With $M(t):=\sup_{s\in[0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\cdot)$ with negative drift, this paper studies the rate of convergence of $\mathbb {P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\mathbb{P}(M(t)>\cdot)$ and $\mathbb{P}(M>\cdot)$. Our main result states that both metrics roughly decay as $\exp(-\vartheta t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when Gärtner--Ellis-type conditions are fulfilled.
The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dime…
We are interested in the quasi-stationarity of the time-inhomogeneous Markov process X t = B t (t + 1) $κ$ where (B t) t$\ge$0 is a one-dimensional Brownian motion and $κ$ $\in$ (0, $\infty$). We first show that the law of X t conditioned not to go out from (--1, 1) until the time t converges weakly towards the Dirac measure $δ$ 0 when $κ$ > 1 2 as t goes to infinity. Then we show that this conditioned probability converges weakly towards the quasi-stationary distribution of an Ornstein-Uhlenbeck process when $κ$ = 1 2. Finally, when $κ$ < 1 2 , it is shown that the conditioned probability con…
We obtain rates of convergence to stationarity in L^1-Wasserstein distance for a d-dimensional reflected Brownian motion (RBM) in the nonnegative orthant that are explicit in the dimension and the system parameters. The results are then applied to a class of RBMs considered in Blanchet-Chen (2016) and to rank-based diffusions including the Atlas model. In both cases, we obtain explicit rates and bounds on relaxation times. In the first case we improve the relaxation time estimates of O(d^4(log d)^2) obtained in Blanchet-Chen (2016) to O((log d)^2). In the latter case, we give the first results…
Extensions of the fractional Brownian fields are constructed over a complete Riemannian manifold. This construction is carried out for the full range of the Hurst parameter $α\in(0,1)$. In particular, we establish existence, distributional scaling (self-similiarity), stationarity of the increments, and almost sure Hölder continuity of sample paths. Stationary counterparts to these fields are also constructed.
We consider pairs of 3-dimensional Brownian paths, started at the origin and conditioned to have no intersections after time zero. We show that there exists a unique measure on pairs of paths that is invariant under this conditioning, while improving the previously known rate of convergence to stationarity.
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed. A framework for such concepts, proposed here, includes a wide class of random equivalence classes.
In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described…
We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Bro…
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Levy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, …
A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional Brownian motion. In this paper, we study some basic properties of this process, its non-Markovian and non-stationarity characteristics, the conditions under which it is a semimartingale, and the main features of its sample paths. We also show that this process could serve to get a good model of certain phenomena, taking not only the sign (like in the case of t…
We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in $\mathbb{R}^d_+$ for which local statistics converge to equilibrium at a rate independent of the dimension $d$. Under suitable assumptions on the reflection matrix, drift and diffusivity coefficients, dimension-independent stretched exponential convergence rates are obtained by estimating contractions in an underlying weighted distance between synchronously coupled RBMs. We also study the Symmetric Atlas model as a first step in obtaining dimension-independent convergence rates for RBMs not satisfying th…
A random dense countable set is characterized (in distribution) by independence and stationarity. Two examples are `Brownian local minima' and `unordered infinite sample'. They are identically distributed; the former ad hoc proof of this fact is now superseded by a general result.
Let $\{B_H(t):t\ge 0\}$ be a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. For the storage process $Q_{B_H}(t)=\sup_{-\infty\le s\le t} \left(B_H(t)-B_H(s)-c(t-s)\right)$ we show that, for any $T(u)>0$ such that $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.
To extend several known centered Gaussian processes, we introduce a new centered mixed self-similar Gaussian process called the mixed generalized fractional Brownian motion, which could serve as a good model for a larger class of natural phenomena. This process generalizes both the well known mixed fractional Brownian motion introduced by Cheridito [10] and the generalized fractional Brownian motion introduced by Zili [31]. We study its main stochastic properties, its non-Markovian and non-stationarity characteristics and the conditions under which it is not a semimartingale. We prove the long…
Let $B=(B_t)_{t\in {\mathbb{R}}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_T)_{t\in {\mathbb{R}}}$ is a Brownian motion independent of $B_T$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $ν$ on ${\mathbb{R}}$ we construct a stopping time $T\ge0$ with the above properties such that $B_T$ has distribution $ν$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of …
In this paper, we consider a two-dimensional sticky Brownian motion. Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. For example, a sticky Brownian motion can be used to model a storage system.with exceptional services. In this paper, we focus on stationary distributions for sticky Brownian motions. The main results obtained here include tail asymptotic properties in boundary stationary distributions, marginal distributions, and joint distributions. The ke…
In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as input.
Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $ρ:\mathbb{R}\to\mathbb{R}$ satisfying $ρ(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_\mathbb{R} |ρ(s)|^dds<\infty$, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_\varepsilon:=\sqrt{\varepsilon}\int_0^{\cdot/\varepsilon}f(Y(s))ds$ converge to those of $σW$ as $\varepsilon\to 0$, where $W$ is a s…
This article presents a review of some old and new results on the long time behavior of reflected diffusions. First, we present a summary of prior results on construction, ergodicity and geometric ergodicity of reflected diffusions in the positive orthant $\mathbb{R}^d_+$, $d \in \mathbb{N}$. The geometric ergodicity results, although very general, usually give implicit convergence rates due to abstract couplings and Lyapunov functions used in obtaining them. This leads us to some recent results on an important subclass of reflected Brownian motions (RBM) (constant drift and diffusion coeffici…
In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation o…
This paper studies the local structure of continuous random fields on $\mathbb R^d$ taking values in a complete separable linear metric space ${\mathbb V}$. Extending seminal work of Falconer, we show that the generalized $(1+k)$-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. These results motivate the further study of the structure of ${\mathbb V}$-valued intrinsic random functions of order $k$ (IRF$_k$,\ $k=0,1,\cdots$). To this end, we focus on the special case where ${\mathbb V}$ is a Hilbert space. Building on th…
Fractional Brownian Motion with Negative Hurst Exponent
Racah Institute of Physics, Hebrew University of Jerusalem（雷哈物理研究所，耶路撒冷希伯来大学）；ELI Beamlines Facility, Extreme Light Infrastructure ERIC（极轻基础设施欧洲研究联盟 ELI光束线设施）
Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(Q_{B_H}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,f_p(t)]}Q_{B_H}(s)>f_p(t))/f_p(t)=\mathscr C(t\log^{1-p} t)^{-1}$, for some $\mathscr C>0$, $\mathbb P(Q_{B_H}(t) > f_p(t)…
