As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor's generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish-Chandra ͑Itzykson-Zuber͒ formula of integral over unitary group is established.

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 Karlin-McGregor 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 particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

Dyson's model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson's model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of Z is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.

Angiogenesis is reportedly enhanced by prostaglandins (PGs). In the present study, we investigated whether or not cyclo‐oxygenase (COX)‐2 mediated angiogenesis in chronic and proliferate granuloma. In rat sponge implants, angiogenesis was gradually developed over a 14‐day experimental period as granuloma formed. This angiogenesis was enhanced by topical injections of human recombinant basic fibroblast growth factor (bFGF). In sponge granuloma, mRNA of COX‐1 was constitutively expressed, whereas that of COX‐2 was increased with neovascularization in parallel with that of vascular endothelial growth factor (VEGF). Topical injections of bFGF increased the expression of COX‐2 mRNA. bFGF‐stimulated angiogenesis was inhibited by indomethacin or selective COX‐2 inhibitors, NS‐398, nimesulide, and JTE‐522. The levels of PGE2 and 6‐keto‐PGF1α in the sponge granuloma were increased with bFGF 13 fold and 9 fold, respectively, and these levels were markedly reduced by NS‐398. The expression of VEGF mRNA in the granuloma was also enhanced by bFGF, and was reduced by NS‐398. Topical injections of PGE2 and beraprost sodium, a PGI2 analogue, increased the expression of VEGF mRNA, with angiogenesis enhancement. The enhanced angiogenesis by bFGF was significantly inhibited by topical injections of VEGF anti‐sense oligonucleotide. These results suggested that COX‐2 may enhance bFGF‐induced neovascularization in sponge granuloma by PG‐mediated expression of VEGF, and that a COX‐2 inhibitor would facilitate the management of conditions involving angiogenesis. British Journal of Pharmacology (2000) 130, 641–649; doi:

One-parameter family of discrete-time quantum-walk models on the square lattice, which includes the Grover-walk model as a special case, is analytically studied. Convergence in the long-time limit t → ∞ of all joint moments of two components of walker's pseudovelocity, X t /t and Y t /t, is proved and the probability density of limit distribution is derived. Dependence of the two-dimensional limit density function on the parameter of quantum coin and initial four-component qudit of quantum walker is determined. Symmetry of limit distribution on a plane and localization around the origin are completely controlled. Comparison with numerical results of direct computer-simulations is also shown.