Infinite Dimensional Fractional Brownian Motion Research Articles

Finite element approximation of the linearized stochastic Cahn–Hilliard equation with fractional Brownian motion

We perform a numerical analysis of the linearized stochastic Cahn–Hilliard equation driven by infinite-dimensional fractional Brownian motion with Hurst index H>12. The equation is discretized using a standard finite element method in space and a fully implicit backward Euler method in time. We prove strong convergence estimates for the considered stochastic Cahn–Hilliard equation. Finally, numerical experiments are performed to confirm the theoretical results.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

Spectral collocation method for stochastic partial differential equations with fractional Brownian motion

In this paper, we consider the numerical approximation of stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with Hurst index H>12. A Fourier spectral collocation approximation is used in space and semi-implicit Euler method is applied for the temporal approximation. Our aim is to investigate the convergence of the proposed method. Optimal strong convergence error estimates in mean-square sense are derived and numerical experiments are presented and confirm theoretical results.

Fractional stochastic heat equation with piecewise constant coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

Local L^p-solution for semilinear heat equation with fractional noise

We study the $L^{p}$-solutions for the semilinear heat equation with unbounded coefficients and driven by a infinite dimensional fractional Brownian motion with self-similarity parameter $H > 1/2$. Existence and uniqueness of local mild solutions are showed.

Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion

<p style='text-indent:20px;'>In this paper, we consider the numerical approximation for a class of fractional stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with hurst index <inline-formula><tex-math id="M1">\begin{document}$ H∈ (\frac{1}{2}, 1) $\end{document}</tex-math></inline-formula>. By using spectral Galerkin method, we analyze the spatial discretization, and we give the temporal discretization by using the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. Under some suitable assumptions, we prove the sharp regularity properties and the optimal strong convergence error estimates for both semi-discrete and fully discrete schemes.

Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations

This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.

Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion

This work was intended as an attempt to motivate the approximation of quasi linear evolution equations driven by infinite-dimensional fractional Brownian motion with Hurst parameter H>12. The spatial approximation method is based on Galerkin and the temporal approximation is based on implicit Euler scheme. An error bound and the convergence of the numerical method are given. The numerical results show usefulness and accuracy of the method.

Stochastic shell models driven by a multiplicative fractional Brownian-motion

We prove existence and uniqueness of the solution of a stochastic shell-model. The equation is driven by an infinite dimensional fractional Brownian-motion with Hurst-parameter H∈(1/2,1), and contains a non-trivial coefficient in front of the noise which satisfies special regularity conditions. The appearing stochastic integrals are defined in a fractional sense. First, we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise, for which we are able to derive important uniform estimates in some functional spaces. Then, thanks to a compactness argument and these estimates, we prove that these variational solutions converge to a limit solution, which turns out to be the unique pathwise mild solution associated to the shell-model with fractional noise as driving process.

Fractional stochastic Volterra equation perturbed by fractional Brownian motion

In this paper, we consider a class of fractional stochastic Volterra equation of convolution type driven by infinite dimensional fractional Brownian motion with Hurst index h∈(0,1). Base on the explicit formula for the scalar resolvent function and the properties of the Mittag–Leffler’s function, the existence and regularity results of the stochastic convolution process are established. Separate proofs are required for the cases of Hurst parameter above and below 12 and it will turn out that the regularity of the solution increases with Hurst parameter h.

Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ: dX(t) = A(X(t))dt+Φ(t)dB H(t), where A is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.

Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems

This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.

Parameter estimation for a two-dimensional stochastic Navier–Stokes equation driven by an infinite dimensional fractional Brownian motion

We study parameter estimation for a two-dimensional stochastic Navier–Stokes equation driven by an infinite dimensional fractional Brownian motion.

Motion of inertial particles in Gaussian fields driven by an infinite-dimensional fractional Brownian motion

We study the motion of an inertial particle in a fractional Gaussian random field. The motion of the particle is described by Newton's second law, where the force is proportional to the difference between the background fluid velocity and the particle velocity. The fluid velocity satisfies a linear stochastic partial differential equation driven by an infinite-dimensional fractional Brownian motion with an arbitrary Hurst parameter H ∈ (0, 1). The usefulness of such random velocity fields in simulations is that we can create random velocity fields with desired statistical properties, thus generating artificial images of realistic turbulent flows. This model also captures the clustering phenomenon of preferential concentration, observed in real world and numerical experiments, i.e. particles cluster in regions of low-vorticity and high-strain rate. We prove almost sure existence and uniqueness of particle paths and give sufficient conditions to rewrite this system as a random dynamical system with a global random pullback attractor. Finally, we visualize the random attractor through a numerical experiment.

Ergodicity of the Infinite Dimensional Fractional Brownian Motion

In this short note we prove that an infinite dimensional fractional Brownian motion B H of any Hurst parameter $${H \in (0, 1)}$$ forms an ergodic metric dynamical system. For the proof we mainly use the fundamental theorems of Kolmogorov.

A FILTERING PROBLEM FOR A LINEAR STOCHASTIC EVOLUTION EQUATION DRIVEN BY A FRACTIONAL BROWNIAN MOTION

A linear unbiased and square mean optimal estimation is obtained for the mild solution process of a stochastic evolution equation with an infinite-dimensional fractional Brownian motion as noise and the noise in the observation process is a finite-dimensional Brownian motion. An innovation process is introduced and the estimation is obtained as a solution of a stochastic differential equation with a finite-dimensional noise. By using an approach based on the equivalence with a deterministic control problem, the estimation for the Fourier coefficients of the signal process is also determined.

Itô formula for the infinite-dimensional fractional Brownian motion

We introduce the stochastic integration with respect to the infinitedimensional fractional Brownian motion. Using the techniques of the anticipating stochastic calculus, we derive an Itô formula for Hurst parameter bigger than $\frac{1}{2}$.

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.

Stochastic evolution equations with fractional Brownian motion

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail.