## Abstract

Mortality introduces an intrinsic time scale into the scale-invariant Brownian motion. This fact has important consequences for different statistics of Brownian motion. Here we are telling three short stories, where spontaneous death, such as radioactive decay, puts a natural limit to "lifetime achievements" of a Brownian particle. In story 1 we determine the probability distribution of a mortal Brownian particle (MBP) reaching a specified point in space at the time of its death. In story 2 we determine the probability distribution of the area A = â«0T x(t)dt of an MBP on the line. Story 3 addresses the distribution of the winding angle of an MBP wandering around a reflecting disk in the plane. In stories 1 and 2 the probability distributions exhibit integrable singularities at zero values of the position and the area, respectively. In story 3 a singularity at zero winding angle appears only in the limit of very high mortality. A different integrable singularity appears at a nonzero winding angle. It is inherited from the recently uncovered singularity of the short-time large-deviation function of the winding angle for immortal Brownian motion.

Original language | American English |
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Article number | 1950172 |

Journal | International Journal of Modern Physics B |

Volume | 33 |

Issue number | 16 |

DOIs | |

State | Published - 30 Jun 2019 |

### Bibliographical note

Publisher Copyright:© 2019 World Scientific Publishing Company.

## Keywords

- Brownian motion
- mortal Brownian motion
- spontaneous decay