We propose a variety of models of random walk, discrete in space and time, suitable for simulating stable random variables of arbitrary index α (0 < α ≤ 2), in the symmetric case. We show that by properly scaled transition to vanishing space and time steps our random walk models converge to the corresponding continuous Markovian stochastic processes, that we refer to as Lévy-Feller diffusion pr...

In this paper, we prove that the probability kernel of a random walk on a trinomial tree converges to the density of a Brownian motion with drift at the rate O(h), where h is the distance between the nodes of the tree. We also show that this convergence estimate is optimal in which the density of the random walk cannot converge at a faster rate. The proof is based on an application of spectral ...

A discrete random walk method on grids was proposed and used to solve the linearized Poisson– Boltzmann equation ~LPBE! @R. Ettelaie, J. Chem. Phys. 103, 3657 ~1995!#. Here, we present an efficient grid-free random walk method. Based on a modified ‘‘walk on spheres’’ algorithm @B. S. Elepov and G. A. Mihailov, Sov. Math. Dokl. 14, 1276 ~1973!# for the LPBE, this Monte Carlo algorithm uses a sur...

Compution of discrete logarithm is of great interest because of its relevance to public key cryptography. It is believed to be a hard problem because no polynomial time solution has been found. One of the generic algorithm in computing discrete logarithm is Pollard rho algorithm (and its derivatives). The algorithm depend on the ”birthday paradox” idea and makes a pseudo-random walk in finite a...

We consider Pollard's rho method for discrete logarithm computation. In the analysis of its running time, the crucial assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not exactly hold for the walk originally suggested by Pollard. We study alternative walks that can be eeciently applied to compute discrete logarithms. We introduce a cl...

A discrete random walk method on grids was proposed and used to solve the linearized Poisson-Boltzmann equation (LPBE) [1]. Here, we present a new and efficient grid-free random walk method. Based on a modified “Walk On Spheres” (WOS) algorithm [11] for the LPBE, this Monte Carlo algorithm uses a survival probability distribution function for the random walker in a continuous and free diffusion...

In this note, we make explicit the limit law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in [5] for a continuous analogue of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectur...

A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...