ABSTRACT
Remark 11.0.1 Differently from what happens in the whole of RN , there never exists an invariant measure µ associated with the semigroup {T (t)} considered in this chapter. Indeed, if this were the case, then∫
T (t)dµ =
∫ Ω
fdµ, (11.0.4)
for any f ∈ Cb(Ω). But, according to Corollary 11.2.2, it holds that 0 ≤ T (t)1l ≤ 1l for any t > 0. Hence, from (11.0.4), it should follow that T (t)1l ≡ 1l which is a contradiction.
