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**Random walks on discrete groups: Boundary and entropy.**
*(English)*
Zbl 0641.60009

This article is of a dual character; it is an introductory survey of the subject described in the title, and also contains many important new results [most of them were announced in a short note by the authors, Sov. Math. Dokl. 20, 1170-1173 (1979); translation from Dokl. Akad. Nauk SSSR 249, 15-18 (1979; Zbl 0442.60069)]. It shows the depth of the relationship between probabilistic properties of random walks and algebraic characteristics of groups (amenability, exponential growth, etc.).

Consider a nondegenerate probability measure \(\mu\) defined on a discrete group G; nondegeneracy means that the support of \(\mu\) generates G as a semigroup. The main results are the following:

(1) The harmonic bounded functions (that is, the solutions of the equation \(h(x)=\sum_{y\in G}h(xy)\mu (y)\) are constant if and only if the entropy is null. The entropy is defined, in the sense of A. Avez, as the number \[ \lim (-1/n)\sum_{x\in G}\mu^{*n}(x) Log \mu^{*n}(x). \] A version of the classical Shannon-McMillan-Breiman theorem is given for this notion of entropy: This entropy is also a limit along the paths of the random walk generated by \(\mu\). In the equivalence stated above, the sufficiency was originally proved by A. Avez [C. R. Acad. Sci., Paris, Sér. A 279, 25-28 (1974; Zbl 0292.60100)]; the necessity and the version of the Shannon-McMillan-Breiman theorem was also proved by the reviewer using the subadditive ergodic theorem [Astérisque 74, 183-201 (1980; Zbl 0446.60059)]. But here the authors use direct and probably simpler arguments.

(2) By an explicit construction the following conjecture of Furstenberg is proved: On every amenable group there exists a probability measure, which can be taken to be symmetric, but this condition is not required, for which harmonic bounded functions are constant. The proof uses only Følner’s condition. This fact was also proved by J. Rosenblatt [Math. Ann. 257, 31-42 (1981; Zbl 0451.60011)], but his proof seems more intricate.

(3) To produce various examples the authors consider the groups which are semidirect products of \({\mathbb{Z}}^ d\) and the Boolean group of the finite subsets of \({\mathbb{Z}}^ d\). These groups are solvable and have exponential growth. Playing with the dimension d the authors exhibit the following phenomena: exponential growth does not imply, in general, nullity of the entropy (counterexample to a conjecture of Avez); the conjecture of Furstenberg, considered above, is not true for measures having finite support; the bounded harmonic functions are not necessarily the same for \(\mu\) and its symmetric.

(4) Relations between amenability, the growth of Følner sets, and the spectral measure associated with a symmetric \(\mu\) are studied. Notions of \(\mu\)-entropy for G-spaces and of Radon-Nikodým transforms are defined (the precise results cannot be stated here).

The paper ends with various remarks, historical comments and suggestions about the future of this field of research.

Consider a nondegenerate probability measure \(\mu\) defined on a discrete group G; nondegeneracy means that the support of \(\mu\) generates G as a semigroup. The main results are the following:

(1) The harmonic bounded functions (that is, the solutions of the equation \(h(x)=\sum_{y\in G}h(xy)\mu (y)\) are constant if and only if the entropy is null. The entropy is defined, in the sense of A. Avez, as the number \[ \lim (-1/n)\sum_{x\in G}\mu^{*n}(x) Log \mu^{*n}(x). \] A version of the classical Shannon-McMillan-Breiman theorem is given for this notion of entropy: This entropy is also a limit along the paths of the random walk generated by \(\mu\). In the equivalence stated above, the sufficiency was originally proved by A. Avez [C. R. Acad. Sci., Paris, Sér. A 279, 25-28 (1974; Zbl 0292.60100)]; the necessity and the version of the Shannon-McMillan-Breiman theorem was also proved by the reviewer using the subadditive ergodic theorem [Astérisque 74, 183-201 (1980; Zbl 0446.60059)]. But here the authors use direct and probably simpler arguments.

(2) By an explicit construction the following conjecture of Furstenberg is proved: On every amenable group there exists a probability measure, which can be taken to be symmetric, but this condition is not required, for which harmonic bounded functions are constant. The proof uses only Følner’s condition. This fact was also proved by J. Rosenblatt [Math. Ann. 257, 31-42 (1981; Zbl 0451.60011)], but his proof seems more intricate.

(3) To produce various examples the authors consider the groups which are semidirect products of \({\mathbb{Z}}^ d\) and the Boolean group of the finite subsets of \({\mathbb{Z}}^ d\). These groups are solvable and have exponential growth. Playing with the dimension d the authors exhibit the following phenomena: exponential growth does not imply, in general, nullity of the entropy (counterexample to a conjecture of Avez); the conjecture of Furstenberg, considered above, is not true for measures having finite support; the bounded harmonic functions are not necessarily the same for \(\mu\) and its symmetric.

(4) Relations between amenability, the growth of Følner sets, and the spectral measure associated with a symmetric \(\mu\) are studied. Notions of \(\mu\)-entropy for G-spaces and of Radon-Nikodým transforms are defined (the precise results cannot be stated here).

The paper ends with various remarks, historical comments and suggestions about the future of this field of research.

### MSC:

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

22D40 | Ergodic theory on groups |

28D20 | Entropy and other invariants |

43A07 | Means on groups, semigroups, etc.; amenable groups |

20F99 | Special aspects of infinite or finite groups |

60G50 | Sums of independent random variables; random walks |

60J50 | Boundary theory for Markov processes |