Dynamics of absorption of a randomly accelerated particle

Consider a randomly accelerated particle moving on the half-line x>0 with a boundary condition at x = 0 that respects the scale invariance of the equations of motion under x→λ³x, v→λv, t→λ²t. If the boundary condition leads to absorption of the particle at x = 0 and if the probability Q(x,v;t) that the particle has not yet been absorbed at time t decays, for long times, as a power law with exponent ϕ, then the power law must have the specific form Q(x,v;t)≈Cx^2ϕ/3U(-2ϕ/3,2/3,v³/9x)t^-ϕ. This is a consequence of scale invariance and the Fokker-Planck equation. Here C is a constant, and U(a,b,z) is a confluent hypergeometric function. The persistence exponents ϕ for several boundary conditions of physical interest follow directly from this result.