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