ABSTRACT
In this chapter we deal with the invariant measures of the semigroup {T (t)} associated with the operator A defined on smooth functions by
Au(x) = N∑
qij(x)Diju(x) + N∑ j=1
bj(x)Dju(x), x ∈ R N . (8.0.1)
We recall that, according to Remark 2.2.10, {T (t)} is a semigroup of contractions in Bb(R
N ). Throughout the chapter, for any measure µ, we write Lpµ for L
p(RN , µ) and
we denote by || · ||p the norm of Lpµ. Moreover, we write W k,p µ for W
k,p(RN , µ) for any k ∈ N and any p ∈ [1,+∞]. By definition, a probability measure µ is an invariant measure for {T (t)} if∫
T (t)f dµ =
f dµ, (8.0.2)
for any f ∈ Bb(RN ) or, equivalently, for any f ∈ C∞c (R N ) (see Lemma 8.1.3).
