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.
Bibliographical noteFunding Information:
I am grateful to Tal Agranov and Naftali R. Smith for useful discussions, and to Naftali R. Smith for help with Fig. 3. This research was supported by the Israel Science Foundation (Grant No. 807/16).
© 2019 World Scientific Publishing Company.
- Brownian motion
- mortal Brownian motion
- spontaneous decay