High-dimensional limits of eigenvalue distributions for general Wishart process (Q2657920)

Let \(\mathcal{S}_N\) be the space of \(N\times N\) symmetric matrices. The generalized Wishart process on \(\mathcal{S}_N\) satisfies the SDE given by \[ d X_t^N = g_N(X_t^N)dB_th_N(X_t^N)+h_N(X_t^N)dB_t^\top g_N(X_t^N)+b_N(X_t^n)dt\,, \] for \(t\geq0\), where \(B_t\) is an \(N\times N\) Brownian motion, and the functions \(b_N, g_N, h_N:\mathbb{R}\to\mathbb{R}\) act on the spectrum of \(X_t^N\). The authors investigate the large-\(N\) limiting distribution of the eigenvalues \(\lambda_1^N(t),\cdots,\lambda_N^N(t)\) of \(X_t^N\) through (a generalisation of) the SDEs satisfied by these eigenvalues. They also derive new conditions for the existence and uniqueness of a strong solution for these latter SDEs. Finally, the special case of self-similarity on the eigenvalues is given particular attention.