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

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

2001-03-09

arxiv.org/abs/cond-mat/0103220v1

Physics

Condensed Matter

Statistical Mechanics

4 figures, submitted to PRE

Scientific paper

10.1103/PhysRevE.64.016120

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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