We present an analytic method for finding the distribution of the area under the stochastic Bessel meander, discuss simulations and its relevance to anomalous diffusion of cold atoms.

Excursions, Meanders and Bridges are three types of random walk processes, special for the unique constraints imposed on the trajectory of the random walker. A bridge is a stochastic path in the time interval [0,Tau], bound to begin and end at the origin x=0. An additional constraint that the path never crosses or even reaches the origin for (0,Tau) makes the process an excursion.  A meander is also a trajectory in [0,Tau], starting from the origin but not bound to end there, however it is constrained not to reach or cross to again for (0,Tau).  

Majumdar and Comtet, in 2004, were the first to study the probability distribution of the area under the Brownian excursion and meander, in the context of the maximal height of fluctuating surfaces. Recently, in our group, we used the areal distribution of Bessel excursions and meanders to explore the anomalous diffusion of cold atoms in Sisyphus cooling, and later in the context of the shape properties of ring polymers. The application of these stochastic paths for the description of physical systems is varied and can potentially be wide spread.