Transformation formulas for fractional Brownian motion

We derive a Molchan–Golosov-type integral transform which changes fractional Brownian motion of arbitrary Hurst index K$K$ into fractional Brownian motion of index H$H$. Integration is carried out over [0,t]$[0,t]$, t>0$t>0$. The formula is derived in the time domain. Based on this transform, we construct a prelimit which converges in L²(P)$L^2\left(\mathbb{P}\right)$-sense to an analogous, already known Mandelbrot–Van Ness-type integral transform, where integration is over (−∞,t]$(-\infty ,t]$, t>0$t>0$.     

Local independence of fractional Brownian motion

Let _σ(t,t′)${\sigma }_{(t,t^{\prime })}$ be the sigma-algebra generated by the differences _Xs−_Xs′$X_s-X_{s^{\prime }}$ with s,s^′∈(t,t^′)$s,s^{\prime }\in (t,t^{\prime })$, where _{(Xt)−∞<t<∞}${\left(X_t\right)}_{-\infty <t<\infty }$ is the fractional Brownian motion with Hurst index H∈(0,1)$H\in (0,1)$. We prove that for any two distinct timepoints _t1$t_1$ and _t2$t_2$ the sigma-algebras _{σ(t1−ε,t1+ε)}${\sigma }_{(t_1-\epsilon ,t_1+\epsilon )}$ and _{σ(t2−ε,t2+ε)}${\sigma }_{(t_2-\epsilon ,t_2+\epsilon )}$ are asymptotically independent as ε↘0$\epsilon \searrow 0$. We show the independence in the strong sense that Shannon’s mutual information between the two σ$\sigma$-algebras tends to zero as ε↘0$\epsilon \searrow 0$. Some generalizations and quantitative estimates are also provided.     

On sojourn of Brownian motion inside moving boundaries

We investigate the large scale structure of certain sojourn sets of one dimensional Brownian motion within two-sided moving boundaries. The macroscopic Hausdorff dimension, upper mass dimension and logarithmic density of these sets, are computed. We also give a uniform macroscopic dimension result for the Brownian level sets.     

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.     

On delayed averages of Brownian motion in Banach spaces

Let {W(t): t ≥ 0} be μ-Brownian motion in a real separable Banach space B, and let a_T be a nondecreasing function of T for which (i) 0 < a_T ≤ T (T ≥ 0), (ii) $\text{a}{}_{\mathrm{T}}\text{T}$ is nonincreasing. We establish a Strassen limit theorem for the net {ξ_T: T ≥ 3}, where

$\text{ξ}{}_{\mathrm{T}}\text{=}\text{W(T + ta}{}_{\mathrm{T}}\text{) ? W(T)}\text{{2a}{}_{\mathrm{T}}\text{[}\text{log}\text{(}\text{T}\text{a}{}_{\mathrm{T}}\text{) +}\text{log log}\text{T]}}\text{1}\text{2}\text{,}\text{0 ? t ? 1}$

     

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

Any solution of the functional equation

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

Particle picture interpretation of some Gaussian processes related to fractional Brownian motion

We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.     

Brownian Chen series and Atiyah-Singer theorem

The purpose of this work is to give a new and short proof of the Atiyah-Singer local index theorem for the Dirac operator on the spin bundle. This proof is obtained by using heat semigroups approximations based on the truncation of Brownian Chen series.     

Grafting hyperbolic metrics and Eisenstein series

The family hyperbolic metric for the plumbing variety {zw = t} and the non holomorphic Eisenstein series are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil–Petersson metric and a local form of symplectic reduction.     

Multiple points of trajectories of multiparameter fractional Brownian motion

Consider 0<α<1 and the Gaussian process Y(t) on ℝ ^N with covariance E(Y(s)Y(t))=|t|^2α+|s|^2α−|t−s|^2α, where |t| is the Euclidean norm of t. Consider independent copies X ¹,…,X ^d of Y and␣the process X(t)=(X ¹(t),…,X ^d (t)) valued in ℝ ^d . When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^{k N /α−( k −1) d} (loglog(1/ɛ)) ^k . If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ ^d (log(1/ɛ) logloglog 1/ɛ) ^k . (This includes the case k=1.) Received: 20 May 1997 / Revised version: 15 May 1998     

Sojourns and future infima of planar Brownian motion

Summary Let {W(t); 0t1} be a two-dimensional Wiener process starting from 0. We are interested in the almost sure asymptotic behaviour, asr tends to 0, of the processesX(r) andY(r), whereX(r) denotes the total time spent byW in the ball centered at 0 with radiusr andY(r) the distance between 0 and the curve {W(t);rt1}. While a characterization of the lower functions ofY was previously established by Spitzer [S], we characterize via integral tests its upper functions as well as the upper and lower functions ofX.     

Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let {X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sup_s∈[0,T]X(s)>u), as u→∞, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T, the case of T having regularly varying tail distribution with parameter λ∈(0,1) and the case of T having slowly varying tail distribution.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Brownian motion and random walk perturbed at extrema

Let b _t be Brownian motion. We show there is a unique adapted process x _t which satisfies dx _t = db _t except when x _t is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of the process. For some ranges of the parameters this is already known. We show that if a random walk close to b _t is perturbed properly, its paths are close to those of x _t . Received: 15 October 1997 / Revised version: 18 May 1998     

On quadratic functionals of the Brownian sheet and related processes

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on [0,1]²$[0,1{]}^2$. In particular: (i) we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes, (ii) we explicitly calculate the Laplace transform of such functionals by means of Karhunen–Loève expansions, (iii) we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75–87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177–193]. Our results extend some classical computations due to Lévy [Wiener's random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171–186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen–Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhäuser Verlag, Basel, 2003, pp. 57–93].     

A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering

Fourier normal ordering (Unterberger, 2009) [34] is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion B$B$ with arbitrary Hurst index α$\alpha$ (in particular, for α≤1/4$\alpha \le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of B$B$ defined in Unterberger (2009) [32]. The regularization procedure is applied to ‘Fourier normal ordered’ iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees (Connes and Kreimer, 1998) [6] for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments.