Oscillatory Fractional Brownian Motion

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. A randomness property of the initial condition increases the long range dependence. We emphasize the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.     

Convergence of branching transport processes to branching brownian motion

Two models are given of branching transport processes that converge to branching Brownian motion starting with one initial particle. The martingale problem method is used.     

Laws of large numbers for supercritical branching Gaussian processes

A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein–Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized.     

Shift Self-Similar Additive Random Sequences Associated with Supercritical Branching Processes

Natural examples of increasing shift self-similar additive random sequences are constructed, which are associated with supercritical branching processes. The rate of growth and the distributional properties of them are studied in terms of the offspring distributions of the supercritical branching processes. The results are applied to two types of laws of the iterated logarithm for a Brownian motion on the unbounded Sierpinski gasket. An extension of the Bingham–Doney–de Meyer theorem on the limits of supercritical branching processes is also proved.     

Maximum likelihood estimation for a branching diffusion process

This paper investigates the properties of the maximum likelihood estimators of the drift and diffusion coefficients under three sampling schemes for a branching diffusion process in which the branching process is a linear birth process and the diffusion is in accordance with the Brownian motion with drift.     

An alternative approach to super-Brownian motion with a locally infinite branching mass

Fleischmann and Mueller (Probab. Theory Related Fields 107 (1997) 325) constructed a super-Brownian motion in R¹ with a locally infinite branching rate function, and they showed that this super-Brownian motion has a strong killing property in the critical case. In this paper, we first construct, via a limiting procedure, a super-Brownian motion which is equivalent to the super-Brownian motion with a locally infinite branching. From this construction, one can easily see the connection between the superprocess and a killed Brownian motion. Next, by taking advantage of the new construction, we give a new proof of the strong killing property of the process.     

On multiplicative excessive functions of a branching Brownian motion

Summary In this note stochastic calculus is used to characterise multiplicative excessive functions of a binary branching Brownian motion with a constant creation rate. Some properties of the martingales given by invariant functions are studied. In particular, it is seen that these positive and unbounded martingales tend a.s. to 0 and are not square integrable. Informally speaking, they exhibit a clustering phenomenon in the underlying supercritical branching Brownian motion.This work was done while the author was visiting the University of British Columbia, Mathematical Department, and was partly supported by a NSERC grant. AMS 1980 subject classifications: primary 60J80, 60J65 secondary 60J60     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Escape Probabilities for Branching Brownian Motion Among Soft Obstacles

We derive asymptotics for the quenched probability that a critical branching Brownian motion killed at a small rate ε in Poissonian obstacles exits from a large domain. Results are formulated in terms of the solution to a semilinear partial differential equation with singular boundary conditions. The proofs depend on a quenched homogenization theorem for branching Brownian motion among soft obstacles.     

Moderate Deviation for the Rightmost Position in a Branching Brownian Motion

We study the moderate deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its moderate deviation function. Firstly, Chauvin and Rouault studied the large deviation probability for the rightmost position in a branching Brownian motion. Recently, Derrida and Shiconsidered lower deviation for the same model. By contrast, Our main result is more extensive.     

Tokunaga and Horton self-similarity for level set trees of Markov chains

The Horton and Tokunaga branching laws provide a convenient framework for studying self-similarity in random trees. The Horton self-similarity is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for elements of a branching system. The stronger Tokunaga self-similarity addresses so-called side branching. The Horton and Tokunaga self-similarity have been empirically established in numerous observed and modeled systems, and proven for two paradigmatic models: the critical Galton–Watson branching process with finite progeny and the finite-tree representation of a regular Brownian excursion. This study establishes the Tokunaga and Horton self-similarity for a tree representation of a finite symmetric homogeneous Markov chain. We also extend the concept of Horton and Tokunaga self-similarity to infinite trees and establish self-similarity for an infinite-tree representation of a regular Brownian motion. We conjecture that fractional Brownian motions are also Tokunaga and Horton self-similar, with self-similarity parameters depending on the Hurst exponent.     

Fractional Lévy Processes as a Result of Compact Interval Integral Transformation

Fractional Brownian motion can be represented as an integral of a deterministic kernel w.r.t. an ordinary Brownian motion either on infinite or compact interval. In previous literature fractional Lévy processes are defined by integrating the infinite interval kernel w.r.t. a general Lévy process. In this article we define fractional Lévy processes using the com pact interval representation.

We prove that the fractional Lévy processes presented via different integral transformations have the same finite dimensional distributions if and only if they are fractional Brownian motions. Also, we present relations between different fractional Lévy processes and analyze the properties of such processes. A financial example is introduced as well.     

Perturbed Brownian motions

Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example). Received: 17 May 1996 / In revised form: 21 January 1997     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Chung's law for integrated Brownian motion

The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

Strong approximations for long memory sequences based partial sums,counting and their Vervaat processes

We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.     

Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.