Tail estimates for one-dimensional non-nearest-neighbor random walk in random environment

Suppose that the integers are assigned i.i.d. random variables {(β gx , . . . , β 1x , α x )} (each taking values in the unit interval and the sum of them being 1), which serve as an environment. This environment defines a random walk {X n } (called RWRE) which, when at x, moves one step of length 1 to the right with probability α x and one step of length k to the left with probability β kx for 1≤ k≤ g. For certain environment distributions, we determine the almost-sure asymptotic speed of the RWRE and show that the chance of the RWRE deviating below this speed has a polynomial rate of decay. This is the generalization of the results by Dembo, Peres and Zeitouni in 1996. In the proof we use a large deviation result for the product of random matrices and some tail estimates and moment estimates for the total population size in a multi-type branching process with random environment.     

Logarithmic speeds for one-dimensional perturbed random walks in random environments

We study the random walk in a random environment on Z⁺={0,1,2,…}${\mathbb{Z}}^{+}=\{0,1,2,\dots \}$, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)^β${(logt)}^{\beta }$, for β∈(1,∞)$\beta \in (1,\infty )$, depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.     

Localization for discrete one-dimensional random word models

We consider Schrödinger operators in whose potentials are obtained by randomly concatenating words from an underlying set according to some probability measure ν on . Our assumptions allow us to consider models with local correlations, such as the random dimer model or, more generally, random polymer models. We prove spectral localization and, away from a finite set of exceptional energies, dynamical localization for such models. These results are obtained by employing scattering theoretic methods together with Furstenberg's theorem to verify the necessary input to perform a multiscale analysis.     

Weak interaction limits for one-dimensional random polymers

 In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces, and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach followed in our earlier work, which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations. We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example (1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and find the scaling as the width tends to infinity. Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60 Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality     

Limit theorems for one-dimensional diffusions and random walks in random environments

Summary The limiting behavior of one-dimensional diffusion process in an asymptotically self-similar random environment is investigated through the extension of Brox's method. Similar problems are then discussed for a random walk in a random environment with the aid of optional sampling from a diffusion model; an extension of the result of Sinai is given in the case of asymptotically self-similar random environments.     

Sharp probability estimates for random walks with barriers

We give sharp, uniform estimates for the probability that a random walk of n steps on the reals avoids a half-line [y,∞) given that it ends at the point x. The estimates hold for general continuous or lattice distributions provided the fourth moment is finite.     

Precise large deviation estimates for a one-dimensional random walk in a random environment

ω_x } (taking values in the interval [1/2, 1)), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when at x, moves one step to the right with probability ω _x , and one step to the left with probability 1 −ωx. Solomon (1975) determined the almost-sure asymptotic speed (= rate of escape) of a RWRE, in a more general set-up. Dembo, Peres and Zeitouni (1996), following earlier work by Greven and den Hollander (1994) on the quenched case, have computed rough tail asymptotics for the empirical mean of the annealed RWRE. They conjectured the form of the rate function in a full LDP. We prove in this paper their conjecture. The proof is based on a “coarse graining scheme” together with comparison techniques. Received: 22 July 1997/Revised version: 15 June 1998     

Front tracking for one-dimensional quasilinear hyperbolic equations with variable coefficients

A new front tracking method is developed for the variable coefficient equation . The method is a generalization of Dafermos' method for the constant coefficient case and is well-defined also for certain discontinuous velocity fields V. We give an explicit inequality stating the stability with respect to flux function, velocity field, and initial data. The numerical method is unconditionally stable and has linear convergence. It is well suited for numerical calculations, as is demonstrated in four examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

A priori estimates for one-dimensional singular integral operators with continuous coefficients

Let the contour γ consist of a finite number of simple closed pairwise nonintersecting curves, satisfying a Lyapunov condition, let S be the operator of singular integration in spacel _p , (γ) (1 <p < ∞), and leta (t), b (t) εC (γ) 1 <p ₁. <p < ∞. The necessary and sufficient condition for A = aI+ bS to be a Φ-operator in space L_p(γ) is that, for all?ε L_p(γ), ∥?∥_p ? const (∥ A?_p + ∥ ? ∥_p1), where ∥?∥_p = ∥?∥L_p (γ).     

Computing Lyapunov constants for random recurrences with smooth coefficients

In recent years, there has been much interest in the growth and decay rates (Lyapunov constants) of solutions to random recurrences such as the random Fibonacci sequence x_n+1=±x_n±x_n−1. Many of these problems involve nonsmooth dynamics (nondifferentiable invariant measures), making computations hard. Here, however, we consider recurrences with smooth random coefficients and smooth invariant measures. By computing discretised invariant measures and applying Richardson extrapolation, we can compute Lyapunov constants to 10 digits of accuracy. In particular, solutions to the recurrence x_n+1=x_n+c_n+1x_n−1, where the {c_n} are independent standard normal variables, increase exponentially (almost surely) at the asymptotic rate (1.0574735537…)ⁿ. Solutions to the related recurrences x_n+1=c_n+1x_n+x_n−1 and x_n+1=c_n+1x_n+d_n+1x_n−1 (where the {d_n} are also independent standard normal variables) increase (decrease) at the rates (1.1149200917…)ⁿ and (0.9949018837…)ⁿ, respectively.     

Stochastic heat equation with random coefficients

We prove the existence of a unique solution for a one-dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two-parameter white noise. The proof is based on a maximal inequality for the Skorohod integral deduced from It?'s formula for this anticipating stochastic integral. Received: 21 November 1997 / Revised version: 20 July 1998     

On algebraic polynomials with random coefficients

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

where , are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form which was previously the most studied.

     

Multiple autoregressive models with random coefficients

This paper derives conditions for the stationarity of a class of multiple autoregressive models with random coefficients. The models considered include as special cases those previously discussed by Andel (Ann. Math. Statist.42 (1971), 755–759; Math. Operationsforsch. Statist.7 (1976), 735–741).