Mortal Brownian motion: three short stories

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 $A={\int }_0^{T}x(t)dt$ of a MBP on the line. Story 3 addresses the distribution of the winding angle of a 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.

arXiv comment7 one-column pages, 4 figures, small changes

References in corpus (9)

• Macroscopic fluctuation theory
• Mortality, Redundancy, and Diversity in Stochastic Search
• Exploration and Trapping of Mortal Random Walkers
• The escape problem for mortal walkers
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• Evanescent continuous time random walks
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• The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles
• Winding statistics of a Brownian particle on a ring

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