdzy, K.. Geometric properties of two-dimensional Brownian paths. Probab. Th. rel. Fields 81 (1989), 485505.
[4]Burdzy, K. and Lawler, G. L.. Non-intersection exponents for Brownian paths. Part II: Estimates and application to a random fractal. Ann. Probab. 18 (1990), 9811009.
[5]Carne, T. K.. Brownian motion and Nevanlinna Theory. Proc. London Math. Soc. (3) 52 (1986), 349368.
[6]Davies, B.. Brownian motion and analytic functions. Ann. Probab. 7 (1979), 913932.
[7]Duplantier, B., Lawler, G. F., Le Gall, J. F. and Lyons, T. J.. The geometry of the Brownian curve, in: Probability's et Analyse stochastique, Tables rondes de St-Chéron Janvier 1992. Bull. Sc. Math. (2) 117 (1993).
[8]Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge University Press, 1934).
[9]Itô, K. and Mckean, H. P.. Diffusion processes and their sample paths (Springer, 1965).
[10]Lai, T. Y. and Wijsman, R. A.. First exit time of a random walk from the boundary f(n) ± cg(n) with applications. Ann. Probab. 7 (1979), 672692.
[11]Lawler, G. L.. Intersection of random walks (Birkhäuser, 1991).
[12]Lyons, T. J.. A synthetic proof of Makarov's law of the iterated logarithm. Bull. LondonMath. Soc. 22 (1990), 159162.
[13]Makarov, N. G.. Probability Methods in the theory of conformal mappings. Algebra i Analiz 1 (1989), 359 (Russian)
[13]Makarov, N. G. English transl.: Leningrad Math. J. 1 (1990), 156.
[14]Nevanlinna, R.. Eindeutige analytische Funktionen (Springer, 1936).
[15]Oksendal, B.. Projection estimates for harmonic measure. Ark. Math. 21 (1983), 191203.
[16]Oksendal, B.. A stochastic proof of an extension of a Theorem of Rado. Proc. Edinburgh Math. Soc. 26 (1983), 333336.
[17]Port, S. C. and Stone, C. J.. Brownian motion and classical potential theory (Academic Press, 1979).
[18]Riesz, F.. Sur une inégalité intégrate. J. London Math. Soc. 5 (1930), 162168.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 28 *
Loading metrics...

Abstract views

Total abstract views: 103 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 12th June 2018. This data will be updated every 24 hours.