by $p(t,x,y),\, x\in\mathbb{R}$, the Gaussian density with variance $t$ and mean $y$, then the function $\varphi_{t}(r,\cdot)$ given by $$\varphi_{t}\left(r,x\right):=\begin{cases}
p\left(\frac{t\left(r-t\right)}{r},x,0\right), & 0<t<r, \ x\in\mathbb{R},\\
0, & r\leq t, \ x\in\mathbb{R},
\end{cases} \label{eq:bbdensity}$$ is equal to the density of $\beta_{t}^{r}$ for $0<t<r$. Furthermore we have the following properties:

1.  (Expectation)  $\mathbf{E}\left[\beta_{t}^{r}\right]=0, \ t\geq0.$

2.  (Covariance)  $\mathbf{E}\left[\beta_{t}^{r}\beta_{s}^{r}\right]=s\wedge t\wedge r-\frac{(s\wedge r)(t\wedge r)}{r}, \ s,t\geq 0$.

3.  (Conditional expectation)   For $t\in[0,r)$ and $t<u$ we have: If $g(\beta^r _u)$ is integrable, then $$\begin{aligned}
    \lefteqn{\mathbf{E}\left[g\left(\beta_{u}^{r}\right)|\beta_{t}^{r}=x\right]}\nonumber\\
    &=&\begin{cases}
    {\displaystyle \int_{\mathbb{R}}}g\left(y\right)p\left(\frac{r-u}{r-t}\left(u-t\right),\, y,\,\frac{r-u}{r-t}x\right)dy, & t<u<r, \ x\in\mathbb{R},\\
    g\left(0\right), & r\leq u, \ x\in\mathbb{R}.
    \end{cases}\label{eq:cond exp ext bb}\end{aligned}$$

4.  $\beta^{r}$ is a Markov process with respect to its completed natural filtration.

5.   $\beta_{t}^{r}=\beta_{t\wedge r}^{r}, \ t\geq0$.

\[BMr\] The process $B^{r}=(B_{t}^{r},\, t\geq0)$ given by $$B_{t}^{r}:=\beta_{t}^{r}+\int_{0}^{t}\frac{\beta_{s}^{r}}{r-s}\, ds\label{eq:BMfrom_extBB}$$ is a Brownian motion stopped at the (deterministic) stopping time $r$, with respect to the completed natural filtration of $\beta^{r}$.

Now we substitute the fixed time $r$ by a random time $\tau$. Let $\tau$ be a strictly positive random variable on $(\Omega,\mathcal{F},\mathbf{P})$. The object which we are interested in is given by the composition of the two mappings $(r,t,\omega)\mapsto\beta_{t}^{r}(\omega)$ and $(t,\omega)\mapsto(\tau(\omega),t,\omega)$. We have the following definition:

The map $\beta=(\beta_{t},\, t\geq0):\,(\Omega,\,\mathcal{F})\rightarrow(C,\,\mathcal{C})$ is defined by $$\beta_{t}\left(\omega\right):=\beta_{t}^{\tau\left(\omega\right)}\left(\omega\right),\quad (t,\omega)\in\mathbb{R}_+\times\Omega\,.\label{eq:beta def 