Speaker
Description
The ‘Arcsine’ laws of Brownian particles in one dimension describe
distributions of three quantities: the time $t_m$ to reach maximum position, the
time $t_r$ spent on the positive side and the time $t_l$ of the last visit to the
origin. Interestingly, the cumulative distribution of all three quantities are the
same and given by Arcsine function. In this paper, we study distribution of
these three times $t_m$, $t_r$ and $t_l$ in the context of single run-and-tumble particle
in one dimension, which is a simple non-Markovian process. We compute
exact distributions of these three quantities for arbitrary time and find that
all three distributions have a delta function part and a non-delta function
part. Interestingly, we find that the distributions of $t_m$ and tr are identical
(reminiscent of the Brownian particle case) when the initial velocities of the
particle are chosen with equal probability. On the other hand, for $t_l$, only
the non-delta function part is the same as the other two. In addition, we find
explicit expressions of the joint distributions of the maximum displacement
and the time at which this maxima occurs.
