Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas

Abstract

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.

Published at http://dx.doi.org/10.1214/07-PS104 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (40)

• Power law violation of the area law in quantum spin chains
• From the area under the Bessel excursion to anomalous diffusion of cold atoms
• On the time to reach maximum for a variety of constrained Brownian motions
• Properties of additive functionals of Brownian motion with resetting
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Diffusions conditioned on occupation measures
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• The Airy distribution: experiment, large deviations and additional statistics
• Patterns in random permutations avoiding the pattern 132
• Large-deviations for spatial diffusion of cold atoms
• On q-functional equations and excursion moments
• On the probability that integrated random walks stay positive
• On the length of a random minimum spanning tree
• Statistics of the first passage time of Brownian motion conditioned by maximum value or area
• An exactly solvable self-convolutive recurrence
• Generating discrete-time constrained random walks and Lévy flights
• Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries
• On the joint distribution of first-passage time and first-passage area of drifted Brownian motion
• Maximum relative height of one-dimensional interfaces : from Rayleigh to Airy distribution
• Generating stochastic trajectories with global dynamical constraints
• Almost square permutations are typically square
• The Distribution of the Area under a Bessel Excursion and its Moments
• On the critical probability in percolation
• Dual representations of Laplace transforms of Brownian excursion and generalized meanders
• Critical behavior in topological ensembles
• Area fluctuations on a subinterval of Brownian excursion
• Area distribution and the average shape of a Lévy bridge
• Statistical fluctuations under resetting: rigorous results
• Interacting partially directed self-avoiding walk: a probabilistic perspective
• Fluctuations of ring polymers
• On the number of large triangles in the Brownian triangulation and fragmentation processes
• Periodic Pólya Urns, the Density Method, and Asymptotics of Young Tableaux
• Area limit laws for symmetry classes of staircase polygons
• Inference on the intraday spot volatility from high-frequency order prices with irregular microstructure noise
• Scalar Conservation Laws with white noise initial data
• Area Distribution of Elastic Brownian Motion
• On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities
• The area under a spectrally positive stable excursion and other related processes
• Persistence and exit times for some additive functionals of skew Bessel processes
• On the joint distribution of the area and the number of peaks for Bernoulli excursions