Consider a family of probability measures {νξ} on a bounded open region D R d with a smooth boundary and a positive parameter set {βξ},all indexed by ξ∈D.For any starting point inside D,we run a diffusion until it first exits D,at which time it stays at the exit point ξ for an independent exponential holding time with rate βξ and then leaves ξ by a jump into D according to the distribution νξ.Once the process jumps inside,it starts the diffusion afresh.The same evolution is repeated independently each time the process jumped into the domain.The resulting Markov process is called diffusion with holding and jumping boundary(DHJ),which is not reversible due to the jumping.In this paper we provide a study of DHJ on its generator,stationary distribution and the speed of convergence.
