Survival of a diffusing particle in a transverse flow field

We consider a particle diffusing in the y-direction, dy/dt = eta(t) where eta(t) is Gaussian white noise, and subject to a transverse flow field in the x-direction, dx/dt = f(y), where x >= 0 and x = 0 is an absorbing boundary. We discuss the time dependence of the survival probability of the particle for a class of functions f(y) that are positive in some regions of space and negative in others.