ar X iv : 0 70 5 . 24 60 v 1 [ m at h . PR ] 1 7 M ay 2 00 7 Noncolliding Brownian Motion and Determinantal Processes ∗
نویسندگان
چکیده
A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the KarlinMcGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding diffusion processes have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. Using the common properties of matrix-kernels, continuity of processes in time is proved and Dirichlet forms are provided.
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