  We consider a non-Gaussian stochastic process where a particle diffuses in
the $y$-direction, $dy/dt=\eta(t)$, subject to a transverse shear flow in the
$x$-direction, $dx/dt=f(y)$. Absorption with probability $p$ occurs at each
crossing of the line $x=0$. We treat the class of models defined by $f(y) = \pm
v_{\pm}(\pm y)^\alpha$ where the upper (lower) sign refers to $y>0$ ($y<0$). We
show that the particle survives up to time $t$ with probability $Q(t) \sim
t^{-\theta(p)}$ and we derive an explicit expression for $\theta(p)$ in terms
of $\alpha$ and the ratio $v_+/v_-$. From $\theta(p)$ we deduce the mean and
variance of the density of crossings of the line $x=0$ for this class of
non-Gaussian processes.
