In analogy to the analyticity condition $\parallel A{e}^{tA}\parallel \le C{t}^{-1}$, t > 0, for a continuous time semigroup ${\left(e^{tA}\right)}_{t\ge 0}$, a bounded operator T is called analytic if the discrete time semigroup ${\left(T^n\right)}_{n\in \mathbb{N}}$ satisfies $\parallel (T-I)T^n\parallel \le C{n}^{-1}$, 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 $\parallel R({\lambda }_0,T)-R({\lambda }_0,S)\parallel$ is small enough for some ${\lambda }_0\in \varrho \left(T\right)\cap \varrho \left(S\right)$, then the type $\omega$ of the semigroup $\left(e^{t(S-I)}\right)$ also controls the analyticity of S in the sense that $\parallel (S-I)S^n\parallel \le C(\omega +n^{-1})e^{\omega n}$, n ∈ ℕ. As an application we generalize...

The aim of this note is to describe the Poisson boundary of the group of invertible triangular matrices with coefficients in a number field. It generalizes to any dimension and to any number field a result of Brofferio concerning the Poisson boundary of random rational affinities.

Nous montrons que toute probabilité de transition sur un espace mesurable correspondant à une chaîne de Markov vérifiant la condition de récurrence de Harris, admet au moins un opérateur potentiel positif ; à partir de là, nous développons une théorie du “potentiel logarithmique” pour ces probabilités de transition, en étudiant notamment de manière approfondie un cône de fonctions dites spéciales.

In this paper we introduce a ⊗-operation over Markov transition matrices, in the context of subshift of finite type, reproducing symbolic properties of the iterates of the critical point on a one-parameter family of unimodal maps. To the *-product between kneading sequences we associate a ⊗-product between the corresponding Markov matrices.

Let P be a Markov kernel on a measurable space E with countably generated σ-algebra, let w:E→[1, +∞[ such that Pw≤Cw with C≥0, and let ${ℬ}_w$ be the space of measurable functions onE satisfying ‖f‖w=sup{w(x)−1|f(x)|, x∈E}<+∞. We prove that Pis quasi-compact on $({ℬ}_w,\parallel ·{\parallel }_w)$ if and only if, for all $f\in {ℬ}_w$, ${\left(\frac{1}{n}{\sum }_{k=1}^n{P}^{k}f\right)}_n$ contains a subsequence converging in ${ℬ}_w$ toΠf=∑di=1μi(f)vi, where the vi’s are non-negative bounded measurable functions on E and the μi’s are probability distributions on E. In particular, when the space of...

We define two splitting procedures of the interval [0,1], one using uniformly distributed points on the chosen piece and the other splitting a piece in half. We also define two procedures for choosing the piece to be split; one chooses a piece with a probability proportional to its length and the other chooses each piece with equal probability. We analyse the probability distribution of the lengths of the pieces arising from these procedures.

We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny's constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state...

We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on ${\mathbb{Z}}^{+}=\{0,1,2,\cdots \}$, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different from the previously studied cases. Our method is based on a martingale technique—the method of Lyapunov functions.

In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields89 (1991) 117–129). This extends also the link at the probabilistic level between riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol...

In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space–time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.