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We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system (RDS). Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) RDS or not. [ABSTRACT FROM AUTHOR]

We prove a stochastic Gronwall lemma of the following type: if Z is an adapted non-negative continuous process which satisfies a linear integral inequality with an added continuous local martingale M and a process H on the right-hand side, then for any p ∈ (0, 1) the pth moment of the supremum of Z is bounded by a constant κp (which does not depend on M) times the pth moment of the supremum of H. Our main tool is a martingale inequality which is due to D. Burkholder. We provide an alternative simple proof of the martingale inequality which provides an explicit numerical value for the constant cp appearing in the inequality which is at most four times as large as the optimal constant. [ABSTRACT FROM AUTHOR]

We show that for a large class of stochastic flows the spatial derivative grows at most exponentially fast even if one takes the supremum over a bounded set of initial points. We derive explicit bounds on the growth rates that depend on the local characteristics of the flow and the box dimension of the set. [ABSTRACT FROM AUTHOR]

In this survey, we provide some tools to obtain estimates for the almost sure exponential growth rate of a stochastic delay differential equation (sdde) which fixes zero. In particular, we are interested in determining whether the solutions of a given sdde are exponentially stable (i.e. have a negative exponential growth rate) or not. We focus on equations without drift, which are a good testground to assess if a method is powerful enough to discriminate between stability and instability when a certain parameter (e.g. noise intensity) varies. The most powerful tool we provide is the method of Lyapunov functionals which is used to obtain upper bounds for p-th moment exponents for small (positive and negative) values of p. [ABSTRACT FROM AUTHOR]

Recently, Cranston, Scheutzow and Steinsaltz showed that the image at time T of a bounded set x under the action of a stochastic flow on R[sup d] with Lipschitz and bounded characteristics is almost surely contained in a ball of radius KT around the origin for all sufficiently large T. Here K is a constant which depends on the dimension d and the characteristics of the flow (but not on the set X). We improve these results in three different ways: our proof is shorter and more transparent, the bound K which we obtain is sharper and we are able to improve the bounds on the tails of the distance of the most distant point in the image at time T to be genuinely sub-Gaussian. [ABSTRACT FROM AUTHOR]

Presents the preface to the June 2005 issue of the journal "Stochastics and Dynamics."