Randomly Started Signals with White Noise

Abstract

It is shown that if $B(t), t \geq 0$, is a Wiener process, $U$ is an independent random variable uniformly distributed on (0, 1), and $\varepsilon$ is a constant, then the distribution of $B(t) + \varepsilon \sqrt{(t - U)^+}, 0 \leq t \leq 1$, is absolutely continuous with respect to Wiener measure on $C\lbrack 0, 1\rbrack$ if $0 < \varepsilon < 2$, and singular with respect to this measure if $\varepsilon > \sqrt 8$.