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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Brownian motion with partial information


Author: Terry R. McConnell
Journal: Trans. Amer. Math. Soc. 271 (1982), 719-731
MSC: Primary 60J65; Secondary 60G46
MathSciNet review: 654858
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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 $ {L^q}$. For what regions $ R$ can we conclude that the unrestricted maximal function belongs to $ {L^q}$? A sufficient condition on $ R$ is that there exist $ p > q$ and a function $ u$, harmonic in $ R$, such that

$\displaystyle \vert x{\vert^p} \leqslant u(x) \leqslant C\vert x{\vert^p} + C,\qquad x \in R,$

for some constant $ C$.

We give applications to analytic and harmonic functions, and to weak inequalities for exit times.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0654858-5
Keywords: Exit time, Hardy space, Brownian motion, harmonic majorization, maximal function
Article copyright: © Copyright 1982 American Mathematical Society