Brownian motion with partial information

McConnell, Terry R.

doi:10.1090/S0002-9947-1982-0654858-5

Brownian motion with partial information
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by Terry R. McConnell PDF

Trans. Amer. Math. Soc. 271 (1982), 719-731 Request permission

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 \[ |x{|^p} \leqslant u(x) \leqslant C|x{|^p} + C,\qquad x \in R,\] for some constant $C$. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.

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