First-passage-time exponent for higher-order random walks: Using Levy flights - art. no. 016120

We present a heuristic derivation of the first-passage-time exponent for th e integral of a random walk [Y. G. Sinai, Theor. Math. Phys. 90, 219 (1992) ]. Building on this derivation, we construct an estimation scheme to unders tand the first-passage-time exponent for the integral of the integral of a random walk, which is numerically observed to be 0.220+/-0.001. We discuss the implications of this estimation scheme for the nth integral of a random walk. For completeness, we also address the n(= infinity) case. Finally, w e explore an application of these processes to an extended, elastic object being pulled through a random potential by a uniform applied force. In so d oing, we demonstrate a time reparametrization freedom in the Langevin equat ion that maps nonlinear stochastic processes into linear ones.