ABSTRACT
In this chapter we present some results concerned with the Ornstein-Uhlenbeck operator, which is the prototype of an elliptic operator with unbounded coefficients. Such an operator is defined on smooth functions ϕ by
(Aϕ)(x) = 1
2 Tr (QD2ϕ(x)) + 〈Bx,Dϕ(x)〉, x ∈ RN , (9.0.1)
where Q and B are N×N constant matrices, with Q (strictly) positive definite and B 6= 0. Here, if not otherwise specified, we consider only the case when Q is strictly positive. In any case, some of the results that we present hold also when Q is a degenerate positive definite matrix and the operator A in (9.0.1) is hypoelliptic, i.e., when the matrix∫ t
esBQesB ∗
ds
is strictly positive definite for any t > 0. Such a condition can be expressed also by saying that the kernel of Q does not contain any invariant subspace of B∗ (see, e.g., [98]). Firstly, in Section 9.1, we show that an explicit formula for the Ornstein-
Uhlenbeck semigroup is available both in the nondegenerate and in the degenerate case. Having such a formula simplifies the study of the main properties of the semigroup. For instance, one can prove uniform estimates for the space derivatives of any order of the function T (t)f when f ∈ Cb(RN ) just differentiating under the integral sign. We do this in Section 9.2 in the nondegenerate case. The case of the degenerate Ornstein-Uhlenbeck operator is much more involved. It has been studied by A. Lunardi in [107]. Here, we limit ourselves to state the main results of [107]. As it has been claimed several times, the Ornstein-Uhlenbeck semigroup is
neither analytic nor strong continuous in Cb(R N ). In particular, T (t)f tends
to f in Cb(R N ) as t tends to 0+, if and only if f ∈ BUC(RN ) and f(etB·)
tends to f uniformly in RN . In Section 9.3, we deal with the invariant measure of {T (t)}. We show that
when the spectrum of the matrix B is contained in the left halfplane, the
Ornstein-Uhlenbeck semigroup admits the Gaussian measure
µ(dx) = 1√
(2pi)NdetQ∞ e−
as the (unique) invariant measure, both in the nondegenerate and in the degenerate case. Here,
Q∞ =
esBQesB ∗
ds. (9.0.2)
The assumptions on the location of the spectrum of the matrix B turns out to be also necessary to guarantee the existence of the invariant measure of {T (t)}. From the results in Chapter 8, we know that the extension of the Ornstein-
Uhlenbeck semigroup to the Lp-spaces associated with the invariant measure µ (in short Lpµ) gives rise to a strongly continuous semigroup for any p ∈ [1,+∞). Moreover, for any f ∈ Lpµ, T (t)f is still given by the same formula as in the
case when f ∈ Cb(RN ). Actually, in the nondegenerate case, {T (t)} is also analytic for any p ∈
(1,+∞). Also in this situation, having an explicit representation formula for T (t)f is of much help. Indeed, we can quite easily show that T (t) maps Lpµ into
W k,pµ (for any k ∈ N) and we can also give precise estimates on the behaviour of the space derivatives of T (t)f in Lpµ when t approaches 0
+. An important feature of the Ornstein-Uhlenbeck semigroup in Lpµ is that a
complete characterization of its infinitesimal generator Lp is available. More precisely, we show that
D(Lp) =W 2,p µ , p ∈ (1,+∞)
and that the graph norm is equivalent to the Euclidean norm ofW 2,pµ . Such a result was firstly proved by A. Lunardi and G. Da Prato in the Hilbert case, and then it has been proved by G. Metafune, D. Pallara, A. Rhandi and R. Schnaubelt for a general p. Since the Ornstein-Uhlenbeck semigroup in Lpµ is compact for any p ∈
(1,+∞), the spectrum of Lp is a discrete set. It has been completely characterized by G. Metafune, D. Pallara and E. Priola in terms of the eigenvalues λ1, . . . , λN of the matrix B. More precisely, they show that
σ(Lp) =
{ λ =
niλi : ni ∈ N ∪ {0}, i = 1, . . . , r
} .
