A REMARK ON THE GENERATOR OF A RIGHT-CONTINUOUS MARKOV PROCESS

Abstract

Given a right-continuous Markov process (Xt)t ≥ 0 on a second countable metrizable space E with transition semigroup (pt)t ≥ 0, we prove that there exists a σ-finite Borel measure μ with full support on E, and a closed and densely defined linear operator [Formula: see text] generating (pt)t ≥ 0 on Lp (E; μ). In particular, we solve the corresponding Cauchy problem in Lp (E; μ) for any initial condition [Formula: see text]. Furthermore, for any real β > 0 we show that there exists a generalized Dirichlet form which is associated to (e-βt pt)t ≥ 0. If the β-subprocess of (Xt)t ≥ 0 corresponding to (e-βt pt)t ≥ 0, β > 0, is μ-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for [Formula: see text]. If (Xt)t ≥ 0 is transient, then β can be chosen to be zero.