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 = η(t), subject to a transverse shear flow in the x-direction, dx/dt = f(y), where x ≥ 0 and x = 0 is an absorbing boundary. We treat the class of models defined by f(y) = ±v±(±y)α where the upper (lower) sign refers to y > 0 (y < 0). We show that the particle survives with probability Q(t) ∼ t-θ with θ = 1/4, independent of α, if v+ = v-. If v+ ≠ v-, however, we show that θ depends on both α and the ratio v+/v-, and we determine this dependence. (letter to the editor)