Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.     

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of excursions of Markov processes. Our setting allows us to solve the Skorokhod embedding problem, in particular, for the age process of excursions of a Markov process, for diffusions and their signed age processes, for Azéma’s martingale and for Bessel processes of dimension smaller than 2.This work is a continuation and an important generalization of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110]. Our methodology is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Itô measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110] for measures with atoms.     

On estimating the derivatives of symmetric diffusions in stationary random environment,with applications to ∇<Emphasis Type="Italic">ϕ</Emphasis> interface model

We consider diffusions on ℝ^d or random walks on ℤ^d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients. We show existence and H?lder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives. In the case of random walks, these estimates are applied to the Ginzburg-Landau ∇ϕ interface model.     

Erratum to: Risk-Sensitive Control with Near Monotone Cost

The infinite horizon risk-sensitive control problem for non-degenerate controlled diffusions is analyzed under a ‘near monotonicity’ condition on the running cost that penalizes large excursions of the process.     

Asymptotics for transient and stationary probabilities for finite and infinite buffer discrete time queues

Consider a discrete time queue with i.i.d. arrivals (see the generalisation below) and a single server with a buffer length m. Let τm be the first time an overflow occurs. We obtain asymptotic rate of growth of moments and distributions of τm as m → ∞. We also show that under general conditions, the overflow epochs converge to a compound Poisson process. Furthermore, we show that the results for the overflow epochs are qualitatively as well as quantitatively different from the excursion process of an infinite buffer queue studied in continuous time in the literature. Asymptotic results for several other characteristics of the loss process are also studied, e.g., exponential decay of the probability of no loss (for a fixed buffer length) in time [0,η], η → ∞, total number of packets lost in [0, η, maximum run of loss states in [0, η]. We also study tails of stationary distributions. All results extend to the multiserver case and most to a Markov modulated arrival process. This revised version was published online in June 2006 with corrections to the Cover Date.     

Testing diffusion processes for non-stationarity

Financial data are often assumed to be generated by diffusions. Using recent results of Fan et al. (J Am Stat Assoc, 102:618–631, 2007; J Financ Econometer, 5:321–357, 2007) and a multiple comparisons procedure created by Benjamini and Hochberg (J R Stat Soc Ser B, 59:289–300, 1995), we develop a test for non-stationarity of a one-dimensional diffusion based on the time inhomogeneity of the diffusion function. The procedure uses a single sample path of the diffusion and involves two estimators, one temporal and one spatial. We first apply the test to simulated data generated from a variety of one-dimensional diffusions. We then apply our test to interest rate data and real exchange rate data. The application to real exchange rate data is of particular interest, since a consequence of the law of one price (or the theory of purchasing power parity) is that real exchange rates should be stationary. With the exception of the GBP/USD real exchange rate, we find evidence that interest rates and real exchange rates are generally non-stationary. The software used to implement the estimation and testing procedure is available on demand and we describe its use in the paper.     

Langevin Diffusions and Metropolis-Hastings Algorithms

We consider a class of Langevin diffusions with state-dependent volatility. The volatility of the diffusion is chosen so as to make the stationary distribution of the diffusion with respect to its natural clock, a heated version of the stationary density of interest. The motivation behind this construction is the desire to construct uniformly ergodic diffusions with required stationary densities. Discrete time algorithms constructed by Hastings accept reject mechanisms are constructed from discretisations of the algorithms, and the properties of these algorithms are investigated.     

Branching particle systems in spectrally one-sided Lévy processes

We investigate the branching structure coded by the excursion above zero of a spectrally positive Lévy process. The main idea is to identify the level of the Lévy excursion as the time and count the number of jumps upcrossing the level. By regarding the size of a jump as the birth site of a particle, we construct a branching particle system in which the particles undergo nonlocal branchings and deterministic spatial motions to the left on the positive half line. A particle is removed from the system as soon as it reaches the origin. Then a measure-valued Borel right Markov process can be defined as the counting measures of the particle system. Its total mass evolves according to a Crump- Mode-Jagers (CMJ) branching process and its support represents the residual life times of those existing particles. A similar result for spectrally negative Lévy process is established by a time reversal approach. Properties of the measurevalued processes can be studied via the excursions for the corresponding Lévy processes.     

Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles

Abstract

We study the long-time behavior of a linear inhomogeneous Boltzmann equation. The collision operator is modeled by a simple relaxation towards the Maxwellian distribution with zero mean and fixed lattice temperature. Particles are moving under the action of an external potential that confines particles, i.e., there exists a unique stationary probability density. Convergence rate towards global equilibrium is explicitly measured based on the entropy dissipation method and apriori time independent estimates on the solutions. We are able to prove that this convergence is faster than any algebraic time function, but we cannot achieve exponential convergence.     

Cyclically stationary Brownian local time processes

Summary. Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray–Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For T the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar Lévy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams’ decomposition of It?’s law of Brownian excursions. Received: 28 June 1995     

First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.      

EXCURSIONS AND LVY SYSTEM OF BOUNDARY PROCESS

In this paper, the authors investigate the joint distribution of end points of excursion away from a closed set straddling on a fixed time and use this result to compute the Levy system and the Dirichlet form of the boundary process.     

EXCURSIONS AND LEVY SYSTEM OF BOUNDARY PROCESS＊＊＊

In this paper, the authors investigate the joint distribution of end points of excursion awayfrom a closed set straddling on a fixed time and use this result to compute the Levy systemand the Dirichlet form of the boundary process.     

Langevin-Type Models I: Diffusions with Given Stationary Distributions and their Discretizations*

We describe algorithms for estimating a given measure known up to a constant of proportionality, based on a large class of diffusions (extending the Langevin model) for which is invariant. We show that under weak conditions one can choose from this class in such a way that the diffusions converge at exponential rate to , and one can even ensure that convergence is independent of the starting point of the algorithm. When convergence is less than exponential we show that it is often polynomial at verifiable rates. We then consider methods of discretizing the diffusion in time, and find methods which inherit the convergence rates of the continuous time process. These contrast with the behavior of the naive or Euler discretization, which can behave badly even in simple cases. Our results are described in detail in one dimension only, although extensions to higher dimensions are also briefly described.     

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in Hubalek and Kuznetsov (Electron. Commun. Probab. 16:84–95, 2011) and Kuznetsov (Ann. Probab. 39(3):1027–1060, 2011). We also derive several new results related to (i) the entrance law of a stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of a stable process reflected at its past infimum, (iii) the distribution of the lifetime of a stable process conditioned to hit zero continuously and (iv) the entrance law and the last passage time of the radial part of a multidimensional symmetric stable process.     

Transformations of diffusions with jumps

The paper deals with some transformations of diffusions with jumps. We consider the class of diffusions with jumps that is closed with respect to composition with invertible, twice continuously differentiable functions. A special random time change gives us again a diffusion with jumps. A result on transformation of a measure is valid for this class of diffusions with jumps. Bibliographty: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 79–100.     

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??In this paper we describe the excursions from a set explicitly for
recurrent Markov chain with discrete time. A new exit system is presented through using a
law conditioned by specifying the starting point and ending point of excursions. In a simple
case, we verify that our conditioned excursion law is a discrete approximation for that of
a diffusion.     

Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities.     

On decay of solutions to nonlinear Schrödinger equations

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.