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Mean First Passage Times of Higher-Dimensional Velocity Jump Processes.

Maria R D'Orsogna1, Alan E Lindsay2, Thomas Hillen3

  • 1University of California at Los Angeles, California State University at Northridge, Department of Mathematics, Los Angeles, California, 91330, USA and Department of Computational Medicine, Los Angeles, California, 90095-1766, USA.

Physical Review Letters
|July 7, 2026
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Summary

We developed a framework for velocity jump processes, crucial in physics and biology. Our method accurately predicts first passage times, even with directional bias, offering new insights into complex systems.

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Area of Science:

  • Physics
  • Biology
  • Finance
  • Stochastic Processes
  • Complex Systems

Background:

  • First passage phenomena are critical in diverse fields, often modeled by diffusive motion.
  • Velocity jump processes, with persistent motion and stochastic velocity changes, are more accurate for many systems.
  • Higher-dimensional first passage properties of velocity jump processes, particularly with directional bias, are underdeveloped.

Purpose of the Study:

  • To develop a general framework for estimating mean first passage time (MFPT) and survival probability moments.
  • To analyze fixed-speed velocity jump processes with varying reorientation behaviors (strong alignment to full anisotropy).
  • To investigate the impact of directional bias on first passage properties in higher dimensions.

Main Methods:

  • Derivation of a universal MFPT form for low Knudsen numbers using bias functions.
  • Analysis of anomalous scaling in the narrow-capture limit.
  • Development of a Langevin representation for first passage statistics.

Main Results:

  • A universal MFPT form was derived, encoding various angular distributions (e.g., von Mises-Fisher, wrapped Cauchy).
  • Directional persistence was shown to induce anomalous scaling in the narrow-capture limit, with finite MFPT where diffusion predicts divergence.
  • The Langevin representation accurately reproduced first passage statistics, validated by numerical simulations.

Conclusions:

  • The developed framework provides a robust method for analyzing first passage phenomena in velocity jump processes.
  • The study highlights the significant impact of directional bias and persistence on system dynamics.
  • Findings offer a more accurate understanding of complex system behaviors across physics, biology, and finance.