Suppose that X is a right process which is associated with a semi-Dirichlet form(ε,D(ε)) on L^2(E;m).Let J be the jumping measure of(ε,D(ε)) satisfying J(E×E-d) <∞.Let u ∈ D(ε)_b:= D(ε)∩ L~∞(E;m),we have the following Fukushima's decomposition u(X_t)-u(X_0) =M_t^u+N_t^u.Define P_t^uf(x)=E_x[e^(N_t^u)f(X_t)].Let Q^u(f,g) =ε(f,g)+ε(u,fg)for f,g∈ D(ε)_b.In the first part,under some assumptions we show that(Q^u,D(ε)_b) is lower semi-bounded if and only if there exists a constant α_0≥0 such that ‖P_t^u‖2≤e^(α_0^t) for every t>0.If one of these assertions holds,then(P_t^u)t≥0 is strongly continuous on L^2(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 A_t be a local continuous additive functional with zero quadratic variation.In the second part,we get the representation of A_t and give two sufficient conditions for P_t^A f(x) = E_x[e^(A_t) f(X_t)]to be strongly continuous.
