Brosamler's formula revisited and extensions

Brosamler's formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain $D\subset R^n$ in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the reflecting Brownian motion). We give new proofs of Brosamler's formula using (path trajectories of) stochastic processes. More precisely, we use a connection between the Dirichlet and the Neumann boundary problems, and the explicit description of the reflecting Brownian motion and its boundary local time in terms of the free Brownian motion. The results are obtained in the case of the Euclidean unit ball in any dimension and in the case of smooth $C^{1,\alpha }$ planar simply connected domains, for continuous boundary data, and then extended to the case of bounded measurable data, respectively integrable boundary data. A new Brosamler-type formula in terms of the free Brownian motion is also given.