Estimation of the mean of a Wiener process observed on an infinite interval

Let $\omega(t)$ be a Wiener process, $\mathbf{M}\omega(t)=0$, $\mathbf{D}\omega(t)=t$, $\varphi(t)$, $t\in[0,\infty]$ be a function form a set $M\subset C_{[0,\infty)}$ and $x(t)=\omega(t)+\varphi(t)$ be the observation process.
In the paper, conditions on the set $M$ are given under which there exist a consistent estimate of $\varphi$.