Invariance Principle For Random Walks Research Articles

Joint Invariance Principles for Random Walks with Positively and Negatively Reinforced Steps

Given a random walk (S_n) with typical step distributed according to some fixed law and a fixed parameter p in (0,1), the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability 1-p, the same step as (S_n) while with probability p, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when p < 1/2 and when the typical step is centred. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, B(t),tp∫0ts-pdBr(s),t-p∫0tspdBc(s)t∈R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( B(t) , \\, t^p \\int _0^t s^{-p} \\mathrm {d}B^r(s) , \\, t^{-p} \\int _0^t s^{p} \\mathrm {d}B^c(s) \\right) _{t \\in \\mathbb {R}^+} \\end{aligned}$$\\end{document}for properly correlated Brownian motions B, B^r, B^c. The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes p in (0,1) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.

Polynomial ballisticity conditions and invariance principle for random walks in strong mixing environments

Polynomial ballisticity conditions and invariance principle for random walks in strong mixing environments

Invariance principles for random walks in random environment on trees

In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees. First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on $\mathbb{R}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in $\mathbb{R}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the recent common ancestor. We prove, after introducing the bias parameter $\beta^{n^{-1/4}}$, for some $\beta>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT). Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.

Quenched invariance principle for random walks on dynamically averaging random conductances

Quenched invariance principle for random walks on dynamically averaging random conductances

Rates of convergence in invariance principles for random walks on linear groups via martingale methods

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general R d {\mathbb R}^d -valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates either in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of the Wasserstein distance and in the almost sure invariance principle for R d {\mathbb R}^d -valued martingales with stationary increments having moments of order p ∈ ( 2 , 3 ] p \in (2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general R d {\mathbb R}^d -valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance G L d ( R ) {\mathrm {GL}}_d(\mathbb {R}) ).

The boundary of random planar maps via looptrees

We study the scaling limits of looptrees associated with Bienaymé–Galton–Watson (BGW) trees, that are obtained by replacing every vertex of the tree by a “cycle” whose size is its degree. First, we consider BGW trees whose offspring distribution is critical and in the domain of attraction of a Gaussian distribution. We prove that the Brownian CRT is the scaling limit of the associated looptrees, thereby confirming a prediction of [18]. Then, we deal with BGW trees whose offspring distribution is critical and heavy-tailed. We show that the scaling limit of the associated looptrees is a multiple of the unit circle. This corresponds to a so-called condensation phenomenon, meaning that the underlying tree exhibits a vertex with macroscopic degree. Here, we rely on an invariance principle for random walks with negative drift, which is of independent interest. Finally, we apply these results to the study of the scaling limits of large faces of Boltzmann planar maps. We complete the results of [50] and establish a phase transition for the topology of these maps in the non-generic critical regime.

Quenched invariance principle for random walks among random degenerate conductances

We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik (Ann. Probab. 43 (2015) 1866–1891) and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.

On the transient (T) condition for random walk in mixing environment

We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. Ann. Probab. 29 (2001) 724–765). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni (Israel J. Math. 148 (2005) 87–113). The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.

An invariance principle for one-dimensional random walks among dynamical random conductances

We study variable-speed random walks on $\mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.

Quenched invariance principle for random walk in time-dependent balanced random environment

Nous demontrons un theoreme de loi limite centrale presque sur pour des marches aleatoires equilibrees non necessairement aux plus proches voisins, dans un milieu aleatoire ergodique. Nous supposons que l’environnement satisfait des conditions d’ergodicite et d’ellipicite appropriees. Notre preuve est basee sur un principe du maximum pour des operateurs aux differences paraboliques.

Quenched invariance principle for random walks with time-dependent ergodic degenerate weights

We study a continuous-time random walk, $X$, on $\mathbb{Z}^{d}$ in an environment of dynamic random conductances taking values in $(0,\infty)$. We assume that the law of the conductances is ergodic with respect to space–time shifts. We prove a quenched invariance principle for the Markov process $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser’s iteration scheme.

Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z} ^d$. Standard conditions for ballisticity and the central limit theorem require ellipticity, and are typically non-local. We use oriented percolation and martingale arguments to find non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting. The use of percolation allows certain non-elliptic models to be treated even though ballisticity has not been proved for elliptic perturbations of these models.

Self-similar scaling limits of Markov chains on the positive integers

We are interested in the asymptotic behavior of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. If $\mathit{X_{n}}$ is such a Markov chain started at n, we establish a limit theorem for $\frac{1}{n}\mathit{X_{n}}$ appropriately scaled in time, where the scaling limit is given by a nonnegative self-similar Markov process. We also study the asymptotic behavior of the time needed by $\mathit{X_{n}}$ to reach some fixed finite set. We identify three different regimes (roughly speaking the transient, the recurrent and the positive-recurrent regimes) in which $\mathit{X_{n}}$ exhibits different behavior. The present results extend those of Haas and Miermont [Bernoulli 17 (2011) 1217–1247] who focused on the case of nonincreasing Markov chains. We further present a number of applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks, nonnegative self-similar Markov processes, invariance principles for random walks conditioned to stay positive and exchangeable coalescence-fragmentation processes.

Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

Quenched invariance principle for random walks on Delaunay triangulations

We consider simple random walks on Delaunay triangulations generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Matern cluster and Matern hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.

Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in $\mathbb{Z}^d$, $d\geq 2$. We first prove an invariance principle (for $d\ge2$) and the transience of the random walks when $d\ge 3$ (recurrence when $d=2$) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.

The quenched invariance principle for random walks in random environments admitting a bounded cycle representation

We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields 129 (2004) 219--244) to the non-reversible setting.

The limit theorems for random walk with state space ℝ in a space-time random environment

We consider a discrete time random walk on real number space in a space-time random environment. We state that when the random environment is i.i.d., under the marginal annealed law, the law of large numbers, iterated law and CLT of the process are correct. Using a martingale approach, we also state an a.s. invariance principle for random walks in general random environment whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain.

Invariance principles for random walks conditioned to stay positive

Let $\{S_n\}$ be a random walk in the domain of attraction of a stable law $\mathcal{Y}$, i.e. there exists a sequence of positive real numbers $(a_n)$ such that $S_n/a_n$ converges in law to $\mathcal{Y}$. Our main result is that the rescaled process $(S_{\lfloor nt\rfloor}/a_n, t\ge 0)$, when conditioned to stay positive, converges in law (in the functional sense) towards the corresponding stable L\'{e}vy process conditioned to stay positive. Under some additional assumptions, we also prove a related invariance principle for the random walk killed at its first entrance in the negative half-line and conditioned to die at zero.

Quenched Invariance Principles for Random Walks with Random Conductances

We prove an almost sure invariance principle for a random walker among i.i.d. conductances in ℤ d , d≥2. We assume conductances are bounded from above but we do not require that they are bounded from below.