Analysis of random walks in dynamic random environments via L2-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on ${\mathbb{Z}}^d$. We assume that the environment has a stationary distribution $\mu$ and satisfies the Poincaré inequality w.r.t. $\mu$. 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 $\mu$. 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 $L^2$- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.     

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Z^d${\mathbb{Z}}^d$, d≥1$d\ge 1$. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ$\mu$. The green particle also jumps at rate 1, but uses different transition kernels p^′$p^{\prime }$ and p^″$p^{″}$ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞$\mu \to \infty$, the speed of the green particle tends to the average jump under p^′$p^{\prime }$. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.     

Large deviations and phase transition for random walks in random nonnegative potentials

We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on Z^d${\mathbb{Z}}^d$. We complement the analysis of M.P.W. Zerner [Directional decay of the Green’s function for a random nonnegative potential on Z^d${\mathbb{Z}}^d$, Ann. Appl. Probab. 8 (1996) 246–280], where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.     

Asymptotic speed of propagation and traveling wave solutions for a lattice integral equation

We investigate the asymptotic speed of propagation and monotone traveling wave solutions for a lattice integral equation which is an epidemic model while the population is distributed on one-dimensional lattice Z$\mathbb{Z}$. It is proved that the asymptotic speed of propagation _c∗$c_{\ast }$ coincides with the minimal wave speed.     

An iterative approximation procedure for the distribution of the maximum of a random walk

Let I(F) be the distribution function (d.f.) of the maximum of a random walk whose i.i.d. increments have the common d.f. F and a negative mean. We derive a recursive sequence of embedded random walks whose underlying d.f.'s F_k converge to the d.f. of the first ladder variable and satisfy FF₁F₂ on [0,∞) and I(F)=I(F₁)=I(F₂)=. Using these random walks we obtain improved upper bounds for the difference of I(F) and the d.f. of the maximum of the random walk after finitely many steps.     

Explicit Bounds for the Return Probability of Simple Random Walks

We give an exact computation of the second order term in the asymptotic expansion of the return probability, P_2n^d(0,0), of a simple random walk on the d-dimensional cubic lattice. We also give an explicit bound on the remainder. In particular, we show that P_2n^d(0,0) < 2 (d/4n)^d/2 where n M=M(d) is explicitly given.     

A second order asymptotic expansion in the local limit theorem for a simple branching random walk in Zd

Consider a branching random walk, where the underlying branching mechanism is governed by a Galton–Watson process and the migration of particles by a simple random walk in ${\mathbb{Z}}^d$. Denote by $Z_n\left(z\right)$ the number of particles of generation $n$ located at site $z\in {\mathbb{Z}}^d$. We give the second order asymptotic expansion for $Z_n\left(z\right)$. The higher order expansion can be derived by using our method here. As a by-product, we give the second order expansion for a simple random walk on ${\mathbb{Z}}^d$, which is used in the proof of the main theorem and is of independent interest.     

Reconstruction of periodic sceneries seen along a random walk

This article was motivated by a question of Kesten. Kesten asked for which random walks periodic sceneries can be reconstructed. Among others, he asked the question for random walks which at each step can move by one or two units to the right. Previously, Howard [C.D. Howard, Distinguishing certain random sceneries on Z$\mathbb{Z}$ via random walks, Statist. Probab. Lett. 34 (2) (1997) 123–132] proved that all periodic sceneries can be reconstructed provided they are observed along the path of a simple random walk.     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron–de Marsily model in ${\mathbb{Z}}^2$ with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.     

Spread of a catalytic branching random walk on a multidimensional lattice

For a supercritical catalytic branching random walk on ${\mathbb{Z}}^d$, $d\in \mathbb{N}$, with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. It is shown that in the result of the proper normalization of the particles positions in the limit there are a.s. no particles outside the closed convex surface in ${\mathbb{R}}^d$ which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation.     

More Randomness of Environment Does Not Always Slow Down a Random Walk

We consider a random walk on in a stationary and ergodic random environment, whose states are called types of the vertices of . We find conditions for which the speed of the random walk is positive. In the case of a Markov chain environment with finitely many states, we give an explicit formula for the speed and for the asymptotic proportion of time spent at vertices of a certain type. Using these results, we compare the speed of random walks on in environments of varying randomness.     

On High-Level Exceedances of Gaussian Fields and the Spectrum of Random Hamiltonians

We study the almost sure asymptotic structure of high-level exceedances by Gaussian random field (x), xV with correlated values, where {V} is a family of -dimensional cubes increasing to Z . The results are applied to the study of the asymptotic behaviour of extreme eigenvalues of random Schrödinger operator in V.     

Structure of random 312‐avoiding permutations

We evaluate the probabilities of various events under the uniform distribution on the set of 312‐avoiding permutations of . We derive exact formulas for the probability that the i^th element of a random permutation is a specific value less than i, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large N when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random 312‐avoiding permutation has k specified decreasing points, and we show that for large N the points below the diagonal look like trajectories of a random walk. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 599–631, 2016     

Stability and asymptoticity of Volterra difference equations: A progress report

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Z$Z$-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried out in Section 6. Finally, we state some problems.