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On a Brownian motion problem of T. Salisbury

Published:1997-03-01
Printed: Mar 1997
• Frank B. Knight
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Abstract

Let $B$ be a Brownian motion on $R$, $B(0)=0$, and let $f(t,x)$ be continuous. T.~Salisbury conjectured that if the total variation of $f(t,B(t))$, $0\leq t\leq 1$, is finite $P$-a.s., then $f$ does not depend on $x$. Here we prove that this is true if the expected total variation is finite.
 MSC Classifications: 60J65 - Brownian motion [See also 58J65]

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