Randomly started signals with white noise (Q5966457)

Let \(\{B(t);t\geq 0\}\) be a standard Wiener process and \(\mu\) be its distribution, i.e, the Wiener measure. Take a random variable U which is independent of \(\{B(t)\}\) and is uniformly distributed in (0,1). For \(\delta >0\), define \(W_{\delta}(t)\), \(t\geq 0\), by \[ W_{\delta}(t)=B(t)+\int^{t}_{0}\delta 2^{-2}(s-U)^{-1/2}I(U\leq s\leq U+1)ds, \] where I is the indicator function. Then the distribution \(\gamma_{\delta}\) of \(\{W_{\delta}(t)\}\) is absolutely continuous with respect to \(\mu\) if \(0<\delta <2\), while \(\gamma_{\delta}\) is singular with respect to \(\mu\) if \(\delta >\sqrt{8}\). For \(\delta\in [2,\sqrt{8}]\) no assertion has yet been obtained.