Convergence of local eigenvector processes of generalized Wigner matrices

We prove convergence of eigenvector processes of the form $(\sqrt{N}\langle \mathbf{u}_k,A_t\mathbf{u}_k\rangle)_{t\in[0,1]}$ where $\mathbf{u}_k$ is a bulk eigenvector of generalized Wigner matrices and $(A_t)$ a family of symmetric matrices with bounded norm and H\"{o}lder regularity. We give explicit examples of limiting processes and prove that a large class of Gaussian process with H\"{o}lder-continuous covariance function can be obtained as such a limit using its Karhunen--Lo\`eve expansion. The proof is based on the multi-dimensional convergence proved Benigni and Cipolloni (2024) and a tightness criterion proved using H\"{o}lder regularity of the observables.