A Representation for Non-Colliding Random Walks

We define a sequence of mappings Γₖ:D₀(R₊)ᵏ→ D₀(R₊)ᵏ and prove the following result: Let N₁,…,Nₙ be the counting functions of independent Poisson processes on R₊ with respective intensities μ₁ < μ₂ < ⋯ < μₙ. The conditional law of N₁,…,Nₙ, given that N₁(t)≤⋯≤ Nₙ(t), for all t≥ 0, is the same as the unconditional law of Γₙ(N). From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov (2001) and Gravner, Tracy and Widom (2001), which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman's representation for the 3-dimensional Bessel process.