Exact L 2 -small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems (original) (raw)

Exact Small Ball Constants for Some Gaussian Processes under the L 2 Norm

Journal of Mathematical Sciences, 2005

We find some logarithmic and exact small deviation asymptotics for the L 2-norms of certain Gaussian processes closely connected with a Wiener process. In particular, processes obtained by centering and integrating Brownian motion and Brownian bridge are examined. Bibliography: 28 titles.

Exact small deviation asymptotics for some Brownian functionals

We find exact small deviation asymptotics with respect to a weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require knowledge of the eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

Theory of Probability & Its Applications, 2005

We find the logarithmic L 2 -small ball asymptotics of some Gaussian processes related to the fractional Brownian motion, fractional Ornstein -Uhlenbeck process (fOU) and their integrated analogues. To that end we use general theorems on spectral asymptotics of integral operators obtained by combining them with the classical theorem of Weyl. In the simplest case of fractional Brownian motion we generalize the result of . We consider also the fractional Lévy's Brownian motion as well as the multiparameter fOU process on the bounded domain.

Short Time Asymptotics of a Certain Infinite Dimensional Diffusion Process

Stochastic Analysis and Related Topics VII, 2001

The main objective of this contribution is to prove the Varadhan type short-time asymptotics of an infinite dimensional diffusion process associated with a certain Dirichlet form. This paper gives a generalization of Fang's results of the Ornstein-Uhlenbeck process on an abstract Wiener space.

$L_2$-small ball asymptotics for a family of finite-dimensional perturbations of Gaussian functions

arXiv: Probability, 2019

In this article we study the small ball probabilities in L_2L_2L2-norm for a family of finite-dimensional perturbations of Gaussian functions. We define three types of perturbations: non-critical, partially critical and critical; and derive small ball asymptotics for the perturbated process in terms of the small ball asymptotics for the original process. The natural examples of such perturbations appear in statistics in the study of empirical processes with estimated parameters (the so-called Durbin's processes). We show that the Durbin's processes are critical perturbations of the Brownian bridge. Under some additional assumptions, general results can be simplified. As an example we find the exact L2L_2L_2-small ball asymptotics for critical perturbations of the Green processes (the processes which covariance function is the Green function of the ordinary differential operator).

Exact L 2 Small Balls of Gaussian Processes

Journal of Theoretical Probability, 2000

We prove a comparison theorem extending Li (6) and develop a complex-analytic approach to treat L 2 small ball probabilities of Gaussian processes. We demonstrate the techniques for the m-times integrated Brownian motions and in examples where one can not apply Li's comparison theorem.

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Probability Theory and Related Fields, 2004

Let (X t (δ) ,t≥0) be the BESQδ process starting at δx. We are interested in large deviations as \({{\delta \rightarrow \infty}}\) for the family {δ−1X t (δ) ,t≤T}δ, – or, more generally, for the family of squared radial OUδ process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.

Limit theorems for reflected Ornstein-Uhlenbeck processes

Statistica Neerlandica, 2014

This paper studies one-dimensional Ornstein-Uhlenbeck processes, with the distinguishing feature that they are reflected on a single boundary (put at level 0) or two boundaries (put at levels 0 and d > 0). In the literature they are referred to as reflected OU (ROU) and doubly-reflected OU (DROU) respectively. For both cases, we explicitly determine the decay rates of the (transient) probability to reach a given extreme level. The methodology relies on sample-path large deviations, so that we also identify the associated most likely paths. For DROU, we also consider the 'idleness process' L t and the 'loss process' U t , which are the minimal nondecreasing processes which make the OU process remain 0 and d, respectively. We derive central limit theorems (CLT s) for U t and L t , using techniques from stochastic integration and the martingale CLT.

On the rate of convergence to equilibrium for two-sided reflected Brownian motion and for the Ornstein–Uhlenbeck process

Queueing Systems

This paper studies the rate of convergence to equilibrium for two diffusion models that arise naturally in the queueing context: two-sided reflected Brownian motion and the Ornstein-Uhlenbeck process. Specifically, we develop exact asymptotics and upper bounds on total variation distance to equilibrium, which can be used to assess the quality of the steady state as an approximation to finite-horizon performance quantities. Our analysis relies upon the simple spectral structure that these two processes possess, thereby explaining why the convergence rate is "pure exponential," in contrast to the more complex convergence exhibited by one-sided reflected Brownian motion.