PUB - Publikationen an der Universität Bielefeld

Given a right-continuous Markov process (X t) t >= 0 on a second countable metrizable space E with transition semigroup (pt) t >= 0, we prove that there exists a sigma- finite Borel measure mu with full support on E, and a closed and densely defined linear operator (L-p; D(L-p)) generating (pt)(t) >= 0 on L-p(E; mu). In particular, we solve the corresponding Cauchy problem in L-p (E; mu) for any initial condition u is an element of D(Lp). Furthermore, for any real beta > 0 we show that there exists a generalized Dirichlet form which is associated to (e(-beta t) pt) t >= 0. If the beta-subprocess of (X-t)(t >= 0) corresponding to (e(-beta t) pt) t >= 0, beta > 0, is mu-special standard then all results from generalized Dirichlet form theory become available, and Fukushima's decomposition holds for u is an element of D (L-2). If (X-t)(t >= 0) is transient, then beta can be chosen to be zero.