On Strong Convergence to Ergodic Projection for Perturbed Substochastic Semigroups

Let (U(t))(t >= 0) be a substochastic Co-semigroup on L-1 space with generator T Let K :D(T) -> L-1 be positive, T-bounded and such that lim(lambda ->+infinity) r(sigma) (K (lambda-T)(-1)) < 1 and integral(Omega) Tf + Kf <= 0 (sic) f is an element of D(T) boolean AND L-+(1). Let (V (t))t >= 0 be the substochastic C-0-semigroup generated by T + K. We show that if some remainder term R-n (t) of the Dyson-Phillips expansion of (V(t))(t >= 0) depends continuously on t in the uniform topology then V(t) converges strongly to its ergodic projection as t -> +infinity The proof relies on a ``0-2'' law for C-0-semigroups by G. Greiner. We characterize also the existence of nontrivial equilibrium points for (V(t))(t >= 0). If the latter is stochastic (i.e. mass-preserving on the positive cone) then, by using compactness arguments, we derive the strong convergence to ergodic projection from a result by K. Pichor and R. Rudnicki relying on ``partially integral'' techniques.