Brownian motion with partial information1

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Abstract

We study the following problem concerning stopped N-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region R. Suppose this modified maximal function belongs to LqFor what regions R can we conclude that the unrestricted maximal function belongs to Lqc! A sufficient condition on R is that there exist p > q and a function w, harmonic in /?, such that \x\p^ u(x) < C\x\p+ C, x Çz R, for some constant C. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.

Original languageEnglish (US)
Pages (from-to)719-731
Number of pages13
JournalTransactions of the American Mathematical Society
Volume271
Issue number2
DOIs
StatePublished - Jun 1982

Keywords

  • Brownian motion
  • Exit time
  • Hardy space
  • Harmonic majorization
  • Maximal function

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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