Brownian Field Research Articles

Stochastic integrals and stochastic differential equations with respect to the fractional Brownian field

Stochastic differential equations on the plane are considered with respect to the fractional Brownian field. We prove the existence and uniqueness of a solution for such equations. These results are based on new estimates obtained for norms in the Besov type spaces for the two-parameter stochastic integral considered with respect to the fractional Brownian field.

Sample path properties of fractional Riesz–Bessel field of variable order

In this paper we consider fractional Riesz–Bessel field of variable order, which is also known as multifractional Riesz–Bessel field. Sample path properties of this random field such as local regularity, locally self-similar property, Hausdorff dimension of the graph, and long∕short range dependent property are studied. The relationship between the multifractional Riesz–Bessel field and the multifractional Brownian field is also established.

Estimation of anisotropic Gaussian fields through Radon transform

We estimate the anisotropic index of an anisotropic fractional Brownian field. For all directions, we give a convergent estimator of the value of the anisotropic index in this direction, based on generalized quadratic variations. We also prove a central limit theorem. First we present a result of identification that relies on the asymptotic behavior of the spectral density of a process. Then, we define Radon transforms of the anisotropic fractional Brownian field and prove that these processes admit a spectral density satisfying the previous assumptions. Finally we use simulated fields to test the proposed estimator in different anisotropic and isotropic cases. Results show that the estimator behaves similarly in all cases and is able to detect anisotropy quite accurately.

Mesoscopic modeling of flow and dispersion phenomena in fractured solids

The problem of hydrodynamic dispersion in porous media is considered and numerical predictions of the mixing degree in a single intersection are provided. The flow field in the intersection and adjacent pores or fractures is calculated using a lattice Boltzmann model for single phase flow. A particle-tracking scheme is used, subsequently, that monitors the migration of solute particles in the area of the intersection taking into account the local flow field and a Brownian field. Mixing is quantified in terms of the probability of solute transfer across the junction into the opposite fracture. To circumvent the problem of large computational times for cases of fast flow compared to diffusion, a lattice Boltzmann advection–diffusion model is used, that offers significant savings on computational time without sacrificing accuracy. It is shown that the solute dispersion in a fracture network is a strong function of the Reynolds number, even if the Peclet number remains constant, due to the extensive recirculation areas that may develop in regions close to the junction.

Polyharmonic smoothing splines and the multidimensional Wiener filtering of fractal-like signals

Motivated by the fractal-like behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensional data. He proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions (RBFs). Unfortunately, this is tedious to implement for images because of the poor conditioning of RBFs and their lack of decay. Here, we present a much more efficient method for the special case of a uniform grid. The key idea is to express Duchon's solution in a fractional polyharmonic B-spline basis that spans the same space as the RBFs. This allows us to derive an algorithm where the smoothing is performed by filtering in the Fourier domain. Next, we prove that the above smoothing spline can be optimally tuned to provide the MMSE estimation of a fractional Brownian field corrupted by white noise. This is a strong result that not only yields the best linear filter (Wiener solution), but also the optimal interpolation space, which is not bandlimited. It also suggests a way of using the noisy data to identify the optimal parameters (order of the spline and smoothing strength), which yields a fully automatic smoothing procedure. We evaluate the performance of our algorithm by comparing it against an oracle Wiener filter, which requires the knowledge of the true noiseless power spectrum of the signal. We find that our approach performs almost as well as the oracle solution over a wide range of conditions.

Some mathematical aspects of the scaling limit of critical two-dimensional systems

It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.

An Efficient Technique for Simulations of Viscoelastic Flows, Derived from the Brownian Configuration Field Method

In this paper we present a new approach for calculating viscoelastic flows. This method is derived from the Brownian configuration field method [M. A. Hulsen, A. P. G. Van Heel, and B. H. A. A. Van Den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79--101] for the simulation of flows of dilute polymeric solutions. The method presented here has the same attractive features as those in a previous work [C. Chauviere, SIAM J. Sci. Comput., 23 (2002), pp. 2123--2140], i.e., improved stability in comparison with traditional methods starting from closed-form constitutive equations. Moreover, the method developed here has the additional advantage of being significantly more CPU efficient. Results are presented by solving the benchmark problem of the flow of an Oldroyd-B fluid past a cylinder in a channel using a spectral element method to demonstrate the advantages of the proposed method.

Les ondelettes à la conquête du drap brownien fractionnaire

A random field depending on two parameters α and β is defined by a fractional integration with respect to the white noise field. A wavelet basis is used to produce a decomposition of it in the space L 2( R 2). The aim of this paper is to produce a simulation algorithm of the fractional Brownian field: the almost convergence, uniform on any compact set, of this decomposition is proved. Finally, the rate of this convergence is determined. To cite this article: A. Ayache et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1063–1068.

A first error indicator for micro–macro simulations of viscoelastic fluids having closed-form constitutive equations

An a posteriori error indicator for viscoelastic flow calculations is proposed by the author for a recently introduced method. This new method is derived from the Brownian configuration field approach and can be applied to models for which closed-form expressions already exist. Numerical results are presented in the context of the spectral element method to show the reliability of the error indicator. The benchmark problem of the flow of an Oldroyd-B fluid past a cylinder in a channel is used and we show that this new method is always more accurate than using a constitutive equation.

A Robust Spectral Element Method for Simulations of Time-Dependent Viscoelastic Flows, Derived from the Brownian Configuration Field Method

This paper is an extension of previous work l4r, where a robust numerical method derived from the Brownian configuration field method l8r was introduced in order to simulate the flows of dilute polymeric solutions. In l4r, we limited our study to solutions of dumbbells having infinite extensibility (Oldroyd-B model), whereas in this paper, we tackle the more difficult problem of dumbbells having finite extensibility (FENE-P model).

A New Method for Micro-Macro Simulations of Viscoelastic Flows

In this paper we develop a robust numerical method derived from the Brownian configuration field method [M. A. Hulsen, A.P. G. Van Heel, and B.H.A. A. Van Den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79--101] for the simulation of flows of dilute polymeric solutions. The statistical properties of the Wiener stochastic process in the stochastic differential equation describing the evolution of the configuration vector are exploited in order to derive a simple expression for the polymeric contribution to the stress. The method is tested numerically by solving the benchmark problem of the flow of an Oldroyd B fluid past a cylinder in a channel using a spectral element method. Results are presented to demonstrate the advantages of the proposed method.

Specific low-shear viscosity of a liquid dispersion of solid particles from variational theory for the Fuchs stability ratio

The contribution of a solid phase to the low-shear viscosity of a solid–liquid dispersion, i.e., the specific viscosity, ηSP, is investigated theoretically by applying a variational procedure. The Fuchs stability ratio has been interpreted as a general steady-state equilibrium constant for aggregation and has been extended to a functional form which describes the motion of two Brownian units in a dispersion. Application of the Euler–Lagrange equation under the validity of an adiabatic-like approximation for the Hamiltonian (approximately Brownian kinetic energy and inertial potential field) yields a constraint that involves specific viscosity, solid volume fraction, φ, interparticle energy and correlation functions of the dispersed phase. The Einstein formula is found as the limit of the Saito equation when an infinitely dilute hard-sphere suspension is considered, while a general closed form expression, ηSP=ηSP(φ), is proposed for a concentrated suspension. It depends on the particle coordination number and affinity, returns the low density expansion predicted by effective-medium-type theories for the viscosity, and can be represented as the sum of two dominant contributions, associated respectively with the first peak of the radial distribution function and the second peak of the total correlation function. Application to experimental data, concerning latex particles in cis-decalin and interacting silica–water systems, is presented and discussed.

A numerical simulation of suspension flow using a constitutive model based on anisotropic interparticle interactions

We report a Brownian configuration field implementation of a recent constitutive equation for suspensions, reported by Phan-Thien et al. 1999. The numerical method is a hybrid technique, which combines a modification of the Brownian configuration field method described by Hulsen et al. 1997 and the adaptive viscosity split stress formulation proposed by Sun et al. 1996. The implementation is used to examine the flow past a sphere in a tube. The relative viscosity derived from the drag force/sedimentation velocity agrees well with a well-known empiricism. In addition, the ratio of the pressure force to the drag on the sphere seems to be weakly dependent on the volume fraction, and is somewhat higher than Brenner's results of 1962, which were derived for Newtonian fluids.

Drap brownien fractionnaire

A random field is defined as a (α, β)-fractional integral of a white noise measure. This field is autosimilar and admits stationary increments. The trajectories of the fractional Brownian field are Holder continuous, and the field is uniformly continuous with respect to the parameters (α, β).

Fractional Brownian motion via fractional Laplacian

Abstract A new and short proof of existence of the fractional Brownian field with exponent α /2, α ∈(0,2], is given in terms of the fractional power of the Laplacian.

Certain Positive-Definite Kernels

In one way or another, the extension of the standard Brownian motion process $\{ {B_t}:t \in [0,\infty )\}$ to a (Gaussian) random field $\{ {B_t}:{\text {t}} \in {\mathbf {R}}_ + ^d\}$ involves a proof of the positive semi-definiteness of the kernel used to generalize $\rho (s,t) = {\text {cov(}}{{\text {B}}_s}{\text {,}}{{\text {B}}_t}{\text {) = s}} \wedge t$ to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Lévy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field.

Certain positive-definite kernels

In one way or another, the extension of the standard Brownian motion process { B t : t ∈ [ 0 , ∞ ) } \{ {B_t}:t \in [0,\infty )\} to a (Gaussian) random field { B t : t ∈ R + d } \{ {B_t}:{\text {t}} \in {\mathbf {R}}_ + ^d\} involves a proof of the positive semi-definiteness of the kernel used to generalize ρ ( s , t ) = cov( B s , B t ) = s ∧ t \rho (s,t) = {\text {cov(}}{{\text {B}}_s}{\text {,}}{{\text {B}}_t}{\text {) = s}} \wedge t to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Lévy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field.

Langevin Equations in Quantized Field Theories

Whenever two kinds of fields ψ and φ are interacting in the Yukawa type ψ*ψφ, we can always regard the φ as a heat bath and the other ψ as a Brownian field, and then obtain the generalized Langevin equation for ψ.