O'Connell's process as a vicious Brownian motion

Abstract

Vicious Brownian motion is a diffusion scaling limit of Fisher's vicious walk model, which is a system of Brownian particles in one dimension such that if two of them meet they kill each other. We consider the vicious Brownian motion conditioned never to collide with each other, and call it the noncolliding Brownian motion. This conditional diffusion process is equivalent to the eigenvalue process of a Hermitian-matrix-valued Brownian motion studied by Dyson. Recently O'Connell introduced a generalization of the noncolliding Brownian motion by using the eigenfunctions (the Whittaker functions) of the quantum Toda lattice in order to analyze a directed polymer model in 1+1 dimensions. We consider a system of one-dimensional Brownian motions with a long-ranged killing term as a generalization of the vicious Brownian motion and construct the O'Connell process as a conditional process of the killing Brownian motions to survive forever.

REVTeX4, 26 pages, 1 figure, corrections and additions made for publication in Phys. Rev. E

References in corpus (10)

• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Exact distribution of the maximal height of p vicious walkers
• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
• Distribution of the time at which N vicious walkers reach their maximal height
• Noncolliding Brownian Motion and Determinantal Processes
• Maximum distributions of bridges of noncolliding Brownian paths
• Vicious Lévy flights
• Vicious walks with long-range interactions
• Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts

Cited by in corpus (9)

• Determinantal structures in the O'Connell-Yor directed random polymer model
• Whittaker functions and related stochastic processes
• Noncolliding Brownian Motion with Drift and Time-Dependent Stieltjes-Wigert Determinantal Point Process
• Survival probability of mutually killing Brownian motions and the O'Connell process
• Random ballistic growth and diffusion in symmetric spaces
• Reciprocal Time Relation of Noncolliding Brownian Motion with Drift
• From elongated spanning trees to vicious random walks
• System of Complex Brownian Motions Associated with the O'Connell Process
• Determinantal Martingales and Interacting Particle Systems