This paper presents a Dyson Brownian Motion (DBM) model for weak measurements in chaotic quantum systems. The authors study the stochastic Schrödinger equation resulting from continuous monitoring of a random Hermitian operator drawn from the Gaussian Unitary Ensemble (GUE). Due to unitary invariance, the dynamics of the eigenvalues of the density matrix decouple from that of the eigenvectors, and are described by stochastic equations. In the presence of an extra dephasing term, the density matrix has a stationary distribution that matches the inverse-Marchenko–Pastur distribution in the large size limit. In the case of perfect measurements, purification eventually occurs, and the authors focus on finite-time dynamics. They find an exact solution for the joint probability distribution of the eigenvalues at each time t and for each size n. Two relevant regimes emerge: at short times, the spectrum is in a Coulomb gas regime with a well-defined continuous spectral distribution in the large n limit. In this case, all moments of the density matrix become self-averaging and the entanglement spectrum can be exactly characterized. At large times, the eigenvalues are exponentially separated, but fluctuations play an essential role. The authors are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime. The paper also discusses the dynamics of the spectrum, mapping it to unconstrained variables and deriving the stationary state at x > 0 and large n. The authors show that the stationary distribution is the inverse-Wishart distribution, and derive the Von Neumann entropy in the stationary state. They also study the finite time dynamics, mapping it to a quantum problem and finding the exact solution for the joint eigenvalue distribution. The paper concludes with the exact results for the unbiased ensemble, including the average of Schur's polynomials and the moments of the eigenvalues and the trace.This paper presents a Dyson Brownian Motion (DBM) model for weak measurements in chaotic quantum systems. The authors study the stochastic Schrödinger equation resulting from continuous monitoring of a random Hermitian operator drawn from the Gaussian Unitary Ensemble (GUE). Due to unitary invariance, the dynamics of the eigenvalues of the density matrix decouple from that of the eigenvectors, and are described by stochastic equations. In the presence of an extra dephasing term, the density matrix has a stationary distribution that matches the inverse-Marchenko–Pastur distribution in the large size limit. In the case of perfect measurements, purification eventually occurs, and the authors focus on finite-time dynamics. They find an exact solution for the joint probability distribution of the eigenvalues at each time t and for each size n. Two relevant regimes emerge: at short times, the spectrum is in a Coulomb gas regime with a well-defined continuous spectral distribution in the large n limit. In this case, all moments of the density matrix become self-averaging and the entanglement spectrum can be exactly characterized. At large times, the eigenvalues are exponentially separated, but fluctuations play an essential role. The authors are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime. The paper also discusses the dynamics of the spectrum, mapping it to unconstrained variables and deriving the stationary state at x > 0 and large n. The authors show that the stationary distribution is the inverse-Wishart distribution, and derive the Von Neumann entropy in the stationary state. They also study the finite time dynamics, mapping it to a quantum problem and finding the exact solution for the joint eigenvalue distribution. The paper concludes with the exact results for the unbiased ensemble, including the average of Schur's polynomials and the moments of the eigenvalues and the trace.