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

J. M. Schwarz·Ron Maimon
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Abstract

We present a heuristic derivation of the first passage time exponent for the integral of a random walk [Y. G. Sinai, Theor. Math. Phys. {\bf 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 $0.220\pm0.001$. We discuss the implications of this estimation scheme for the $n{\rm th}$ integral of a random walk. For completeness, we also address the $n=\infty$ 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 reparameterization freedom in the Langevin equation that maps nonlinear stochastic processes into linear ones.

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J. M. Schwarz, Ron Maimon. 2001-03-09. First passage time exponent for higher-order random walks:Using Levy flights. https://doi.org/10.1103/physreve.64.016120

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