Dynamic Random Environment Research Articles

Quenched large deviations in renewal theory

In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

Brownian Particle in the Curl of 2-D Stochastic Heat Equations

We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp log\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{\\log }$$\\end{document}-super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) ω̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\omega }$$\\end{document}. We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of ω̲\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\underline{\\omega }$$\\end{document}. Adapting their method, we show that if s≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\ge 1$$\\end{document}, with s=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s=1$$\\end{document} corresponding to the standard stochastic heat equation, then the particle stays log\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{\\log }$$\\end{document}-super diffusive, whereas if s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s<1$$\\end{document}, corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s<1$$\\end{document}, we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the log\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log $$\\end{document}-corrections to diffusivity. We also provide examples which display loga\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log ^a$$\\end{document}-super diffusive behaviour for a∈(0,1/2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in (0,1/2]$$\\end{document}.

A scaling limit of the parabolic Anderson model with exclusion interaction

AbstractWe consider the (discrete) parabolic Anderson model , , , where the ξ‐field is ‐valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by‐product of our main result we obtain precise asymptotics for the survival probability of a simple random walk that is killed at a scale dependent rate when meeting an exclusion particle. Our proof relies on the discrete theory of regularity structures of Erhard and Hairer and on novel sharp estimates of joint cumulants of arbitrary large order for the exclusion process. We think that the latter is of independent interest and may find applications elsewhere.

A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment

In this short note, we prove that v(−ε)=−v(ε). Here, v(ε) is the speed of a one-dimensional random walk in a dynamic reversible random environment, that jumps to the right (resp. to the left) with probability 1∕2+ε (resp. 1∕2−ε) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where v(ε),v(−ε) are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.

Collisions of random walks in dynamic random environments

We study dynamic random conductance models on Z2 in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.

Symmetric exclusion as a random environment: Invariance principle

We establish an invariance principle for a one-dimensional random walk in a dynamic random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes: a Brownian motion and a Gaussian process with stationary increments.

Random walk in cooling random environment: ergodic limits and concentration inequalities

In previous work by Avena and den Hollander [3], a model of a random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a given sequence of times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure, in dimension one. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over diverging periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moment bounds and concentration estimates. Along the way we derive two results of independent interest, namely, concentration inequalities for the random walk in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the cooling scheme, and by hinting at what can be done in higher dimensions. We argue that, while the cooling scheme does not affect the speed in the strong law of large numbers nor the rate function in the large deviation principle, it does affect the fluctuation properties.

Regular and Singular Continuous Time Random Walk in Dynamic Random Environment

Regular and Singular Continuous Time Random Walk in Dynamic Random Environment

Solving dynamic normal distribution stochastic decision-making problems based on time degree and vertical projection distance

How to effectively aggregate time-series information has long been a significant issue in the field of decision-making method and decision support system. This paper studies a dynamic normal distribution stochastic decision-making method that is based on the time degree and vertical projection distance. A dynamic normal distribution number weighted arithmetic average (DNDNWAA) operator is introduced, and a time sequence weight calculation model is constructed that fully considers the subjective preference of the historical information of the decision-maker. An attribute weight-determining model based on vertical projection distance is presented against the characteristics of normally distributed stochastic variables. The original dynamic normal distribution stochastic decision-making information is aggregated via the aggregation operator under the normally distributed stochastic variables. The aggregated comprehensive stochastic decision-making information based on stochastic probability distribution theory is converted into interval numbers, and the interval number possibility degree model is applied to provide a solution ordering result. Finally, the validity and rationality of the method proposed in this paper are verified by analyzing numerical examples. The proposed method can guide decision-makers to make better decisions in dynamic random information environment.

Random Walk on a Perturbation of the Infinitely-Fast Mixing Interchange Process

We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $\gamma>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $\gamma$ large enough. This statement is thus a perturbation of the case $\gamma =+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i)$ We deal with any dimension $d\geq 1$; $(ii)$ Each particle of the interchange process carries a transition vector chosen according to an arbitrary law $\mu$; $(iii)$ We show that $X_t/t$ is not only in the same direction of the annealed drift, but that it is also close to it.

Analysis of random walks in dynamic random environments via [formula omitted]-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on Zd. We assume that the environment has a stationary distribution μ and satisfies the Poincaré inequality w.r.t. μ. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L2- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.

Random walk on random walks: Low densities

We consider a random walker in a dynamic random environment given by a system of independent discrete-time simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or nonlazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.

Random walk on random walks: higher dimensions

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

Averaged vs. quenched large deviations and entropy for random walk in a dynamic random environment

We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level-1 contractions in both settings. Motivation for this work comes from variational descriptions of the quenched free energy of directed polymer models where the same Markov process entropy appears.

Law of large numbers for random walks on attractive spin-flip dynamics

We prove a law of large numbers for certain random walks on certain attractive dynamic random environments when initialised from all sites equal to the same state. This result applies to random walks on Zd with d≥1. We further provide sufficient mixing conditions under which the assumption on the initial state can be relaxed, and obtain estimates on the large deviation behaviour of the random walk.As prime example we study the random walk on the contact process, for which we obtain a law of large numbers in arbitrary dimension. For this model, further properties about the speed are derived.

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete Laplacian, and \xi is a dynamic random environment that drives the equation. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\kappa, split into two at rate \xi \vee 0, and die at rate (-\xi) \vee 0. Our focus is on the Lyapunov exponents \lambda_p(\kappa) = \lim_{t\to\infty} \frac{1}{t} \log \E([u(0,t)]^p)^{1/p}, p \in \N, and \lambda_0(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t). We investigate what happens when \kappa\Delta is replaced by \Delta^\cK, where \cK = \{\mathcal{K}(x,y)\colon\,x,y\in\Z^d,\,x \sim y\} is a collection of random conductances between neighbouring sites replacing the constant conductances \kappa in the homogeneous model. We show that the associated annealed Lyapunov exponents are given by the formula \lambda_p(\cK) = \sup\{\lambda_p(\kappa) \colon\,\kappa\in\Supp(\cK)\}, where \Supp(\cK) is the set of values taken by the \cK-field. We also show that for the associated quenched Lyapunov exponent this formula only provides a lower bound. Our proof is valid for three classes of reversible \xi, and for all \cK satisfying a certain clustering property, namely, there are arbitrarily large balls where \cK is almost constant and close to any value in \Supp(\cK). What our result says is that the Lyapunov exponents are controlled by those pockets of \cK where the conductances are close to the value that maximises the growth in the homogeneous setting.

Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.

Random walks in dynamic random environments and ancestry under local population regulation

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining. Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

Absolute continuity and weak uniform mixing of random walk in dynamic random environment

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the “environment seen from the walker-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

Random Walk in Dynamic Random Environment with Long-Range Space Correlations

Random Walk in Dynamic Random Environment with Long-Range Space Correlations