Fake exponential Brownian motion

We construct a fake exponential Brownian motion, a continuous martingale different from classical exponential Brownian motion but with the same marginal distributions, thus extending results of Albin and Oleszkiewicz for fake Brownian motions. The ideas extend to other diffusions.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

Berry-Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion

In this paper, we derive explicit bounds for the Kolmogorov distance in the CLT and we prove the almost sure CLT for the quadratic variation of the sub-fractional Brownian motion. We use recent results on the Stein method combined with the Malliavin calculus and an almost sure CLT for multiple integrals.     

Large deviations for Brownian motion in a random scenery

We prove large deviations principles in large time, for the Brownian occupation time in random scenery . The random field is constant on the elements of a partition of ^d into unit cubes. These random constants, say consist of i.i.d. bounded variables, independent of the Brownian motion {B_s,s0}. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched' and ``annealed' settings.Mathematics Subject Classification (2000):60F10, 60J55, 60K37     

Random walks and Brownian motion on cubical complexes

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition kernels of the random walks converge to that for Brownian motion. The proof involves pulling back onto the complex the distribution of Brownian sample paths on a single cube, combined with a distribution on walks between cubes. The main application lies in analysing sets of evolutionary trees: several tree spaces are cubical complexes and we briefly describe our results and applications in this context.     

Survey on normal distributions,central limit theorem,Brownian motion and the related stochastic calculus under sublinear expectations

This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Itô’s type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.     

The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

Brownian motion in twisted domains

The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out units along the unit normal to the curve when the traveler is units away from the origin. The function is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case , . When , a twisted domain can reasonably be interpreted as a ``twisted cone.'

     

Global fluctuations for 1D log-gas dynamics

We study in this article the hydrodynamic limit in themacroscopic regime of the coupled system of stochastic differential equations,

(0.1)$d{\lambda }_t^i=\frac{1}{\sqrt{N}}d{W}_t^i?V^{\prime }\left({\lambda }_t^i\right)dt+\frac{\beta }{2N}\sum _{j\ne i}\frac{dt}{{\lambda }_t^i?{\lambda }_t^j},i=1,\dots ,N,$

with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta$-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty$ is known to satisfy a mean-field Mc-Kean–Vlasov equation. We prove that, for suitable initial conditions, fluctuations around the limit are Gaussian and satisfy an explicit PDE.The proof is very much indebted to the harmonic potential case treated in Israelsson (2001). Our key argument consists in showing that the time-evolution generator may be written in the form of a transport operator on the upper half-plane, plus a bounded non-local operator interpreted in terms of a signed jump process.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

Brownian motion in the framework of quaternion analysis

Abstract

We study Brownian motion in the setting of quaternion analysis. We give a quaternion version of the Itô’s integral.     

Functional Limit Theorems for Multiparameter Fractional Brownian Motion

We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy’s modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.      

Sharp integrability for Brownian motion in parabola-shaped regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains , 0<α<1. Together with some simple good-λ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for α=1/2 by Bañuelos et al. (Ann. Probab. 29 (2001) 882) and for general α by Li (Ann. Probab. 31 (2003) 1078). A sharp version of this result is also proved in higher dimensions.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

Jordan curves in the level sets of additive Brownian motion

This paper studies the topological and connectivity properties of the level sets of additive Brownian motion. More precisely, for each excursion set of this process from a fixed level, we give an explicit construction of a closed Jordan curve contained in the boundary of this excursion set, and in particular, in the level set of this process.

     

Central limit theorem for traces of large random symmetric matrices with independent matrix elements

We study Wigner ensembles of symmetric random matricesA=(a _ij ),i, j=1,...,n with matrix elementsa _ij ,ij being independent symmetrically distributed random variables

Central limit theorem for the heat kernel measure on the unitary group

We prove that for a finite collection of real-valued functions f₁,…,fn on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of under the properly scaled heat kernel measure at a given time on the unitary group U(N) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N−1. In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.