Sur un théorème spectral et son application aux noyaux lipchitziens

Hennion, Hubert

doi:10.1090/S0002-9939-1993-1129880-8

Sur un théorème spectral et son application aux noyaux lipchitziens
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Proc. Amer. Math. Soc. 118 (1993), 627-634 Request permission

Abstract:

The aim of this note is to point out that the Ionescu Tulcea and Marinescu theorem can be reinforced, using a Nussbaum formula for the essential spectral radius of an operator. In this stronger version, this theorem is suitable for the spectral analysis of lipschitzian, positive, not necessarily markovian kernels, Ruelle theorem follows. As an application to Probability Theory, a large deviation theorem is proved.

References

Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674, DOI 10.1007/978-1-4684-9172-2
Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/61), 22–130. MR 209909, DOI 10.1007/BF01343363
Wolfgang Doeblin and Robert Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France 65 (1937), 132–148 (French). MR 1505076, DOI 10.24033/bsmf.1270

Linear operators. Part

Y. Guivarc’h and J. Hardy, Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), no. 1, 73–98 (French, with English summary). MR 937957
Y. Guivarc’h and A. Raugi, Products of random matrices: convergence theorems, Random matrices and their applications (Brunswick, Maine, 1984) Contemp. Math., vol. 50, Amer. Math. Soc., Providence, RI, 1986, pp. 31–54. MR 841080, DOI 10.1090/conm/050/841080
Hubert Hennion, Dérivabilité du plus grand exposant caractéristique des produits de matrices aléatoires indépendantes à coefficients positifs, Ann. Inst. H. Poincaré Probab. Statist. 27 (1991), no. 1, 27–59 (French, with English summary). MR 1098563
C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140–147 (French). MR 37469, DOI 10.2307/1969514
Gerhard Keller, Generalized bounded variation and applications to piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete 69 (1985), no. 3, 461–478. MR 787608, DOI 10.1007/BF00532744
Émile Le Page, Théorèmes limites pour les produits de matrices aléatoires, Probability measures on groups (Oberwolfach, 1981) Lecture Notes in Math., vol. 928, Springer, Berlin-New York, 1982, pp. 258–303 (French). MR 669072
S. V. Nagaev, Some limit theorems for stationary Markov chains, Teor. Veroyatnost. i Primenen. 2 (1957), 389–416 (Russian, with English summary). MR 0094846
Roger D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473–478. MR 264434
David Ruelle, Thermodynamic formalism, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. The mathematical structures of equilibrium statistical mechanics. MR 2129258, DOI 10.1017/CBO9780511617546
David Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 175–193 (1991). MR 1087395, DOI 10.1007/BF02699133
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469