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We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\, p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor. Motivated by the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy inequality, we also study the domination inequality under an additional monotonicity assumption. In this special case, a constant which stays bounded for $p$ near $1$ was proven by Pratelli and Lenglart. We provide the sharp constant for this case., Comment: In the introduction, a falsely cited reference was removed. Some misprints were corrected

During the past decades, the question of existence and properties of a random attractor of a random dynamical system generated by an S(P)DE has received considerable attention, for example by the work of Gess and R\"ockner. Recently some authors investigated sufficient conditions which guarantee synchronization, i.e. existence of a random attractor which is a singleton. It is reasonable to conjecture that synchronization and negativity (or non-positivity) of the top Lyapunov exponent of the system should be closely related since both mean that the system is contracting in some sense. Based on classical results by Ruelle, we formulate positive results in this direction. Finally we provide two very simple but striking examples of one-dimensional monotone random dynamical systems for which 0 is a fixed point. In the first example, the Lyapunov exponent is strictly negative but nevertheless all trajectories starting outside of 0 diverge to $\infty$ or $-\infty$. In particular, there is no synchronization (not even locally). In the second example (which is just the time reversal of the first), the Lyapunov exponent is strictly positive but nevertheless there is synchronization., Comment: 8 pages

In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. [1]. We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class J are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in J which are neither long-tailed nor dominatedly-varying-tailed is not only non-empty but even quite rich in the sense that it has a nonempty intersection with several other well-established classes. In addition, the integrated tail distribution of some particular of these distributions shows that the Pakes-Veraverbeke-Embrechts Theorem for the class J in [1] does not hold trivially., Comment: 12 pages

We consider a stochastic functional differential equation with an arbitrary Lipschitz diffusion coefficient depending on the past. The drift part contains a term with superlinear growth and satisfying a dissipativity condition. We prove tightness and Feller property of the segment process to show existence of an invariant measure., Comment: 9 pages

Packet-switching WDM ring networks with a hotspot transporting unicast, multicast, and broadcast traffic are important components of high-speed metropolitan area networks. For an arbitrary multicast fanout traffic model with uniform, hotspot destination, and hotspot source packet traffic, we analyze the maximum achievable long-run average packet throughput, which we refer to as \textit{multicast capacity}, of bi-directional shortest-path routed WDM rings. We identify three segments that can experience the maximum utilization, and thus, limit the multicast capacity. We characterize the segment utilization probabilities through bounds and approximations, which we verify through simulations. We discover that shortest-path routing can lead to utilization probabilities above one half for moderate to large portions of hotspot source multi- and broadcast traffic, and consequently multicast capacities of less than two simultaneous packet transmissions. We outline a one-copy routing strategy that guarantees a multicast capacity of at least two simultaneous packet transmissions for arbitrary hotspot source traffic.

We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow.

We derive a high-resolution formula for the quantization and entropy coding approximation quantities for fractional Brownian motion, respective to the supremum norm and L^p[0,1]-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our proof uses an explicit construction of efficient codebooks based on a particular entropy constrained coding scheme. This procedure can be used to construct close to optimal high resolution quantizers., Comment: 25 pages

"We consider a class of stochastic linear functional differential systems driven by semimartingales with stationary ergodic increments. We allow smooth convolution-type dependence of the noise terms on the history of the state. Using a stochastic variational technique we construct a compactifying stochastic semiflow on the state space. A multiplicative Ruelle-Oseledec ergodic theorem then gives the existence of a discrete Lyapunov spectrum and a saddle-point property in the hyperbolic case."

We provide sufficient conditions for weak synchronization by noise for order-preserving random dynamical systems on Polish spaces. That is, under these conditions we prove the existence of a weak point attractor consisting of a single random point. This generalizes previous results in two directions: First, we do not restrict to Banach spaces and second, we do not require the partial order to be admissible nor normal. As a second main result and application we prove weak synchronization by noise for stochastic porous media equations with additive noise., Comment: 25 pages