Perturbation of analytic operators and temporal regularity of discrete heat kernels

• Autorzy: Sönke Blunck
• Języki publikacji: EN

Abstrakty

EN

In analogy to the analyticity condition $∥ Ae^{tA}∥ ≤ Ct^{-1}$, t > 0, for a continuous time semigroup $(e^{tA})_{t ≥ 0}$, a bounded operator T is called analytic if the discrete time semigroup $(T^n)_{n ∈ ℕ}$ satisfies $∥ (T-I)T^{n}∥ ≤ Cn^{-1}$, n ∈ ℕ. We generalize O. Nevanlinna's characterization of powerbounded and analytic operators T to the following perturbation result: if S is a perturbation of T such that $∥ R(λ_0,T)-R(λ_0,S)∥$ is small enough for some $λ_{0} ∈ ϱ(T) ∩ ϱ(S)$, then the type $ω$ of the semigroup $(e^{t(S-I)})$ also controls the analyticity of S in the sense that $∥(S-I)S^{n}∥ ≤ C(ω+n^{-1})e^{ωn}$, n ∈ ℕ. As an application we generalize and give a simple proof of a result by M. Christ on the temporal regularity of random walks T on graphs of polynomial volume growth. On arbitrary spaces Ω of at most exponential volume growth we obtain this regularity for any powerbounded and analytic operator T on $L_{2}(Ω)$ with a heat kernel satisfying Gaussian upper bounds.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 86
• Numer: 2
• Strony: 189-201

Daty

wydano

2000

otrzymano

2000-01-11

Twórcy

autor

Sönke Blunck

• Département de Mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France

Bibliografia

• [BB] M. T. Barlow and R. F. Bass, Random walks on graphical Sierpiński carpets, in preparation.
• [C] M. Christ, Temporal regularity for random walk on discrete nilpotent groups, J. Fourier Anal. Appl., Kahane Special Issue (1995), 141-151.
• [C-SC] T. Coulhon et L. Saloff-Coste, Puissances d'un opérateur régularisant, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), 419-436.
• [H-SC] W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on graphs, Ann. Probab. 21 (1993), 673-709.
• [J] O. D. Jones, Transition probabilities for the simple random walk on the Sierpiński graph, Stochastic Process. Appl. 61 (1996), 45-69.
• [N1] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, Basel, 1993.
• [N2] O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in: Linear Operators, J. Janas, F. H. Szafraniec and J. Zemánek (eds.), Banach Center Publ. 38, Inst. Math., Polish Acad. Sci., 1997, 247-264.
• [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, 1983.