×

On the perturbation of positive semigroups. (English) Zbl 1385.47016

Motivated by an application to heat kernel estimates, the authors prove a domination theorem for perturbations of positive operator semigroups on \(L^2\)-spaces. For the sake of simplicity, let us discuss the main result of the paper here only over finite measure spaces, although the paper deals with the more general \(\sigma\)-finite case. Let \((\Omega,\mu)\) be a finite measure space and let \(T_0\) and \(T\) be two positive \(C_0\)-semigroups on \(L^2(\Omega,\mu)\) with generators \(A_0\) and \(A\), respectively; for what follows, it is convenient to think of \(A\) as a perturbation of \(A_0\). The authors consider the question whether the estimate \(T(t) \leq T_0(t) + tc \; {1}_\Omega \otimes {1}_\Omega\) holds for all \(t \geq 0\) and a constant \(c \in \mathbb{R}\). The main result of the paper asserts that the validity of this estimate can be characterised by a condition on the semigroup generators or, alternatively, by a condition on the resolvent of the generators. As a consequence of their perturbation theorem, the authors obtain heat kernel estimates which generalise results from [M.T.Barlow et al., J. Reine Angew.Math.626, 135–157 (2009; Zbl 1158.60039)].

MSC:

47D06 One-parameter semigroups and linear evolution equations
47B65 Positive linear operators and order-bounded operators
47B34 Kernel operators
47A55 Perturbation theory of linear operators

Citations:

Zbl 1158.60039
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramovich, YA; Aliprantis, CD, An invitiation to operator theory, No. 50 (2002), Procivence
[2] Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H., Moustakas, U., Nagel, R., Neubrander, F.: One-parameter Semigroups of Positive Operators. Springer, Berlin (1986) · Zbl 0585.47030
[3] Barlow, M.T., Bass, R.F., Chen, Z.Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963-1999 (2009) · Zbl 1166.60045 · doi:10.1090/S0002-9947-08-04544-3
[4] Barlow, M.T., Grigor’yan, A., Kumagai, T.: Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626, 135-157 (2009) · Zbl 1158.60039
[5] Chen, Z.Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Prob. Theory Relat. Fields 140(1-2), 277-317 (2008) · Zbl 1131.60076
[6] Foondun, M.: Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part. Electron. J. Prob. 14(11), 314-340 (2009) · Zbl 1190.60069
[7] Grigor’yan, A., Hu, J.: Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174(1), 81-126 (2008) · Zbl 1154.47034 · doi:10.1007/s00222-008-0135-9
[8] Meyer, P.-A.: Renaissance, recollectement, mélanges, ralentissement de processus de Markov. Ann. Inst. Fourier 25(3-4), 464-497 (1975) · Zbl 0304.60041
[9] Ouhabaz, E.M.: Analysis of Heat Equations on Domains. Princeton University Press, Princeton (2005) · Zbl 1082.35003
[10] Segal, I.E.: Equivalences of measure spaces. Am. J. Math. 73(2), 275-313 (1951) · Zbl 0042.35502 · doi:10.2307/2372178
[11] Stollmann, P., Voigt, J.: Perturbation of dirichlet forms by measures. Potential Anal. 5, 109-138 (1996) · Zbl 0861.31004 · doi:10.1007/BF00396775
[12] Wingert, D.: Evolutionsgleichungen und obere Abschätzungen an die Lösungen des Anfangswertproblems. Doctoral Thesis (2011). http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-107849. Accessed 29 Oct 2014
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.