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Motivated by recent developments on random polymer models we propose ageneralisation of reflected Brownian motion (RBM) in a polyhedral domain. Thisprocess is obtained by replacing the singular drift on the boundary by acontinuous one which depends, via a potential U, on the position of the processrelative to the domain. It was shown by Harrison and Williams (1987) that RBMin a polyhe- dral domain has an invariant measure in product form if a certainskew-symmetry condition holds. We show that (modulo technical assumptions) thegeneralised RBM has an invariant measure in product form if (and essentiallyonly if) the same skew-symmetry condition holds, independent of the choice ofpotential. The invari- ant measure of course does depend on the potential.Examples include TASEP-like particle systems, generalisations of Brownianmotion with rank-dependent drift and diffusions connected to the generalisedPitman transform.

Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition