Telegraph Process with Elastic Boundary at the Origin

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability α, or reflected upwards, with probability 1?α. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.     

Random Flights in Higher Spaces

We consider in this paper random flights in ℝ ^d performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S ₁ ^d . We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ³ are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2. Related motions with random velocity in spaces of lower dimension are analyzed and their distributions derived.     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.AMS 2000 Subject Classification: 60J65, 60G40, 93E30     

On the Generalized Telegraph Process with Deterministic Jumps

We consider a semi-Markovian generalization of the integrated telegraph process subject to jumps. It describes a motion on the real line characterized by two alternating velocities with opposite directions, where a jump along the alternating direction occurs at each velocity reversal. We obtain the formal expressions of the forward and backward transition densities of the motion. We express them as series in the case of Erlang-distributed random times separating consecutive jumps. Furthermore, a closed form of the transition density is given for exponentially distributed times, with constant jumps and random initial velocity. In this case we also provide mean and variance of the process, and study the limiting behaviour of the probability law, which leads to a mixture of three Gaussian densities.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

Cyclic Bernoulli polling

We introduce, analyse and optimize the class of Bernoulli random polling systems. The server movescyclically among N channels (queues), butChange-over times between stations are composed ofwalking times required to move from one channel to another andswitch-in times that are incurredonly when the server actually enters a station to render service. The server uses aBernoulli random mechanism to decide whether to serve a queue or not: upon arrival to channeli, it switches in with probabilityp _i , or moves on to the next queue (w.p. 1 —p _i ) without serving any customer (e.g. packet or job). The Cyclic Bernoulli Polling (CBP) scheme is independent of the service regime in any particular station, and may be applied to any service discipline. In this paper we analyse three different service disciplines under the CBP scheme: Gated, Partially Exhaustive and Fully Exhaustive. For each regime we derive expressions for (i) the generating functions and moments of the number of customers (jobs) at the various queues at polling instants, (ii) the expected number of jobs that an arbitrary departing job leaves behind it, and (iii) the LST and expectation of the waiting time of a cutomer at any given queue. The fact that these measures of performance can be explicitly obtained under the CBP is an advantage over all parameterized cyclic polling schemes (such as the k-limited discipline) that have been studied in the literature, and for which explicit measures of performance are hard to obtain. The choice of thep _i 's in the CBP allows for fine tuning and optimization of performance measures, as well as prioritization between stations (this being achieved at a low computational cost). For this purpose, we develop a Pseudo-conservation law for amixed system comprised of channels from all three service disciplines, and define a Mathematical Program to find the optimal values of the probabilities {p _i } _i ^N =1 so as to minimize the expected amount of unfinished work in the system. Any CBP scheme for which the optimalp _i 's are not all equal to one, yields asmaller amount of the expected unfinished work in the system than that in the standard cyclic polling procedure with equivalent parameters. We conclude by showing that even in the case of a single queue, it is not always true thatp ₁=1 is the best strategy, and derive conditions under which it is optimal to havep ₁ < 1.Supported by a Grant from the France-Israel Scientific Cooperation (in Computer Science and Engineering) between the French Ministry of Research and Technology and the Israeli Ministry of Science and Technology, Grant Number 3321190.     

Sharp Asymptotics for the Partition Function of Some Continuous-time Directed Polymers

This paper is concerned with two related types of directed polymers in a random medium. The first one is a d-dimensional Brownian motion living in a random environment which is white-noise in time and homogeneous in space. The second is a continuous-time discrete-space Markov process on ℤ ^d , in a random environment with similar properties as in continuous space, albeit defined only on . The case of a space-time white noise environment can be achieved in this second setting. By means of some Gaussian tools, we estimate the free energy of these models at low temperature, and give some further information on the strong disorder regime of the objects under consideration. Frederi Viens’ research partially supported by NSF grants no.: DMS 0204999 and DMS 0606615.     

Random Ishikawa Iterative Sequence with Applications

Abstract

The purpose of this paper is to construct a random Ishikawa iterative sequence for random strongly pseudo-contractive operator T in separable Banach spaces and to study that under suitable conditions this random iterative sequence converges to a random fixed point to T.     

Approximation for the Normal Inverse Gaussian Process Using Random Sums

Abstract

We approximate the normal inverse Gaussian (NIG) process with random summations. The random sum we introduce is a random walk subordinated to the first passage time of another independent random walk; the model is interpreted as an internal mechanism at small scale that generates the NIG process. The main result is a functional limit theorem of weak convergence in the Skorohod topology.     

A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle (Ann. Probab. 20, 125–136, 1992) give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk (X _n ) in random environment on a regular tree, which is closely related to Mandelbrot’s (C. R. Acad. Sci. Paris 278, 289–292, 1974) multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent such that behaves asymptotically like . The value of ν is explicitly formulated in terms of the distribution of the environment.      

On Generating Random Intervals and Hyperrectangles

Abstract

We look at the problem of generating a random hyperrectangle in a unit hypercube such that each point of the hypercube has probability p of being covered. For random intervals of [0, 1], we compare various methods based either on the distribution of the length or on the distribution of the midpoint. It is shown that no constant length solution exists. Nice versatility is achieved when a random interval is chosen from among the spacings defined by a Dirichlet process. A small simulation is included.     

Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model

In this paper, in a time-homogeneous diffusion setting, we study a sequence of last passage times of generalized drawdown processes before the first passage time of another monitoring generalized drawdown process. These quantities are closely related to consecutive small market downward movements before a final big market crash modeled by the first passage time of the monitoring generalized drawdown process. Our theoretical framework simultaneously incorporates the first passage time, the absolute and relative drawdown times, and generalizes the scope of (Zhang and Hadjiliadis Methodol Comput Appl Probab 14(8):739–752 2012). We explicitly determine the path decompositions bridging these random times, the stochastic differential equations (SDE) governing each of the path fragments, and derive the Laplace transform of durations between any two of these random times. Applications include designing various risk measures summarizing several market information (e.g. historical maximum/minimum, absolute drawdown, relative drawdown) altogether. They allow us to study, in a microscopic way, a sequence of smaller drawdowns (signals) before a big drawdown (market crash) actually realizes. They can have applications to the detection of a potential market crash from signals of market drawdowns. We illustrate our results using a drifted Brownian motion.     

Weakly Convex and Concave Random Maps with Position Dependent Probabilities

Abstract

A random map is a discrete‐time dynamical system in which one of a number of transformations is randomly selected and applied in each iteration of the process. In this paper, we study random maps with position dependent probabilities on the interval. Sufficient conditions for the existence of absolutely continuous invariant measures for weakly convex and concave random maps with position dependent probabilities is the main result of this note.     

Asymptotics of Damped Periodic Motions with Random Initial Speed

We consider motion on the circle, possibly with friction and external forces, the initial velocity being a large random variable. We prove that under various assumptions the probability law of the stopping position of the motion converges to a distribution depending only on the motion equation. Here the time of stopping is either a constant or the first time instant at which the velocity vanishes, and the initial velocity is of the form αU + β, where U is a fixed random variable and α and/or β tend to infinity.     

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

Abstract

We study the limit at zero of the first-passage time density of a one-dimensional diffusion process over a moving boundary and we also deal with the inverse first-passage time problem, which consists of determining the boundary shape when the first-passage density is known. Our results generalize the analogous ones already known for Brownian motion. We illustrate some examples for which the results are obtained analytically and by a numerical procedure.     

Energetic and Dynamic Properties of a Quantum Particle in a Spatially Random Magnetic Field with Constant Correlations along one Direction

We consider an electrically charged particle on the Euclidean plane subjected to a perpendicular magnetic field which depends only on one of the two Cartesian co-ordinates. For such a “unidirectionally constant” magnetic field (UMF), which otherwise may be random or not, we prove certain spectral and transport properties associated with the corresponding one-particle Schr?dinger operator (without scalar potential) by analysing its “energy-band structure”. In particular, for an ergodic random UMF we provide conditions which ensure that the operator’s entire spectrum is almost surely absolutely continuous. This implies that, along the direction in which the random UMF is constant, the quantum-mechanical motion is almost surely ballistic, while in the perpendicular direction in the plane one has dynamical localization. The conditions are verified, for example, for Gaussian and Poissonian random UMF’s with non-zero mean-values. These results may be viewed as “random analogues” of results first obtained by A. Iwatsuka [Publ. RIMS, Kyoto Univ. 21 (1985) 385] and (non-rigorously) by J.E. Müller [Phys. Rev. Lett. 68 (1992) 385].

Heinz BAUER (31 January 1928 - 15 August 2002)

of Erlangen-Nürnberg

Communicated by Frank den Hollander submitted 30/12/04, accepted 13/06/05     

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].     

Hitting time results for Maker‐Breaker games

We study Maker‐Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property \begin{align*}\mathcal{P}\end{align*}. We focus on three natural target properties for Maker's graph, namely being k ‐vertex‐connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the k ‐vertex connectivity game exactly at the time the random graph process first reaches minimum degree 2k; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree 2; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree 4. The latter two statements settle conjectures of Stojakovi? and Szabó. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011