Markov processes on the Lipschitz boundary for the Neumann and Robin problems

We investigate the Markov process on the boundary of a bounded Lipschitz domain associated to the Neumann and Robin boundary value problems. We first construct L p -semigroups of sub-Markovian contractions on the boundary, generated by the boundary conditions, and we show that they are induced by the transition functions of the forthcoming processes. As in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected Brownian motion. The Robin problem is treated with a Kato type L p -perturbation method, using the Revuz correspondence. An exceptional (polar) set occurs on the boundary. We make the link with the Dirichlet forms approach.