Fast Sets and Points for Fractional Brownian Motion

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

Fractal Measures of the Sets Associated to Gaussian Random Fields

This paper summarizes recent results about the Hausdorff measure of the image, graph and level sets of Gaussian random fields, the packing dimension and packing measure of the image of fractional Brownian motion, the local times and multiple points of Gaussian random fields. Some open problems are also pointed out.

Propagation of Singularities in a Semi Fractional Brownian Sheet ∗

Let X be a semi fractional Brownian sheet, that is a centred and continuous Gaussian random field with E{X(s, t)X(ŝ, t̂ )} = (t ∧ t̂ )(sα + ŝα − |s − ŝ|α)/2. We prove for α ∈ [1, 2) the propagation of certain singularities into the fractional direction of X. Here, singularities are times where the law of the iterated logarithm fails, such as fast points.

Propagation of Singularities in the Semi-Fractional Brownian Sheet

Let X be a semi-fractional Brownian sheet, that is a centred and continuous Gaussian random field with E[X(s, t)X(ŝ, t̂ )] = (t∧ t̂ )(sα+ ŝα−|s− ŝ|α)/2. We provide, for α ∈ (0, 2), an analysis of the propagation of singularities into the fractional direction of X. Here, singularities are times where the law of the iterated logarithm fails, such as fast points.