Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J65, 60G46

Retrieve articles in all journals with MSC: 60J65, 60G46

Additional Information

Keywords: Exit time, Hardy space, Brownian motion, harmonic majorization, maximal function
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society