First-passage-time exponent for higher-order random walks:  Using Lévy flights

We present a heuristic derivation of the first-passage-time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. 90, 219 (1992)]. Building on this derivation, we construct an estimation scheme to understand 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=∞ case. Finally, we explore an application of these processes to an extended, elastic object being pulled through a random potential by a uniform applied force. In so doing, we demonstrate a time reparametrization freedom in the Langevin equation that maps nonlinear stochastic processes into linear ones.