Construction of a solution of random transport equation with boundary condition

$(t, x)\in[0, T]\times R^{1}$ , $ T<\infty$ , where $\dot{B}_{t}(\omega)$ is the white noise. He constructed a solution of Cauchy problem of equation (0.1) with given initial data (0.2) $u(0, x;\omega)=\phi(x)$ . His main tools are a stochastic integral which he defined and the concept of the differentiation $\frac{\partial X_{t}}{\partial B_{t}}$ of a stochastic process $X_{t}$ with respect to the Brownian motion $B_{t}$ . Here, in this paper, we consider a natural extension of his equation: