We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.
We construct the analogue of the local time -- at a fixed point $x$ -- for Markov processes indexed by Levy trees. We start by proving that Markov processes indexed by Levy trees satisfy a special Markov property which can be thought as a spatial version of the classical Markov property. Then, we construct the analogue of the local time by an approximation procedure and we characterize the support of its Lebesgue-Stieltjes measure. We also give an equivalent construction in terms of a special family of exit local times. Finally, combining these results, we show that the points at which the Mar…
A family of Feller branching diffusions $Z^x$, $x \ge 0$, with nonlinear drift and initial value $x$ can, with a suitable coupling over the {\em ancestral masses} $x$, be viewed as a path-valued process indexed by $x$. For a coupling due to Dawson and Li, which in case of a linear drift describes the corresponding Feller branching diffusion, and in our case makes the path-valued process Markovian, we find an SDE solved by $Z$, which is driven by a random point measure on excursion space. In this way we are able to identify the infinitesimal generator of the path-valued process. We also establi…
We study continuous processes indexed by a special family of graphs. Processes indexed by vertices of graphs are known as probabilistic graphical models. Burdzy and Pal in their paper proposed a continuous version of graphical models -- processes indexed by time-like graphs. We construct a more general family of continuous processes on a wider family of graphs, find new properties and solve the conjecture of uniqueness of the distribution for the process indexed by graphs with infinite number of vertices. We present new results showing connections to the stochastic heat equation, Markov random…
We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides wi…
We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly Hölder continuous in space and time. We calculate the respective Hölder exponents, which extend the well-known results on the Hölder exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the anal…
Consider a continuous time particle system $η^t=(η^t(k),k\in \mathbb{L})$, indexed by a lattice $\mathbb{L}$ which will be either $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, a segment $\{1,\cdots, n\}$, or $\mathbb{Z}^d$, and taking its values in the set $E_κ^{\mathbb{L}}$ where $E_κ=\{0,\cdots,κ-1\}$ for some fixed $κ\in\{\infty, 2,3,\cdots\}$. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix $T$. These are standard settings, satisfied by the TASEP, the voter models, the contact…
We discuss a Gaussian multiplicative chaos (GMC) structure underlying a family of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on a space $Γ$ of directed pathways crossing a diamond fractal with Hausdorff dimension two. The laws of these random continuum path measures arise in a critical weak-disorder limiting regime for discrete directed polymers on disordered hierarchical graphs. For the analogous subcritical continuum polymer model in which the diamond fractal has Hausdorff dimension less than two, the random path measures can be constructed as subcritical GMCs through coupl…
We study the fractal properties of the stationary distrubtion π for a simple Markov process on R. We will give bounds for the Hausdorff dimension of π, and lower bounds for the multifractal spectrum of π. Additionally, we will provide a method for numerically estimating these bounds.
We investigate the transportation cost-information inequalities for bifurcating Markov chains which are a class of processes indexed by binary tree. These processes provide models for cell growth when each individual in one generation gives birth to two offsprings in the next one. Transportation cost inequalities provide useful concentra-tion inequalities. We also study deviation inequalities for the empiri-cal means under relaxed assumptions on the Wasserstein contraction of the Markov kernels. Applications to bifurcating non linear autore-gressive processes are considered: deviation inequali…
Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for additive functionals of BMC under $L^2$-ergodic conditions with three different regimes. This completes the pointwise approach developed in a previous work. As application, we study the elementary case of symmetric bifurcating autoregressive process, which justify the non-trivial hypothesis considered on the kernel transition of the BMC. We illustrate in this example the phase tran…
We prove an analog of the classical Zero-One Law for both homogeneous and nonhomogeneous Markov chains (MC). Its almost precise formulation is simple: given any event $A$ from the tail $σ$-algebra of MC $(Z_n)$, for large $n$, with probability near one, the trajectories of the MC are in states $i$, where $P(A|Z_n=i)$ is either near $0$ or near $1$. A similar statement holds for the entrance $σ$-algebra, when $n$ tends to $-\infty$. To formulate this second result, we give detailed results on the existence of nonhomogeneous Markov chains indexed by $\mathbb Z_-$ or $\mathbb Z$ in both the finit…
We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.
The main purpose of this article is to establish moderate deviation principles for additive functionals of bifurcating Markov chains. Bifurcating Markov chains are a class of processes which are indexed by a regular binary tree. They can be seen as the models which represent the evolution of a trait along a population where each individual has two offsprings. Unlike the previous results of Bitseki, Djellout \& Guillin (2014), we consider here the case of functions which depend only on one variable. So, mainly inspired by the recent works of Bitseki \& Delmas (2020) about the central limit theo…
From the datum of an integer partition and a classical Lie algebra, one can define a Markov chain on an associated multiplicative graph. For each classical family A, C, B, D, we thus obtain a sequence of Markov chain which is indexed by the rank of the considered algebra. In this article we show that, for each type, the transition kernel of the Markov chain has a limit when the rank tends to infinity. Moreover, the limit kernel does not depend on the considered type.
We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy. Applications to Markov chains, Gibbs states and the stochastic scrabble are given.
We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidth is selected by a method inspired by the works of Goldenshluger and Lepski [18]. Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty.
Consider a sequence $(η^N(t) :t\ge 0)$ of continuous-time, irreducible Markov chains evolving on a fixed finite set $E$, indexed by a parameter $N$. Denote by $R_N(η,ξ)$ the jump rates of the Markov chain $η^N_t$, and assume that for any pair of bonds $(η,ξ)$, $(η',ξ')$ $\arctan \{R_N(η,ξ)/R_N(η',ξ')\}$ converges as $N\uparrow\infty$. Under a hypothesis slightly more restrictive (cf. \eqref{mhyp} below), we present a recursive procedure which provides a sequence of increasing time-scales $θ^1_N, \dots, θ^{\mf p}_N$, $θ^j_N \ll θ^{j+1}_N$, and of coarsening partitions $\{\ms E^j_1, \dots, \ms E…
Consider a system of interacting particles indexed by the nodes of a graph whose vertices are equipped with marks representing parameters of the model such as the environment or initial data. Each particle takes values in a countable state space and evolves according to a (possibly non-Markovian) continuous-time pure jump process whose jump intensities depend only on its own state (or history) and marks as well as the states (or histories) and marks of particles and edges in its neighborhood in the graph. Under mild conditions on the jump intensities, it is shown that the trajectories of the i…
A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one…
We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph $G = (V,E)$. The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form …
We consider a Markov chain $(x_n)$ whose kernel is indexed by a scaling parameter $γ>0$, refered to as the step size. The aim is to analyze the behavior of the Markov chain in the doubly asymptotic regime where $n\to\infty$ then $γ\to 0$. First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putti…
Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (2007), where the considered additive functionals are sums of martingale increments, and only one regime app…
Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistence and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations…
