Survival probability of mutually killing Brownian motions and the O'Connell process

Abstract

Recently O'Connell introduced an interacting diffusive particle system in order to study a directed polymer model in 1+1 dimensions. The infinitesimal generator of the process is a harmonic transform of the quantum Toda-lattice Hamiltonian by the Whittaker function. As a physical interpretation of this construction, we show that the O'Connell process without drift is realized as a system of mutually killing Brownian motions conditioned that all particles survive forever. When the characteristic length of interaction killing other particles goes to zero, the process is reduced to the noncolliding Brownian motion (the Dyson model).

v2: AMS-LaTeX, 20 pages, 2 figures, minor corrections made for publication in J. Stat. Phys

References in corpus (3)

• Baxter operator and Archimedean Hecke algebra
• Whittaker functions and related stochastic processes
• O'Connell's process as a vicious Brownian motion

Cited by in corpus (7)

• 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
• 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