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We consider the symmetric tridiagonal matrix-valued process associated withGaussian beta ensemble (G$\beta$E) by putting independent Brownian motions andBessel processes on the diagonal entries and upper (lower)-diagonal ones,respectively. Then, we derive the stochastic differential equations that theeigenvalue processes satisfy, and we show that eigenvalues of their (indexed)principal minor sub-matrices appear in the stochastic differential equations.By the Cauchy's interlacing argument for eigenvalues, we can characterize thesufficient condition that the eigenvalue processes never collide with eachother almost surely, by the dimensions of the Bessel processes.

Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble