First-passage statistics for Lévy flights with a drift

Abstract We provide the first-passage statistics of Lévy flights in a one-dimensional semi-infinite domain, considering the effects of a constant drift µ while the jump length distribution is symmetric. By solving the space-fractional Fokker–Planck equation governing the evolution of the probability density function (PDF), we derive expressions for both the survival probability and the first-passage time for given values of the stability index α and drift parameter µ . Our findings are further validated by comparing them with simulations from the stochastic Langevin equation driven by α -stable noise. Additionally, we make use of the Skorokhod theorem for processes with stationary independent increments and show that our numerical results are in good agreement with the analytical expressions for the PDF of first-passage times and survival probabilities. Specifically, in the asymptotic long-time limit, we show that for α = 2 (Brownian motion), the first-passage time distribution follows the classical Lévy–Smirnov form and decays exponentially, irrespective of the drift direction. The survival probability exhibits a distinct asymptotic behaviour: for positive drift, it decreases exponentially, while for negative drift, it saturates to a finite value, indicating a nonzero probability for the particle to escape to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> . For α = 1 (Cauchy process), the survival probability and the first-passage time density exhibit a power-law decay, with µ -dependent exponents. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we find that the long-time asymptotic remains consistent with the drift-free case, but with a µ -dependent prefactor. Finally, for α -stable processes with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , the first-passage time density follows a power-law decay, whose exponent depends on both α and the drift direction; concurrently, the survival probability decays in power-law form for positive drift, while in the case of negative drift, it saturates to a finite value.