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# On a Brownian motion problem of T. Salisbury

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.