Solution of partial differential equations by a modified random walk

A new Monte Carlo technique is applied to solve difference equations of elliptic and parabolic partial differential equations with given boundary values. Fixed random walk is extended to modified random walk, whereby a random walk is made on a maximum square. The average number of steps and the computational time in a modified random walk is much less than in a fixed random walk. Numerical examples support the utility of this method.     

Two-boundary problems for a random walk

We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1485–1509, November, 2007.     

Quenched tail estimate for the random walk in random scenery and in random layered conductance

We discuss the quenched tail estimates for the random walk in random scenery. The random walk is the symmetric nearest neighbor walk and the random scenery is assumed to be independent and identically distributed, non-negative, and has a power law tail. We identify the long time asymptotics of the upper deviation probability of the random walk in quenched random scenery, depending on the tail of scenery distribution and the amount of the deviation. The result is in turn applied to the tail estimates for a random walk in random conductance which has a layered structure.     

一维紧邻时间随机环境下可逗留随机游动的有关性质

给出了可数状态空间中时间随机环境下可逗留随机游动的一个统一模型,对于一维紧邻时间随机环境下的随机游动,在一定的条件下,讨论它的极限性质和中心极限定理,该结论类似于空间随机环境下的随机游动的有关结论.     

Chung-type law of the iterated logarithm for continuous time random walk

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish a Chung-type law of the iterated logarithm for continuous time random walk with jumps and waiting times in the domains of attraction of stable laws.     

Random Walks on Trees with Finitely Many Cone Types

This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk.     

Slow movement of a random walk on the range of a random walk in the presence of an external field

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ( $d\ge 5$ ). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved.     

Quenched Large Deviations for Multidimensional Random Walk in Random Environment with Holding Times

We consider a random walk in random environment with random holding times, that is, the random walk jumping to one of its nearest neighbors with some transition probability after a random holding time. Both the transition probabilities and the laws of the holding times are randomly distributed over the integer lattice. Our main result is a quenched large deviation principle for the position of the random walk. The rate function is given by the Legendre transform of the so-called Lyapunov exponents for the Laplace transform of the first passage time. By using this representation, we derive some asymptotics of the rate function in some special cases.     

The monotonicity of the probability of return into the source in models of branching random walks

The paper discusses two models of a branching random walk on a many-dimensional lattice with birth and death of particles at a single node being the source of branching. The random walk in the first model is assumed to be symmetric. In the second model an additional parameter is introduced which enables “artificial” intensification of the prevalence of branching or walk at the source and, as the result, violating the symmetry of the random walk. The monotonicity of the return probability into the source is proved for the second model, which is a key property in the analysis of branching random walks.     

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in , of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupiainen. Under this assumption, we prove that the exit measure of the random walk in a random environment from a large ball, approaches the exit measure of a simple random walk from the same ball, in the sense that the variational distance between smoothed versions of these measures converges to zero. We also prove the transience of the random walk in random environment. The analysis is based on propagating estimates on the variational distance between the exit measure of the random walk in random environment and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities. Partially supported by NSF grant DMS-0503775.     

Random Walks on Directed Covers of Graphs

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.     

Critical survival barrier for branching random walk

We consider a branching random walk with an absorbing barrier, where the associated one-dimensional random walk is in the domain of attraction of an α-stable law. We shall prove that there is a barrier and a critical value such that the process dies under the critical barrier, and survives above it. This generalizes previous result in the case that the associated random walk has finite variance.     

Tubular recurrence

We introduce the directed-edge-reinforced random walk and prove that the process is equivalent to a random walk in random environment. Using Oseledec"s multiplicative ergodic theorem, we obtain recurrence and transience criteria for random walks in random environment on graphs with a certain linear structure and apply them to directed-edge-reinforced random walks. This revised version was published online in August 2006 with corrections to the Cover Date.     

Simple random walk on the line in random environment

Summary We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin.     

The survival of branching annihilating random walk

Summary Branching annihilating random walk is an interacting particle system on . As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown here that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non-attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by Gray and Griffeath is used.     

A random walk with a branching system in random environments

We consider a branching random walk in random environments, where the particles are reproduced as a branching process with a random environment (in time), and move independently as a random walk on Z with a random environment (in locations). We obtain the asymptotic properties on the position of the rightmost particle at time n, revealing a phase transition phenomenon of the system.     

Intersections of random walks with random sets

Two general theorems about the intersections of a random walk with a random set are proved. The result is applied to the cases when the random set is a (deterministic) half-line and a two-sided random walk. Research supported by NSF Grant DMS-8702879 and an Alfred P. Sloan Research Fellowship.     

Edge-reinforced random walk on one-dimensional periodic graphs

In the present paper, linearly edge-reinforced random walk is studied on a large class of one-dimensional periodic graphs satisfying a certain reflection symmetry. It is shown that the edge-reinforced random walk is recurrent. Estimates for the position of the random walker are given. The edge-reinforced random walk has a unique representation as a random walk in a random environment, where the random environment is given by random weights on the edges. It is shown that these weights decay exponentially in space. The distribution of the random weights equals the distribution of the asymptotic proportion of time spent by the edge-reinforced random walker on the edges of the graph. The results generalize work of the authors in Merkl and Rolles (Ann Probab 33(6):2051–2093, 2005; 35(1):115–140, 2007) and Rolles (Probab Theory Related Fields 135(2):216–264, 2006) to a large class of graphs and to periodic initial weights with a reflection symmetry.     

Quenched point-to-point free energy for random walks in random potentials

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and steps of the walk. The potential is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of entropy. We establish regularity properties of the point-to-point free energy, and link it to the infinite volume point-to-line free energy and quenched large deviations of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.     

Multidimensional random walk with reflections

Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. One-dimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.