Brownian Sheet Research Articles

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H 1, H 2 ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.

Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Oksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black–Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.

Convergence faible vers une classe de processus Gaussiens en norme de Besov anisotropique

We prove an approximation result of Donsker's type in anisotropic Besov spaces for a class of two parameter Gaussian processes. The particular case of the fractional Brownian sheet was considered in [X. Bardina, C. Florit, Approximation in law to the d-parameter fractional Brownian sheet based on the functional invariance principle, Preprint, 2003] for the Banach space of continuous functions.

Le cocycle brownien de degré deux comme bruit blanc sur les droites affines de l'espace

Inspired by the work of Čencov [N.N. Čencov, Le mouvement brownien à plusieurs paramètres de M. Lévy et le bruit blanc généralisé, Teor. Veroyatnost. i Primenen. 3 (1957) 281–282. [1]] on multiparameter Lévy's Brownian motion (see [P. Lévy, Processus Stochastiques et Mouvement Brownien. Suivi d'une note de M. Loève, Gauthier-Villars, Paris, 1948. [4]]), we show that the white noise on the straight lines in dimension three generates a cocycle of degree two, whose restrictions to plans have the law of plane Brownian sheets. We check that this degree two cocycle is the same than the one studied by us in a preceding work. Consequently we give here another proof of the existence of the degree two Brownian cocycle in dimension three. Moreover this construction is adapted to the numerical simulation. To cite this article: J. Depauw, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Sectorial Local Non-Determinism and the Geometry of the Brownian Sheet

We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$, then with probability one $\text{dim}_H B(F) = 2 \text{dim} F$ for all Borel sets of $R_+^N$, where $\text{dim}_H$ could be everywhere replaced by the ``Hausdorff,'' ``packing,'' ``upper Minkowski,'' or ``lower Minkowski dimension.'' (3) Let $M_k$ be the set of $k$-multiple points of $B$. If $N \leq d/2$ and $ Nk > (k-1)d/2$, then $\text{dim}_H M_k = \text{dim}_p M_k = 2 Nk - (k-1)d$, a.s. The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $N>d/2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $(N,1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).

Properties of local-nondeterminism of Gaussian and stable random fields and their applications

In this survey, we first review various forms of local nondeterminism and sectorial local nondeterminism of Gaussian and stable random fields. Then we give sufficient conditions for Gaussian random fields with stationary increments to be strongly locally nondeterministic (SLND). Finally, we show some applications of SLND in studying sample path properties of (N,d)-Gaussian random fields. The class of random fields to which the results are applicable includes fractional Brownian motion, the Brownian sheet, fractional Brownian sheets and so on.

Approximation in law to the $d$-parameter fractional Brownian sheet based on the functional invariance principle

We show a result of approximation in law of the d -parameter fractional Brownian sheet in the space of the continuous functions on [0,T]^d . The construction of these approximations is based on the functional invariance principle.

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 . 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].

Brownian sheet and reflectionless potentials

In this paper, the investigation into stochastic calculus related with the KdV equation, which was initiated by S. Kotani [Construction of KdV-flow on generalized reflectionless potentials, preprint, November 2003] and made in succession by N. Ikeda and the author [Quadratic Wiener functionals, Kalman–Bucy filters, and the KdV equation, Advanced Studies in Pure Mathematics, vol. 41, pp. 167–187] and S. Taniguchi [On Wiener functionals of order 2 associated with soliton solutions of the KdV equation, J. Funct. Anal. 216 (2004) 212–229] is continued. Reflectionless potentials give important examples in the scattering theory and the study of the KdV equation; they are expressed concretely by their corresponding scattering data, and give a rise of solitons of the KdV equation. Ikeda and the author established a mapping ψ of a family G 0 of probability measures on the one-dimensional Wiener space to the space Ξ 0 of reflectionless potentials. The mapping gives a probabilistic expression of reflectionless potential. In this paper, it will be shown that ψ is bijective, and hence G 0 and Ξ 0 can be identified. The space Ξ 0 was extended to the one Ξ of generalized reflectionless potentials, and was used by V. Marchenko to investigate the Cauchy problem for the KdV equation and by S. Kotani to construct KdV-flows. As an application of the identification of G 0 and Ξ 0 via ψ , taking advantage of the Brownian sheet, it will be seen that convergences of elements in G 0 realizes the extension of Ξ 0 to Ξ .

Degree two Brownian Sheet in Dimension three

In this work, we construct a degree two Brownian Sheet in dimension three which is obtained from the ordinary Brownian Sheet in ℝ2 in the same way that Paul Levy has obtained his two-parameter Brownian motion from the ordinary Brownian motion.

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

Let BH = {BH(t ), t ∈ ℝN} be an (N, d)-fractional Brownian sheet with index H = (H1, . . . , HN) ∈ (0, 1)N. The uniform and local asymptotic properties of BH are proved by using wavelet methods. The Hausdorff and packing dimensions of the range BH ([0, 1]N), the graph Gr BH ([0, 1]N) and the level set are determined.

Functional Central Limit Theorem for Double-Indexed Summation Process

We establish necessary and sufficient conditions for the weak convergence of a double-indexed summation process to a Brownian sheet in some Holder spaces.

Singularity functions for fractional processes: application to the fractional Brownian sheet

In this paper almost sure convergence and asymptotic normality of generalized quadratic variation are studied. The main result in this paper extend classical results from Baxter and Gladyshev so that they can be applied to fractional Gaussian processes. An application to the estimation of the true axes of a fractional Brownian sheet is also obtained.

The influence of a log-type small ball factor in the study of the lower classes

Let { B H 1 , H 2 ( t 1 , t 2 ) , t 1 ⩾ 0 , t 2 ⩾ 0 } be a fractional Brownian sheet with indexes 0 < H 1 , H 2 < 1 . When H 1 = H 2 : = H , there is a logarithmic factor in the small ball function of the sup-norm statistic of B H , H . First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of B H , H . Then we characterize the sufficiency part of the lower classes of these statistics by an integral test. Finally, when we consider the sup-norm statistic, the influence of the log-type small ball factor in the necessity part is measured by a second integral test.

Optimality of an explicit series expansion of the fractional Brownian sheet

We show that an explicit series expansion of the fractional Brownian motion derived by Dzhaparidze and Van Zanten (Probab. Theory Related Fields 130 (1) (2004) 39) is rate-optimal in the sense that the expected uniform norm of the truncated series vanishes at the optimal rate as the truncation point tends to infinity. The same it true for the expansion of the general fractional Brownian sheet that is obtained by a canonical extension.

Identification of an isometric transformation of the standard Brownian sheet

We use quadratic variations to identify almost surely the coordinate system where the standard Brownian sheet is defined. This identification is carried out with the help of an algorithmic-like procedure.

Level crossings of a two-parameter random walk

We prove that the number Z ( N ) of level crossings of a two-parameter simple random walk in its first N × N steps is almost surely N 3 / 2 + o ( 1 ) as N → ∞ . The main ingredient is a strong approximation of Z ( N ) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm for simulating the level sets of the Brownian sheet.

Constructing a Brownian Sheet with Values in a Compact Riemannian Manifold

In this paper, we propose a new method of constructing a two-parameter random field W x (s, t ), x ∈ M , with values in a compact Riemannian manifold M possessing the property that the random processes W x ( · ,t )a ndW x M (s, · ) are Brownian motions on the manifold M with parameters t and s , respectively, issuing from the point x .( By aBrownian motion on a manifold M with parameter t we mean the diffusion process generated by the operator −(t/2)∆M , where ∆M is the Laplace operator on the manifold M.) For the case in which the manifold is a compact Lie group, the two-parameter random field constructed in the paper coincides with the Brownian sheet defined by Malliavin (1) in 1991. (Malliavin called this random field a Brownian motion with values in C((0, 1) ,M ) , which is the set of continuous functions defined on the closed interval (0, 1) and taking values in M.) Nevertheless, for the case in which the manifold is a compact Lie group, the method proposed in the present paper essentially differs from that used in Malliavin's paper. 1. FIRST STEP IN THE CONSTRUCTION OF THE RANDOM FIELD W x Suppose that M is a d-dimensional compact Riemannian manifold without boundary isomet- rically embedded in R m .B y aBrownian sheet with values in R m we mean the family of m independent standard Brownian sheets. Suppose that Wt,s is an n-dimensional Brownian sheet. Consider Wt,s as a process taking values in the space C((0, 1), R m ) . We denote this process by the symbol Wt . We introduce the following notation: if E is a locally convex space, then E t denotes C((0 ,t ) ,E ); if y ∈ C((0, 1), R m ) is a continuous function, then W y denotes the distribution of the process W y = y +Wt .I fψ ∈ C((0, 1), R m ) , then we define the process (W y )t = ψ(t )+ W y . Suppose that W y is the distribution of this process and E y,ψ is the expectation with respect to the measure W y . Further, Ue(M ) denotes the e -neighborhood of the manifold M. We consider W y for functions y and ψ satisfying the conditions: y(0) ∈ M , ψ(0) = 0 . The goal of this section is to prove the existence of a limit (given below) with respect to the family of bounded continu- ous cylindrical functions, where by a cylindrical function C((0, 1) × (0, 1), R m ) → R we mean a

Hausdorff dimension of the graph of the Fractional Brownian Sheet

Let \{B^{(\alpha)}(t)\}_{t\in\mathbb{R}^{d}} be the Fractional Brownian Sheet with multi-index \alpha=(\alpha_1,\ldots, \alpha_d) , 0&lt; \alpha_i&lt; 1 . In [14], Kamont has shown that, with probability 1 , the box dimension of the graph of a trajectory of this Gaussian field, over a non-degenerate cube Q\subset\mathbb{R}^{d} is equal to d+1-\min(\alpha_1,\ldots,\alpha_d) . In this paper, we prove that this result remains true when the box dimension is replaced by the Hausdorff dimension or the packing dimension.