×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
References:
[1] Arendt, W., Gaussian estimates and interpolation of the spectrum in Lp, Differential integral equations, 7, 1153-1168, (1994) · Zbl 0827.35081
[2] Bakry, D., L’hypercontractivité et son utilisation en théorie des semi-groupes, (), 1-114 · Zbl 0856.47026
[3] Bogachev, V.; Röckner, M.; Schmuland, B., Generalized mehler semigroups and applications, Probab. theory related fields, 105, 193-225, (1996) · Zbl 0849.60066
[4] Chojnowska-Michalik, A.; Goldys, B., Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces, Probab. theory related fields, 102, 331-356, (1995) · Zbl 0859.60057
[5] Chojnowska-Michalik, A.; Goldys, B., Nonsymmetric ornstein – uhlenbeck semigroup as second quantized operator, J. math. Kyoto univ., 36, 481-498, (1996) · Zbl 0882.47013
[6] A. Chojnowska-Michalik, and, B. Goldys, Symmetric Ornstein-Uhlenbeck generators: characterizations and identification of domains, preprint, 2001. · Zbl 0997.47036
[7] Chojnowska-Michalik, A.; Goldys, B., Generalized ornstein – uhlenbeck semigroups: littlewood – paley – stein inequalities and the P. A. Meyer equivalence of norms, J. funct. anal., 182, 243-279, (2001) · Zbl 0997.47036
[8] Da Prato, G.; Lunardi, A., On the ornstein – uhlenbeck operator in spaces of continuous functions, J. funct. anal., 131, 94-114, (1995) · Zbl 0846.47004
[9] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, Encyclopedia of mathematics and its applications 44, (1992), Cambridge University Press Cambridge · Zbl 0761.60052
[10] Da Prato, G.; Zabczyk, J., Ergodicity for infinite dimensional systems, (1996), Cambridge University Press Cambridge · Zbl 0849.60052
[11] Davies, E.B., Heat kernels and spectral theory, (1989), Cambridge University Press Cambridge · Zbl 0699.35006
[12] Davies, E.B.; Simon, B., L1 properties of intrinsic Schrödinger semigroups, J. funct. anal., 65, 126-146, (1986) · Zbl 0613.47039
[13] Engel, K.; Nagel, R., One-parameters semigroup for linear evolution equations, (2000), Springer Berlin
[14] Fuhrman, M., Analyticity of transition semigroups and closability of bilinear forms in Hilbert spaces, Studia math., 115, 53-71, (1995) · Zbl 0830.47033
[15] Goldys, B., On analyticity of ornstein – uhlenbeck semigroups, Rend. mat. acc. lincei, s. 9, 10, 131-140, (1999) · Zbl 1026.47506
[16] Hörmander, L., Hypoelliptic second order differential operators, Acta math., 119, 147-171, (1967) · Zbl 0156.10701
[17] Lanconelli, E.; Polidoro, S., On a class of hypoelliptic evolution operators, Rend. sem. mat. univ. politec Torino pdes, 52, 26-63, (1994) · Zbl 0811.35018
[18] Lunardi, A., On the ornstein – uhlenbeck operator in L2 spaces with respect to invariant measures, Trans. amer. math. soc., 349, 155-169, (1997) · Zbl 0890.35030
[19] Lunardi, A., Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients, Ann. scuola norm. sup. Pisa, 24, 133-164, (1997) · Zbl 0887.35062
[20] Metafune, G., Lp-spectrum of ornstein – uhlenbeck operators, Ann. scuola norm. sup. Pisa, 30, 97-124, (2001) · Zbl 1065.35216
[21] Metafune, G.; Prüss, J.; Rhandi, A.; Schnaubelt, R., The domain of the ornstein – uhlenbeck operator on an Lp space with an invariant measure, Ann. scuola norm. sup. Pisa, 31, (2002) · Zbl 1170.35375
[22] Meyer, P.A., Note sur le processus d’ornstein – uhlenbeck, Séminaire de probabilités XVI, (1982), Springer Berlin, p. 95-133
[23] Priola, E., The fundamental solution for a degenerate parabolic Dirichlet problem, (2000), University of Torino Turin, p. 95-133
[24] Röckner, M., Lp analysis of finite and infinite dimensional diffusion operators, (), 65-116 · Zbl 0944.60078
[25] Seidman, T., How violent are fast controls?, Control signals systems, 1, 89-95, (1988) · Zbl 0663.49018
[26] Zabczyk, J., Mathematical control theory: an introduction, (1992), Birkhäuser Basel · Zbl 1071.93500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.