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Spectrum of Ornstein-Uhlenbeck operators in \(L ^{p}\) spaces with respect to invariant measures. (English) Zbl 1027.47036
Let \(A:= \sum q_{ij} D_{ij}+ \sum b_{ij} X_j D_i\) denote a generator of an Ornstein-Uhlenbeck process on a finite-dimensional real vector space \(\mathbb{R}^N\), and denote by \((T(t))_{t\geq 0}\) the corresponding Markov semigroup. Put \(B= (b_{ij})\). If \(\text{Spec}(B)\subseteq \{\text{Re }\lambda< 0\}\), i.e. \((e^{-tB})_{t> 0}\) is contractive, then there exists an invariant measure denoted by \(\mu\). \((T(t))\) defines constraction operator semigroups on \(L^p_\mu= L^p(\mathbb{R}^N, \mu)\) for \(1\leq p< \infty\), with infinitesimal generators \((A_p, D_p)\). The aim of the paper under review is to determine the spectrum of \((A_p, D_p)\).
For \(1< p<\infty\), the spectrum is discrete (since \(A_p\) has a compact resolvent) and is determined by the spectrum of \(B\), \(\text{Spec}(A_p)= \{\sum n_i\lambda_i: n_i\in \mathbb{N}\}\) where \(\text{Spec}(B)= \{\lambda_i\}\). The eigenfunctions are polynomials with span dense in \(L^p_\mu\) (cf. Theorem 3.1). In fact, \(\text{Spec}(A_p)\) depends only on the drift term \(L\) of \(A\), an observation which allows (in Section 4) to determine the algebraic multiplicity of eigenvalues \(\gamma\) of \(A_p\).
Section 5 shows that the situation is completely different for \(p= 1\). In this case (Theorem 5.1), \(\text{Spec}(A_p)\) coincides with the left half-plane of \(\mathbb{C}\).

47D07 Markov semigroups and applications to diffusion processes
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