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

The " knotting " properties of Brownian motion are investigated. Because the path of Brownian motion in 3-space intersects itself, the topological definition of a knot does not apply. A modified concept is defined; implication in a knot-tube. It is shown that 3-space Brownian motion is implicated in infinitely many disjoint knot-tubes in every time interval. As a corollary every segment of the ...

Let X be a drifted fractional Brownian motion with Hurst index H > 1/2. We prove that there exists a fractional backward representation of X , i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when H = 1/2. We then apply our result to stochastic differential equations driven b...

An identity in distribution due to F. Knight for Brownian motion is extended in two di erent ways: rstly by replacing the supremum of a re ecting Brownian motion by the range of an unre ected Brownian motion, and secondly by replacing the re ecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excurs...

An identity in distribution due to F. Knight for Brownian motion is extended in two diierent ways: rstly by replacing the supremum of a reeecting Brownian motion by the range of an unreeected Brownian motion, and secondly by replacing the reeecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excurs...

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.