The harmonic measures of Lucy Garnett. (English) Zbl 1031.58003

Candel, Alberto

Adv. Math. 176, No. 2, 187-247 (2003).

The author makes a careful study of harmonic measures on foliated spaces. This type of measures was introduced by L. Garnett [J. Funct. Anal. 51, 285-311 (1983; Zbl 0524.58026)]. Here, for any generalized Laplacian \(\Delta\) along the leaves of any foliated space \(M\), a measure \(m\) on \(M\) is said to be harmonic when \(\int_M\Delta f\cdot m=0\) for all \(f\in C_c(M)\). If \(M\) is compact, the existence of a harmonic probability measure is shown by using the local maximum principle and the Hahn-Banach theorem. Suppose that \(M\) is compact and let \(D_t\), \(t>0\), be the diffusion semigroup defined by \(\Delta\). An important property is that \(D_t\) maps continuous functions on \(M\) to continuous functions. This is a theorem in Garnett’s paper, but her arguments seem to work only when there is no holonomy. This gap is corrected in this paper with a different approach, where the Hille-Yosida theorem is the essential tool.
Several properties of harmonic measures are reconsidered here. In particular, they can be characterized by their invariance under \(D_t\), and by certain local description. This local description is used to define the modular form, which vanishes if and only if the measure is totally invariant. Inspired by the work of E. Ghys [Ann. Math. (2) 141, 387-422 (1995; Zbl 0843.57026)], the author considers the Markov process associated to \(\Delta\), which restricts to the corresponding Brownian motion on each leaf; thus he considers the space \(\Omega(M)\) of paths \(\omega:[0,\infty)\to L\) on all leaves \(L\) of \(M\) endowed with the Wiener measures. The shift operators \(\theta_t(\omega)(s)=\omega(s+t)\) define a semigroup \(\theta=\{\theta_t\mid t\geq 0\}\) of transformations of \(\Omega(M)\), which permits to study the recurrence properties of the foliated space \(M\) as if it was given by a one dimensional flow. For instance, the Wiener measures on the leaves can be combined with any measure \(m\) on \(M\) yielding a measure \(\mu\) on \(\Omega(M)\), which turns out to be \(\theta\)-invariant if and only if \(m\) is harmonic; moreover, in this case, \(m\) is ergodic if and only if \(\mu\) is ergodic with respect to \(\theta\). For harmonic probability measures, the author proves an ergodic theorem, shows that \(D_t\) is mixing on \(L^2(M)\), and obtains the ergodic decomposition in the style of Krylov and Bogoyubov.
Finally, the author studies the asymptotic properties of cocycles on \(\Omega(M)\); for instance, any differential form of degree one defines such a cocycle. This is used two show two main results: the first one states that, if \(m\) is an ergodic harmonic measure which is not totally invariant, then the von Neumann algebra of \((M,m)\) contains an essential factor of type III; and the second one shows the existence of resilient leaves under the presence of some transverse quasi-conformal structure.
The author announces other applications in forthcoming papers.

Reviewer: Jesus A.Álvarez López (Santiago de Compostela)

MSC:

58C35	Integration on manifolds; measures on manifolds
57R30	Foliations in differential topology; geometric theory
60J65	Brownian motion
58J65	Diffusion processes and stochastic analysis on manifolds

Keywords:

foliated space; harmonic measures; Brownian motion

Citations:

Zbl 0524.58026; Zbl 0843.57026

References:

[1]	J.A. Álvarez-López, A. Candel, Generic geometry of leaves, preprint.; J.A. Álvarez-López, A. Candel, Generic geometry of leaves, preprint. · Zbl 1426.57001
[2]	Bonatti, C.; Langevin, R.; Moussu, R., Feullitages de \(CP(n)\): de l’holonomie hyperbolique pour les minimaux exceptionnels, Inst. Hautes Études Sci. Publ. Math., 75, 123-134 (1992) · Zbl 0782.32023
[3]	A. Candel, Asymptotic behavior of solutions to differential equations in the complex domain, in preparation.; A. Candel, Asymptotic behavior of solutions to differential equations in the complex domain, in preparation.
[4]	A. Candel, Uniformization of euclidean surface laminations, preprint.; A. Candel, Uniformization of euclidean surface laminations, preprint. · Zbl 0785.57009
[5]	A. Candel, L. Conlon, Foliations I, in: Graduate Studies in Mathematics, Vol. 23, Amer. Math. Soc., Providence, RI, 2000.; A. Candel, L. Conlon, Foliations I, in: Graduate Studies in Mathematics, Vol. 23, Amer. Math. Soc., Providence, RI, 2000. · Zbl 0936.57001
[6]	Chavel, I., Eigenvalues in Riemannian Geometry (1984), Academic Press: Academic Press Orlando, FL · Zbl 0551.53001
[7]	Cheng, S. Y.; Li, P.; Yau, S.-T., On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math., 103, 1021-1063 (1980) · Zbl 0484.53035
[8]	Chung, K. L., A Course in Probability Theory (1974), Academic Press: Academic Press San Diego · Zbl 0159.45701
[9]	Connes, A., Cyclic cohomology and the transverse fundamental class of a foliation, (Araki, H.; Effros, E., Geometric Methods in Operator Algebras (Kyoto, 1983). Geometric Methods in Operator Algebras (Kyoto, 1983), Pitman Research Notes in Mathematics Series, Vol. 123 (1986), Longman Scientific Technical: Longman Scientific Technical Harlow), 52-144 · Zbl 0647.46054
[10]	G. Duminy, L’invariant de Godbillon-Vey d’un feuilletage se localise sur las feuilles ressort, preprint.; G. Duminy, L’invariant de Godbillon-Vey d’un feuilletage se localise sur las feuilles ressort, preprint.
[11]	Dunford, N.; Schwarz, J., Linear Operators, Part I: General Theory (1958), Interscience Publishers Inc: Interscience Publishers Inc New York · Zbl 0084.10402
[12]	Dynkin, E., Theory of Markov Processes (1960), Pergamon Press: Pergamon Press New York · Zbl 0096.11704
[13]	Dynkin, E., Markov Processes, Grundlehren der mathematischen Wissenschaften, Vols. 121, 122 (1965), Springer: Springer Berlin · Zbl 0132.37901
[14]	Garnett, L., Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., 51, 285-311 (1983) · Zbl 0524.58026
[15]	Ghys, É.; Langevin, R.; Walczak, P., Entropie geometrique des feuilletages, Acta Math., 160, 105-142 (1988) · Zbl 0666.57021
[16]	Ghys, É., Topologie des feuilles génériques, Ann. Math., 141, 387-422 (1995) · Zbl 0843.57026
[17]	É. Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, Vol. 8, Soc. Math. France, Paris, 1999, pp. 49-95.; É. Ghys, Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, Vol. 8, Soc. Math. France, Paris, 1999, pp. 49-95. · Zbl 1018.37028
[18]	Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0562.35001
[19]	Hille, E.; Phillips, R., Functional Analysis and Semi-groups. Functional Analysis and Semi-groups, Colloquium Publications, Vol. 31 (1957), American Mathematical Society: American Mathematical Society New York · Zbl 0078.10004
[20]	Hurder, S.; Katok, A., Secondary classes and transverse measure theory for a foliation, Bull. Amer. Math. Soc., 11, 347-350 (1984) · Zbl 0593.58015
[21]	Hurder, S.; Katok, A., Ergodic theory and Weil measures for foliations, Ann. Math., 126, 221-275 (1987) · Zbl 0645.57021
[22]	Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), North-Holland/Kodansha, Elsevier Science Publishers: North-Holland/Kodansha, Elsevier Science Publishers Tokyo · Zbl 0684.60040
[23]	Itô, K.; McKean, H. P., Diffusion Processes and their Sample Paths (1974), Springer: Springer Berlin · Zbl 0285.60063
[24]	Kaimanovich, V., Brownian motion on foliations: entropy, invariant measures, mixing, Funct. Anal. Appl., 22, 326-328 (1989) · Zbl 0675.58039
[25]	Kingman, J. F.C., The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. B, 30, 499-510 (1968) · Zbl 0182.22802
[26]	Krieger, W., On ergodic flows and the isomorphism of factors, Math. Ann., 223, 19-70 (1976) · Zbl 0332.46045
[27]	Kryloff, N.; Bogoliouboff, N., La théorie générale de la measure dans son application à l’étude des systèmes dynamiques de la mécanique non linéare, Ann. Math., 38, 65-113 (1937) · JFM 63.1002.01
[28]	Ladyzhenskaya, O.; Ural’tseva, N., Linear and Quasilinear Elliptic Equations (1968), Academic Press: Academic Press New York · Zbl 0164.13002
[29]	Nemytskii, N.; Stepanov, V., Qualitative Theory of Differential Equations (1989), Dover Publications: Dover Publications New York
[30]	Orey, S., Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities, Van Nostrand Reinhold Mathematical Studies, Vol. 34 (1971), Van Nostrand Reinhold Co: Van Nostrand Reinhold Co New York · Zbl 0295.60054
[31]	Ornstein, D.; Sucheston, L., An operator theorem on \(L_1\) convergence to zero with applications to Markov kernels, Ann. Math. Statist., 41, 1631-1639 (1970) · Zbl 0284.60068
[32]	Oxtoby, J., Ergodic sets, Bull. Amer. Math. Soc., 58, 116-136 (1952) · Zbl 0046.11504
[33]	Port, S.; Stone, Ch., Brownian Motion and Classical Potential Theory (1978), Academic Press: Academic Press New York · Zbl 0413.60067
[34]	Sacksteder, R., Foliations and pseudogroups, Amer. J. Math., 87, 79-102 (1965) · Zbl 0136.20903
[35]	Stroock, D.; Varadhan, S. R., Multidimensional Diffusion Processes, Grundlehren der mathematischen Wissenschaften, Vol. 233 (1979), Springer: Springer New York · Zbl 0426.60069
[36]	Weil, A., Collected Papers I (1979), Springer: Springer New York · Zbl 0424.01027
[37]	Winkler, W., A note on continuous parameter zero-two law, Ann. Probab., 1, 341-344 (1973) · Zbl 0261.60051
[38]	Yosida, K., Functional Analysis (1972), Springer: Springer New York · Zbl 0152.32102