Homogenization on nested fractals

Summary We study the homogenization problem on nested fractals. LetX _t be the continuous time Markov chain on the pre-nested fractal given by puttingi.i.d. random resistors on each cell. It is proved that under some conditions, converges in law to a constant time change of the Brownian motion on the fractal asn, where is the contraction rate andt _E is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.Dedicated to Professor Masatoshi Fukushima on his 60th birthdayResearch partially supported by the Yukawa Foundation     

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.     

Persistence probabilities for stationary increment processes

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron–de Marsily model in ${\mathbb{Z}}^2$ with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.     

The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups

We establish Lamperti representations for semi-stable Markov processes in locally compact groups. We also study the particular cases of processes with values in R$\mathbb{R}$ and C$\mathbb{C}$ under the hypothesis that they do not visit 0. These Lamperti representations yield some properties of these semi-stable Markov processes.     

Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant

Summary This work is concerned with the existence and uniqueness of a class of semimartingale reflecting Brownian motions which live in the non-negative orthant of ^d . Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the orthant the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d-1)-dimensional faces that form the boundary of the orthant, the bounded variation part of the process increases in a given direction (constant for any particular face) so as to confine the process to the orthant. For historical reasons, this pushing at the boundary is called instantaneous reflection. In 1988, Reiman and Williams proved that a necessary condition for the existence of such a semimartingale reflecting Brownian motion (SRBM) is that the reflection matrix formed by the directions of reflection be completely-L. In this work we prove that condition is sufficient for the existence of an SRBM and that the SRBM is unique in law. It follows from the uniqueness that an SRBM defines a strong Markov process. Our results have potential application to the study of diffusions arising as approximations tomulti-class queueing networks.Research supported in part by NSF Grants DMS 8657483, 8722351 and 9023335, and a grant from AT&T Bell Labs. In addition, R.J. Williams was supported in part during the period of this research by an Alfred P. Sloan Research Fellowship     

SDEs with constraints driven by semimartingales and processes with bounded p-variation

We study the existence, uniqueness and stability of solutions of general stochastic differential equations with constraints driven by semimartingales and processes with bounded $p$-variation. Applications to SDEs with constraints driven by fractional Brownian motion and standard Brownian motion are given.     

Multiscale diffusion approximations for stochastic networks in heavy traffic

Stochastic networks with time varying arrival and service rates and routing structure are studied. Time variations are governed by, in addition to the state of the system, two independent finite state Markov processes X and Y. The transition times of X are significantly smaller than typical inter-arrival and processing times whereas the reverse is true for the Markov process Y. By introducing a suitable scaling parameter one can model such a system using a hierarchy of time scales. Diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. In particular, it is shown that, under certain conditions, properly normalized buffer content processes converge weakly to a reflected diffusion. The drift and diffusion coefficients of this limit model are functions of the state process, the invariant distribution of X, and a finite state Markov process which is independent of the driving Brownian motion.     

Excursion theory for rotation invariant Markov processes

Summary Let (X _t,P ^x) be a rotation invariant (RI) strong Markov process onR ^d{0} having a skew product representation [|X _t |, ], where ( _t ) is a time homogeneous, RI strong Markov process onS ^d–1, |X _t|, and _t are independent underP ^x andA _t is a continuous additive functional of |X _t|. We characterize the rotation invariant extensions of (X _t,P ^x) toR ^d. Two examples are given: the diffusion case, where especially the Walsh's Brownian motion (Brownian hedgehog) is considered, and the case where (X _t,P ^x) is self-similar.     

A Markov additive risk process in dimension 2 perturbed by a fractional Brownian motion

We consider the following theoretical reinsurance ruin problem. An insurance company has two types of independent claims, respectively modeled by a Markov additive process (large claims) and a fractional Brownian motion (small claims) with Hurst parameter H∈[1/2,1)$H\in [1/2,1)$, and chooses to reinsure both of them according to a quota share policy. This leads to studying a bivariate risk process. We study two types of ruins, corresponding to either ruin of one of the risk processes, or of both. We obtain asymptotics of the corresponding ruin probabilities when initial reserves tend to infinity along a direction.     

Fluctuations of Omega-killed spectrally negative Lévy processes

In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases $\omega \left(x\right)=q$ and $\omega \left(x\right)=q{\mathbf{1}}_{(a,b)}\left(x\right)$, we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate $\omega \left(x\right)$ when the Lévy surplus process is at level $x<0$. Finally, we apply these results to obtain some exit identities for spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.     

Extremes of q-Ornstein–Uhlenbeck processes

Two limit theorems are established on the extremes of a family of stationary Markov processes, known as $q$-Ornstein–Uhlenbeck processes with $q\in (?1,1)$. Both results are crucially based on the weak convergence of the tangent process at the lower boundary of the domain of the process, a positive self-similar Markov process little investigated so far in the literature. The first result is the asymptotic excursion probability established by the double-sum method, with an explicit formula for the Pickands constant in this context. The second result is a Brown–Resnick-type limit theorem on the minimum process of i.i.d. copies of the $q$-Ornstein–Uhlenbeck process: with appropriate scalings in both time and magnitude, a new semi-min-stable process arises in the limit.     

A strong uniform approximation of fractional Brownian motion by means of transport processes

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H$H$, and we derive a rate of convergence, which becomes better when H$H$ approaches 1/2$1/2$. The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.     

Small noise and long time phase diffusion in stochastic limit cycle oscillators

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor $\epsilon \searrow 0$ – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp ?\left(c{\epsilon }^{?2}\right)$, $c>0$, and we show both that on the time scale ${\epsilon }^{?2}$ the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.     

Sur le comportement asymptotique des potentiels d'une chaîne de Markov sur ℝ d

Résumé Dans cet article j'étudie le comportement à l'infini des potentiels des chaînes de Markov sur ^d (d3) proches du mouvement brownien, tout spécialement le cas des marches aléatoires, ainsi que des critères de transience et de récurrence inspirés de la méthode utilisée.

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We study the asymptotic behaviour of potentials of Markov chains on ^d (d3), closed to Brownian motion, and particularly the case of random walks. Following a similar approach, we give transience and recurrence criteria.

     

Some bivariate stochastic models arising from group representation theory

The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued second-order differential operator). The matrix-valued spherical functions associated to the compact symmetric pair $(\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right),\mathrm{diag}\mathrm{SU}\left(2\right))$ will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright–Fisher models involving only mutation effects.     

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark–Ocone derivative to their continuous counterparts. Moreover, given a sequence $\left(X^n\right)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $\sigma \left(B\right)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients.     

A class of path-valued Markov processes and its applications to superprocesses

Summary Let ( _s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with ( _s ), which is closely related to the so-called superprocess with spatial motion ( _s ). In particular, a subsetH of the state space of ( _s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when ( _s ) is a diffusion process, we recover certain results recently obtained by Dynkin.