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 --> lambda (3)x, v --> lambdav, t --> lambda (2)t. 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 phi, then the power law must have the specific form Q(x, v; t) approximate to Cx(2 phi /3) U (-2/3 phi, 2/3, v3/9x)t(-phi). 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 phi for several boundary conditions of physical interest follow directly from this result.