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We provide an integral formula for the Poisson kernel of halfspaces for Brownian motion in real hyperbolic space H n . This enables us to find asymptotic properties of the kernel. We also show convergence to the Poisson kernel of the whole space H n . For n = 3, 4 or 6 we compute explicit formulas for the Poisson kernel itself. 2000 Mathematics Subject Classification: Primary: 60J45. Secondary: 58J65.doi:10.4171/rmi/487 fatcat:6zcy4a7q6jczzkpytvq53zsyji