Killing random motions with finite velocity

Abstract

In this paper, we study finite-velocity random motions subject to a killing mechanism acting at direction changes: at each change of direction, the motion is killed with a fixed probability and remains at its current position thereafter. We first consider the one-dimensional case of a standard telegraph process. We derive the exact distribution of the position at the killing time and show that it is Laplace distributed. We also characterize the law of the process conditional on survival up to a fixed time, proving that it coincides with that of a standard telegraph process driven by a thinned Poisson process with reduced switching intensity. We further derive the full time-dependent distribution of the killed process, including both its absolutely continuous and singular components. We then extend the analysis to a planar finite-velocity random motion with infinitely many possible directions. In this setting, we derive the distribution of the killed process and obtain an explicit representation of its density in terms of modified Bessel functions and a series involving Bessel polynomials.

Diffusion limits of cyclic finite-velocity random motions along vector fields
Olga Aryasova, Ilya Pavlyukevich, Yusif Sualah
arXiv31 Aug 2026PreprintCC BY 4.0Full textProbabilityarXiv
Heat Kernel Estimates for Random Walks with Killing on Transient Graphs
Emily Dautenhahn, Anthony Graves-McCleary
arXiv13 Oct 2026PreprintCC BY 4.0Full textProbabilityarXiv
Unified description of random motions and generalized Fokker-Planck equations
Luca Angelani
arXiv24 Sep 2026PreprintarXiv licenceFull textMathematical physics Statistical & nonlinear physicsarXiv
The velocity is deterministic in random walk in 2-valued i.i.d. random environment
Mark Holmes, Thomas S. Salisbury
arXiv29 Oct 2026PreprintCC BY 4.0Full textProbabilityarXiv
Richardson dispersion and anomalous dissipation for the random velocity model with controlled sweeping
Charles Fefferman, Svitlana Mayboroda
arXiv5 Nov 2026PreprintCC0Full textAnalysis & PDEs Mathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{orsingher2026killing,
  title = {{Killing random motions with finite velocity}},
  author = {Enzo Orsingher and Manfred Marvin Marchione},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15094},
  eprint = {2610.15094},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15094},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Killing random motions with finite velocity
AU  - Enzo Orsingher
AU  - Manfred Marvin Marchione
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15094
UR  - https://arxiv.org/abs/2610.15094
AB  - In this paper, we study finite-velocity random motions subject to a killing mechanism acting at direction changes: at each change of direction, the motion is killed with a fixed probability and remains at its current position thereafter. We first consider the one-dimensional case of a standard telegraph process. We derive the exact distribution of the position at the killing time and show that it is Laplace distributed. We also characterize the law of the process conditional on survival up to a fixed time, proving that it coincides with that of a standard telegraph process driven by a thinned Poisson process with reduced switching intensity. We further derive the full time-dependent distribution of the killed process, including both its absolutely continuous and singular components. We then extend the analysis to a planar finite-velocity random motion with infinitely many possible directions. In this setting, we derive the distribution of the killed process and obtain an explicit representation of its density in terms of modified Bessel functions and a series involving Bessel polynomials.
ER  -

Abstract

Related

Cite