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Michael Scheutzow

Publications and source records attributed to Michael Scheutzow.

68 records · Page 4Linked to original sources
arXiv2010

Evolution systems of measures for stochastic flows

A new concept of {\em an evolution system of measures for stochastic flows} is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for asymptotically compact stochastic flows is obtained. For a white noise stochastic flow, there exists a one to one correspondence between evolution systems of measures for a stochastic flow \emph{and} evolution systems of measures for the associated Markov transition semigroup. As an application, an alternative approach for evolution systems of measures of 2D stochastic Navier-Stokes equations with a time-periodic forcing term is presented.

math.DS↗
arXiv2010

Pinning of interfaces in random media

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though the local evolution law is viscous (in particular, the velocity of the interface in the model is linear in the driving force).

math.AP↗
arXiv2010

Wait-and-see strategies in polling models

We consider a general polling model with $N$ stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work ("wait-and-see") and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the server waits idly is set to be a fixed, deterministic parameter for each station. Switchover times and service times are allowed to follow some general distribution, respectively. In some cases, which can be characterised, this strategy yields strictly lower average queueing delay than for the exhaustive strategy, which corresponds to setting the "wait-and-see credit" equal to zero for all stations. This extends results of Peköz (Probability in the Engineering and Informational Sciences 13 (1999)) and of Boxma et al. (Annals of Operations Research 112 (2002)). Furthermore, we give a lower bound for the delay for {\it all} strategies that allow the server to wait at the stations even though no work is present.

math.PR↗
arXiv2009

Existence and uniqueness of solutions of stochastic functional differential equations

We provide sufficient conditions on the coefficients of a stochastic functional differential equation with bounded memory driven by Brownian motion which guarantee existence and uniqueness of a maximal local and global strong solution for each initial condition. Our results extend those of previous works. For local existence and uniqueness, we only require the coefficients to be continuous and to satisfy a one-sided local Lipschitz (or monotonicity) condition. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochastic versions of Gronwall's Lemma, the final one - which we call Dereich-Lemma - provides tail bounds for Hoelder norms of stochastic integrals.

math.PR↗
arXiv2009

Attractors and Expansion for Brownian Flows

We show that a stochastic flow which is generated by a stochastic differential equation on $\R^d$ with bounded volatility has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant $β$ outside a large ball. Using a similar approach, we provide a lower bound for the linear growth rate of the inner radius of the image of a large ball under a stochastic flow in case the drift component in the direction away from the origin is larger than a certain strictly positive constant $β$ outside a large ball. To prove the main result we use chaining techniques in order to control the growth of the diameter of subsets of the state space under the flow.

math.PR↗
arXiv2009

Asymptotic coupling and a weak form of Harris' theorem with applications to stochastic delay equations

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to recent work on SPDEs on which this work builds. The emphasis here is on stochastic differential delay equations.Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations.

math.PR↗
arXiv2009

Lack of strong completeness for stochastic flows

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If -- in addition -- the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition $x$, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently $x$, then the maximal flow is called {\em strongly complete}. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\em not} strongly complete thus answering the question in the negative.

math.PR↗
arXiv2008

Dispersion of volume under the action of isotropic Brownian flows

We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive.

math.PR↗
arXiv2008

Criteria for strong and weak random attractors

The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors.

math.PR↗
arXiv2008

Multicast Capacity of Optical WDM Packet Ring for Hotspot Traffic

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.

cs.IT↗
arXiv2008

Chaining Techniques and their Application to Stochastic Flows

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.

math.PR↗
arXiv2007

High resolution quantization and entropy coding of jump processes

We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\mathbb{R}^d$-valued compound Poisson processes.

math.PR↗
arXiv2005

High-resolution quantization and entropy coding for fractional Brownian motion

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.

math.PR↗