On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Let M =G/H be an irreducible homogeneous compact manifold of dimension n equipped with its canonical Riemannian metric. Let γ be the lowest nonzero eigenvalue of the Laplace operator. Let μ be the normalized Haar measure and μ _t be the heat diffusion measure, i.e., the law of Brownian motion started at a fixed origin in M. We show that the total variation distance between μ_t and μ is not small for t ≪λ ⁻¹ logn.This is sharp, up to a factor of two, in the case of compact irreducible simply connected symmetric spaces.     

On weak convergence to Brownian motion

Summary We consider a minimal form of the usual conditions for the dependent central limit theorem and invariance principle for near martingales. We show that these conditions imply convergence to Brownian motion in a way that is slightly stronger than weak convergence in D[0,). On the other hand, if a sequence of processes with paths in D[0,) converges to Brownian motion in this way, then we can always find a sequence of partitions of the time axis that is such that these conditions hold for the corresponding array of increments.     

Brownian motion reflected on Brownian motion

 We study Brownian motion reflected on an ``independent' Brownian path. We prove results on the joint distribution of both processes and the support of the parabolic measure in the space-time domain bounded by a Brownian path. We show that there exist two different natural local times for a Brownian path reflected on a Brownian path. Received: 25 October 2000 / Revised version: 30 March 2001 / Published online: 20 December 2002     

Traps for reflected Brownian motion

Consider an open set , d ≥ 2, and a closed ball . Let denote the expectation of the hitting time of B for reflected Brownian motion in D starting from x ∈ D. We say that D is a trap domain if . A domain D is not a trap domain if and only if the reflecting Brownian motion in D is uniformly ergodic. We fully characterize the simply connected planar trap domains using a geometric condition and give a number of (less complete) results for d > 2. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0303310. Research partially supported by NSF grant DMS-0201435.     

Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion

Any solution of the functional equation

On the infimum attained by the reflected fractional Brownian motion

Let {B _H (t):t≥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 \(T(u)=o(u^{\frac {2H-1}{H}})\) , $$\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u)\sim\mathbb P(Q_{B_{H}}(0)>u),$$ as \(u\to \infty \) . This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

Large deviations for multi-dimensional reflected fractional Brownian motion

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.     

Rates of convergence for increments of Brownian motion

The results of this paper concern rates of convergence for increments of Brownian motion. As a by-product we give some improvements of a result of Bolthausen dealing with Strassen's law of the iterated logarithm.     

On the rate of convergence to equilibrium in the Becker-Döring equations

We provide an explicit rate of convergence to equilibrium for solutions of the Becker-Döring equations using the energy/energy-dissipation relation. The main difficulty is the structure of equilibria of the Becker-Döring equations, which do not correspond to a Gaussian measure, such that a logarithmic Sobolev-inequality is not available. We prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e^−ct1/3.     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.     

Filtering of a reflected Brownian motion with respect to its local time

We consider a filtering problem when the state process is a reflected Brownian motion _Xt$X_t$ and the observation process is its local time _Λs${\Lambda }_s$, for s≤t$s\le t$. For this model we derive an approximation scheme based on a suitable interpolation of the observation process _Λt${\Lambda }_t$. The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale.     

Markov modulation of a two-sided reflected Brownian motion with application to fluid queues

In this paper, we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states.     

Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

We study the weak convergence of the family of processes {V _n (t)} _n??? defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {?? _n (u)} _n??? is a family of processes converging in law to a Brownian motion, as n????. We consider two cases of {?? _n }. First, we construct ?? _n based on the well-known Donsker??s theorem and show that {V _n (t)} _n??? converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n????. Second, we construct ?? _n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {V _n (t)} _n???.     

On the sub-mixed fractional Brownian motion

Let {SHt, t ≥ 0} be a linear combination of a Brownian motion and an independent sub-fractional Brownian motion with Hurst index 0 H 1. Its main properties are studied.They suggest that SHlies between the sub-fractional Brownian motion and the mixed fractional Brownian motion. We also determine the values of H for which SHis not a semi-martingale.