2012/01/26 by Neil O'Connell, Janosch Ortmann, O'Connell, Neil +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1201.5586
arxiv created 2012/09/10 · arxiv updated 2012/09/11
Motivated by recent developments on random polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. It was shown by Harrison and Williams (1987) that RBM in a polyhe- dral domain has an invariant measure in product form if a certain skew-symmetry condition holds. We show that (modulo technical assumptions) the generalised RBM has an invariant measure in product form if (and essentially only if) the same skew-symmetry condition holds, independent of the choice of potential. The invari- ant measure of course does depend on the potential. Examples include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.