Levy Brownian Motion Research Articles

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).

Stochasticp-adic equations of mathematical physics

Properties and probability characteristics of solutions of the stochastic p-adic equations of mathematical physics containing the Vladimirov pseudodifferential operators and their generalizations are discussed. The p-adic analogues of the Brownian sheet and the Levy Brownian motion are constructed.

On Estimating the Mean Value of Levy's Brownian Motion

The Gaussian field under consideration is a constant plus Levy's Brownian motion. The maximum likelihood estimate of the mean is constructed explicitly in the case of observation of the field outside some ball.

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Levy Brownian motion.

An integrated fractional Fourier transform

For 0 < α < 2, an integrated fractional Fourier transform F α of Wiener type, closely related to fractional Levy Brownian motion X α, is established. By means of this transform, transform, the reproducing Kernel Hilbert space and the covariance operator of X α are characterized and the connection between X α and the Laplacian is shown. As a further application a Blaschke-Fuglede-type theorem for a fractional Radon transform is derived.

Brownian motion generated by the Levy Laplacian

An existence and uniqueness theorem for the the heat equation associated to the Levy-Laplacian is proved and a simple explicit formula is derived. The associated {\it Levy heat semigroup} is used to construct a classical Markov process called the {\it Levy Brownian motion}

An action functional for L�vy Brownian motion

Stoll's construction [7] of Levy Brownian motion l on ℝd as a white noise integral is used to obtain an action functional I(x) defined for the ‘surfaces’ x of l. This provides a ‘Cameron-Martin’ formula for translation of Levy measure λ, and also a large deviation principle for scaled Levy measures λδ. Proofs follow the lines of [2], where nonstandard techniques were used to give natural proofs of the corresponding results for Wiener measure.

A nonstandard construction of Lévy Brownian motion

A nonstandard construction of Levy Brownian motion on ℝd is presented, which extends R.M. Anderson's nonstandard representation of Brownian motion. It involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of Levy Brownian motion. Moreover, a new invariance principle can be deduced in a similar way as Donsker's invariance principles follows from Anderson's construction.

Maximum in the Levy-Baxter Theorem for Gaussian Random Fields

The range of almost sure limits of $F$-variation for a class of Gaussian random fields is considered by adopting a class of sequences of partitions in the parameter space of the random field. The application to Levy's Brownian motion explains, in the case of two-dimensional parameters, that the almost sure limit given by Berman is the maximum in a range.

Harmonic analysis of local times and sample functions of Gaussian processes

v(A) = u(x-'(A)) on the linear Borel sets A. If v is absolutely continuous with respect to the usual linear Borel measure, then its derivative is called the local time of x. In this paper we deduce some properties of the variation of x from smoothness properties of the local time when u is fixed and x is a sample function of a Gaussian process. The main results are Theorems 5.1 and 5.2 which imply: If X(t), 0 C It - s 1, for some I, 0< f < 2, then the sample functions (a) are of unbounded y-variation for y < 2/1; (b) nowhere satisfy a Holder condition of order 2/(2m + E + 1) if : < 2/(2m + E + 1), where m is a nonnegative integer and 0 < E < 1; (c) are nowhere differentiable if : < 1; (d) return infinitely often to every neighborhood of almost all the points they visit. The result (c) was stated without proof in [5] for a stationary process whose covariance function r is not twice differentiable at the origin. A related result was proved by Yeh [7] under the additional assumption that X(.) is continuous. The paper of Marcus [4] contains results on Holder conditions for processes with stationary increments satisfying the condition that E(X(t) - X(0))2 is concave in some neighborhood of 0, or related conditions. Using a previous result on analytic local times [2], we prove in ?6 that Levy's Brownian motion over Hilbert space has sample functions unbounded over certain compact ellipsoids (cf. [1]). 1. Square integrable local times. Put