A limit theorem for perturbed operator semigroups with applications to random evolutions

Let U(t) and S(t) be strongly continuous contraction semigroups on a Banach space L with infinitesimal operators A and B, respectively. Suppose the closure of A + αB generates a semigroup Tα(t). The behavior of Tα(t) as α goes to infinity is examined. In particular, suppose S(t) converges strongly to P. If the closure of PA generates a semigroup T(t) on R(P), then Tα(t) goes to T(t) on R(P). If PA = 0 and if BVf = −f for feN(P), conditions are given that imply Tα(αt) converges on R(P) to a semigroup generated by the closure of PAVA. The results are used to obtain new and known limit theorems for random evolutions, which in turn give approximation theorems for diffusion processes.