Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

In the present talk, we will discuss the relationship between eigenvalue processes of Hermi-tian matrix-valued stochastic processes and finite particle systems of noncolliding diffusion. A part of the results reported in our recent paper [21] is presented. 1 Bru’s Theorem We denote the space of N × N Hermitian matrices by H(N), the group of N × N unitary matrices by U(N), and the group of N ×N real orthogonal matrices by O(N). We use the notations S(N) and A(N) for the spaces of N × N real symmetric and real antisymmetric matrices and S(N;C) and A(N;C) for the spaces of N×N complex symmetric and complex antisymmetric matrices. 1.1 Hermitian matrix-valued stochastic processes We consider complex-valued processes ξij(t) ∈ C, 1 ≤ i, j ≤ N, t ∈ [0,∞), with the condition ξji(t) ∗ = ξij(t), and define the matrix-valued processes by Ξ(t) = (ξij(t))1≤i,j≤N ∈ H(N). We denote by U(t) = (uij(t))1≤i,j≤N the family of unitary matrices which diagonalize Ξ(t) so that U(t)†Ξ(t)U(t) = Λ(t) = diag{λ1(t), λ2(t), · · · , λN(t)}, where {λi(t)}Ni=1 are eigenvalues of Ξ(t) and we assume their increasing order λ1(t) ≤ λ2(t) ≤ · · · ≤ λN(t). Define Γij(t), 1 ≤ i, j ≤ N, by Γij(t)dt = (U(t)†dΞ(t)U(t))ij(U(t)†dΞ(t)U(t))ji, where dΞ(t) = (dξij)1≤i,j≤N. The indicator function 1{ω} gives 1{ω} = 1 if the condition ω is satisfied, and 1{ω} = 0 otherwise