On generalized feynman-kac transformation for markov processes associated with semi-dirichlet forms

Suppose that X is a right process which is associated with a semi-Dirichlet form (ɛ, D(ɛ)) on L2(E; m). Let J be the jumping measure of (ɛ, D(ɛ)) satisfying J(E × E − d) < ro. Let u G D(ɛ)b: = D(ɛ) ∩ L∞ (E; m), we have the following Fukushima's decomposition ũ (Xt) − ũ (Xo) = Mu t + Nut. Define Put f (x) = Ex[eNutf (Xt)]. Let Qu(f, g) = ɛ (f, g) + ɛ (u, fg) for f, g ∈ D(ɛ)b. In the first part, under some assumptions we show that (Qu, D(ɛ)b) is lower semi-bounded if and only if there exists a constant α o ≥ 0 such that ||Ptu ||2 ≤ eα ot for every t > 0. If one of these assertions holds, then (Ptu)t≥ o is strongly continuous on L2(E; m). If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J(E × E − d) < ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of Atand give two sufficient conditions for PtAf (x) = Ex [eAtf (Xt)] to be strongly continuous.