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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 $$\mathbb{C}$$.

##### MSC:
 47D07 Markov semigroups and applications to diffusion processes
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