Survival of a diffusing particle in a transverse shear flow: a first-passage problem with continuously varying persistence exponent

We consider a particle diffusing in the y-direction, dy/dt = eta(t), subject to a transverse shear flow in the x-direction, dx/dt = f(y), where x greater than or equal to 0 and x = 0 is an absorbing boundary. We treat the class of models defined by f(y) = +/-v(+/-)(+/-y)(alpha) where the upper (lower) sign refers to y > 0 (y < 0). We show that the particle survives with probability Q(t) similar to t(-theta) with theta = 1/4, independent of alpha, if v(+) = v(-). If v(+) not equal v(-), however, we show that theta depends on both alpha and the ratio v(+)/v(-), and we determine this dependence.