On the maximum increase and decrease of one-dimensional diffusions

In this paper the joint distribution of the maximum increase and the maximum decrease up to a first hitting time is calculated for a regular one-dimensional diffusion. Moreover, it is shown that the process given by the maximum decrease when the hitting level is the “time” parameter is a pure jump Markov process and its generator is found. As examples, Brownian motion and three dimensional Bessel process are analyzed more in detail.     

On the character of convergence to Brownian local time. I

Summary Many results are known about the convergence of some processes to Brownian local time. Among such processes are the process of occupation times of Brownian motion, the number of downcrossings of Brownian motion over smaller and smaller intervals before timet, the number of visits of the recurrent integer-valued random walk to some point duringn steps and others. In this paper we consider the asymptotic behaviour of the differences between Brownian local time and some of the processes which converge to it.     

Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Combined random number generator via the generalized Chinese remainder theorem

The combined random number (RN) generator has been considered by many scholars as a good RN generator. One promising type of combined RN generator, recommended by L'Ecuyer (Oper. Res. 44 (1996) 816; 47 (1999) 159), is the combined multiple recursive generator (MRG). This paper analyzes the combined MRG via the Chinese remainder theorem. A new combined generator based on the generalized Chinese remainder theorem and on the Ore algorithm (Amer. Math. Monthly 59 (1952) 365) is presented. The proposed combined generator improves the combined MRG in terms of both the suitability for various types of RN generators and the restriction on the moduli of the individual generators. Therefore, the proposed combined generator is an ideal RN generator and is most recommended.     

Some Limit Theorems for Heights of Random Walks on a Spider

A simple random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker and the Brownian motion can go on the legs in n steps. The heights on the legs are also investigated when the number of legs goes to infinity.     

On representations of solutions of 1–dimensional stochastic differential equations with reflecting boundary conditions

In this paper we give representations of the solution of 1–dimensional stochastic differential equation (SDE for short) with reflecting barrieres. To this we construct the solution of deterministic Skorohod equation with two reflecting boundaries and show which can be expressed by the operator “sup inf”. Then the solution of given SDE can be represented by a form that depend on a reflecting Brownian motion determined by solving the deterministic Skorohod eqyation     

On goodness-of-fit tests for multiple recursive random number generators

This paper considers the problem of reducing the computational time in testing uniformity for a full period multiple recursive generator (MRG). If a sequence of random numbers generated by a MRG is divided into even number of segments, say 2s, then the multinomial goodness-of-fit tests and the empirical distribution function goodness-of-fit tests calculated from the ith segment are the same as those of the (s + i)th segment. The equivalence properties of the goodness-of-fit test statistics for a MRG and its associated reverse and additive inverse MRGs are also discussed.     

Random walk of a “drunk company”

We study the collective behavior of a system of Brownian agents each of which moves orienting itself to the group as a whole. This system is the simplest model of the motion of a “united drunk company.” For such a system, we use the functional integration technique to calculate the probability of transition from one point to another and to determine the time dependence of the probability density to find a member of the “drunk company” near a given point. It turns out that the system exhibits an interesting collective behavior at large times and this behavior cannot be described by the simplest mean-field-type approximation. We also obtain an exact solution in the case where one of the agents is “sober” and moves along a given trajectory. The obtained results are used to discuss whether such systems can be described by different theoretical approaches.     

Genealogy of extremal particles of branching Brownian motion

Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to a certain random mechanism. By virtue of the groundbreaking work by M. Bramson on the convergence of solutions of the Fisher‐KPP equation to traveling waves, the law of the rightmost particle in the limit of large times is rather well understood. In this work, we address the full statistics of the extremal particles (first‐, second‐, third‐largest, etc.). In particular, we prove that in the large t‐limit, such particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time 0, or within a distance of order 1 from time t. The approach relies on characterizing, up to a certain level of precision, the paths of the extremal particles. As a byproduct, a heuristic picture of branching Brownian motion “at the edge” emerges, which sheds light on the still unknown limiting extremal process. © 2011 Wiley Periodicals, Inc.     

Strong approximation for the general Resten-Spitzer random walk in independent random scenery

This paper is to prove that, if a one-dimensional random walk can be approximated by a Brownian motion, then the related random walk in a general independent scenery can be approximated by a Brownian motion in Brownian scenery.     

Harmonic functions on Walsh’s Brownian motion

We examine a variation of two-dimensional Brownian motion introduced by Walsh that can be described as Brownian motion on the spokes of a (rimless) bicycle wheel. We construct the process by randomly assigning angles to excursions of reflecting Brownian motion. Hence, Walsh’s Brownian motion behaves like one-dimensional Brownian motion away from the origin, but differently at the origin as it is immediately sent off in random directions. Given the similarity, we characterize harmonic functions as linear functions on the rays satisfying a slope-averaging property. We also classify superharmonic functions as concave functions on the rays satisfying extra conditions.     

Reverse multiple recursive random number generators

Every multiple recursive generator (MRG) has an associated reverse MRG that generates the same sequence but in reverse order. An explicit formula for the recurrence of this reverse MRG is derived based on finite field arithmetic. These two MRGs are equal in both periodicity and spectral value. This property could be exploited to reduce the number of candidates when searching for good MRG parameters.     

Strong approximation for the general Kesten-Spitzer random walk in independent random scenery

This paper is to prove that, if a one-dimensional random wa lkcan be approximated by a Brownian motion, then the related random walk in a g eneral independent scenery can be approximated by a Brownian motion in Brownian scenery.     

On the interaction of geometry,analysis and stochastics on a body

It is reasonable to expect that the geometrical feature of a body has influence to spectral asymptotics of its “natural” Laplacian as well as to the behavior of its “natural” Brownian motion. In fact, such an interaction can be expressed by a so–called “Einstein relation” implicating Hausdorff, spectral and walk dimension. These quantities express geometric, analytic and stochastic aspects of a set. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit theorems for projections of random walk on a hypersphere

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein–Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a “functional” generalisation of Poincaré’s observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli–Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.     

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.     

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behavior of uniform random maps with a prescribed face‐degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov–Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q‐angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so‐called “discrete snakes” obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.     

Reflected brownian motion in a wedge: Semimartingale property

In this paper, the object of study is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The basic question considered here is “When is this process a semimartingale?”. It is first shown that a related process, defined by specifying the corner of the wedge to be an absorbing state, rather than an instantaneous one, is a semimartingale. Conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by Vardhan and Williams (“Brownian motion in a wedge with oblique reflection”, Comm. Pure Appl. Math., to appear). Under these conditions, it is shown that starting away from the corner, the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge. This equivalence is also shown to hold starting from the corner, except in one unresolved case for which the wedge angle exceeds π and the directions of reflection are exactly opposed.     

Markov processes conditioned on their location at large exponential times

Suppose that ${\left(X_t\right)}_{t\ge 0}$ is a one-dimensional Brownian motion with negative drift $?\mu$. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like $X$ conditioned to hit 0, after which time it behaves as $X$ killed at the last time $X$ visits 0. Equivalently, the limit process has the dynamics of the killed “bang–bang” Brownian motion that evolves like Brownian motion with positive drift $+\mu$ when it is negative, like Brownian motion with negative drift $?\mu$ when it is positive, and is killed according to the local time spent at 0.An extension of this result holds in great generality for a Borel right process conditioned to be in some state $a$ at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the “bang–bang” construction for a general Markov process.As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.