Razumikhin-type theorems on general decay stability of stochastic functional differential equations with infinite delay

To the best of the authors’ knowledge, there are no results based on the so-called Razumikhin technique via a general decay stability, for any type of stochastic differential equations. In the present paper, the Razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay. The obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay. Some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given. An example is presented to illustrate the usefulness of the theory.     

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

Let {BH₁,H₂(t₁,t₂),t₁?0,t₂?0} be a fractional Brownian sheet with indexes 0<H₁,H₂<1. When H₁=H₂:=H, there is a logarithmic factor in the small ball function of the sup-norm statistic of BH,H. First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of BH,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.     

A strong uniform approximation of fractional Brownian motion by means of transport processes

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H$H$, and we derive a rate of convergence, which becomes better when H$H$ approaches 1/2$1/2$. The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

The generalized Levinger transformation

In this paper, we present new results relating the numerical range of a matrix A with the generalized Levinger transformation L(A,α,β)=αH_A+βS_A, where H_A and S_A, are, respectively the Hermitian and skew-Hermitian parts of A. Using these results, we then derive expressions for eigenvalues and eigenvectors of the perturbed matrix A+L(E,α,β), for a fixed matrix E and α, β are real parameters.     

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2$H>1/2$ and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.     

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

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.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Regularity of the sample paths of a class of second-order spde's

We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (Electron. J. Probab. 8 (2003) 1), we prove that the solutions belong to a fractional L²-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller. For spatial covariances given by Riesz kernels, we show that our results are optimal.     

Stochastic conservation laws: Weak-in-time formulation and strong entropy condition

This article is an attempt to complement some recent developments on conservation laws with stochastic forcing. In a pioneering development, Feng and Nualart [8] have developed the entropy solution theory for such problems and the presence of stochastic forcing necessitates introduction of strong entropy condition. However, the authors' formulation of entropy inequalities are weak-in-space but strong-in-time. In the absence of a priori path continuity for the solutions, we take a critical outlook towards this formulation and offer an entropy formulation which is weak-in-time and weak-in-space.     

Ito's- and Tanaka's-type formulae for the stochastic heat equation: The linear case

In this paper, we consider the linear stochastic heat equation with additive noise in dimension one. Then, using the representation of its solution X as a stochastic convolution of the cylindrical Brownian motion with respect to an operator-valued kernel, we derive Itô's- and Tanaka's-type formulae associated to X.     

The stochastic wave equation with fractional noise: A random field approach

We consider the linear stochastic wave equation with spatially homogeneous Gaussian noise, which is fractional in time with index H>1/2$H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in Dalang (1999) [10], where the noise is white in time. Under this condition, we show that the solution is L²(Ω)$L^2\left(\Omega \right)$-continuous. Similar results are obtained for the heat equation. Unlike in the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.     

On collision local time of two independent fractional ornstein-uhlenbeck processes

In this article, we study the existence of collision local time of two independent d-dimensional fractional Ornstein-Uhlenbeck processes X_t~(H_1)and _t~(H_2),with different parameters H_i∈(0, 1), i = 1, 2. Under the canonical framework of white noise analysis,we characterize the collision local time as a Hida distribution and obtain its' chaos expansion.     

Fractional integral operators in the complex matrix variate case

A formal definition of fractional integrals in the complex matrix variate case is given here. This definition will encompass all the various fractional integral operators introduced by various authors in the real scalar and matrix cases. The new definition is introduced in terms of M-convolutions of products and ratios of matrices in the complex domain. Their connections to statistical distribution theory, Mellin convolutions, M-transforms and Mellin transform are pointed out. Some basic properties are given and a pathway extension of the new definition is also given. The pathway extension will provide a switching mechanism to move among three different families of functions.     

An Itô–Stratonovich formula for Gaussian processes: A Riemann sums approach

The aim of this paper is to establish a change of variable formula for general Gaussian processes whose covariance function satisfies some technical conditions. The stochastic integral is defined in the Stratonovich sense using an approximation by middle point Riemann sums. The change of variable formula is proved by means of a Taylor expansion up to the sixth order, and applying the techniques of Malliavin calculus to show the convergence to zero of the residual terms. The conditions on the covariance function are weak enough to include processes with infinite quadratic variation, and we show that they are satisfied by the bifractional Brownian motion with parameters (H,K)$(H,K)$ such that 1/6<HK<1$1/6<HK<1$, and, in particular, by the fractional Brownian motion with Hurst parameter H∈(1/6,1)$H\in (1/6,1)$.     

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.