A Lindbladian for holographic Brownian motion

A bstract We derive a Lindbladian description of holographic Brownian motion in the high-temperature regime. Starting from the influence functional for a trailing string endpoint, we identify the corresponding quantum master equation and prove that it is completely positive and trace-preserving. We determine the coefficients of the Lindbladian explicitly for two holographic backgrounds: the BTZ black hole and the AdS 5 black brane, restricting in the latter case to the endpoint fluctuation along the x 1 -direction. We then analyze the time evolution of phase-space moments, energy relaxation, and steady states.