arXiv subjects

Michael Scheutzow

Publications and source records attributed to Michael Scheutzow.

At least 55 records · Page 3Linked to original sources

arXiv2016

Synchronization by noise for order-preserving random dynamical systems

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.

math.PR↗

arXiv2015

Weak synchronization for isotropic flows

We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide a sufficient condition on the boundary behavior of $r$ at $0$ which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on $S^{d-1}$ as well as isotropic Ornstein-Uhlenbeck flows on $\mathbb{R}^d$.

math.PR↗

arXiv2015

A new class of large claim size distributions: Definition, properties, and ruin theory

We investigate a new natural class $\mathcal{J}$ of probability distributions modeling large claim sizes, motivated by the `principle of one big jump'. Though significantly more general than the (sub-)class of subexponential distributions $\mathcal{S}$, many important and desirable structural properties can still be derived. We establish relations to many other important large claim distribution classes (such as $\mathcal{D}$, $\mathcal{S}$, $\mathcal{L}$, $\mathcal {K}$, $\mathcal{OS}$ and $\mathcal{OL}$), discuss the stability of $\mathcal{J}$ under tail-equivalence, convolution, convolution roots, random sums and mixture, and then apply these results to derive a partial analogue of the famous Pakes-Veraverbeke-Embrechts theorem from ruin theory for $\mathcal{J}$. Finally, we discuss the (weak) tail-equivalence of infinitely-divisible distributions in $\mathcal{J}$ with their Lévy measure.

math.PR↗

arXiv2015

On the structure of a class of distributions obeying the principle of a single big jump

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.

math.PR↗

arXiv2015

Asymptotics for Lipschitz percolation above tilted planes

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $γ$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well as $γ\to π/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $γ\to π/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor.

math.PR↗

arXiv2014

On a transformation between distributions obeying the principle of a single big jump

Beck et al. (2013) introduced a new distribution class J which contains many heavy-tailed and light-tailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavy-tailed distributions to light-tailed ones, we find some light-tailed distributions, which belong to the class J but do not belong to the convolution equivalent distribution class and which are not even weakly tail equivalent to any convolution equivalent distribution. This fact helps to understand the structure of the light-tailed distributions in the class J and leads to a negative answer to an open question raised by the above paper.

math.PR↗

arXiv2014

Blow-up of a stable stochastic differential equation

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete - in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e.~there exist (random) initial conditions for which the solutions explode in finite time.

math.PR↗

arXiv2014

A coupling approach to Doob's theorem

We provide a coupling proof of Doob's theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure $μ$ converge to $μ$ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for $μ$-almost all initial conditions.

math.PR↗

arXiv2013

Capacity and Delay Analysis of Next-Generation Passive Optical Networks (NG-PONs) - Extended Version

Building on the Ethernet Passive Optical Network (EPON) and Gigabit PON (GPON) standards, Next-Generation (NG) PONs (i) provide increased data rates, split ratios, wavelengths counts, and fiber lengths, as well as (ii) allow for all-optical integration of access and metro networks. In this paper we provide a comprehensive probabilistic analysis of the capacity (maximum mean packet throughput) and packet delay of subnetworks that can be used to form NG-PONs. Our analysis can cover a wide range of NG-PONs through taking the minimum capacity of the subnetworks making up the NG-PON and weighing the packet delays of the subnetworks. Our numerical and simulation results indicate that our analysis quite accurately characterizes the throughput-delay performance of EPON/GPON tree networks, including networks upgraded with higher data rates and wavelength counts. Our analysis also characterizes the trade-offs and bottlenecks when integrating EPON/GPON tree networks across a metro area with a ring, a Passive Star Coupler (PSC), or an Arrayed Waveguide Grating (AWG) for uniform and non-uniform traffic. To the best of our knowledge, the presented analysis is the first to consider multiple PONs interconnected via a metro network.

cs.IT↗

arXiv2013

A Stochastic Gronwall Lemma

We prove a stochastic Gronwall lemma of the following type: if $Z$ is an adapted nonnegative 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 $p \in (0,1)$ the $p$-th moment of the supremum of $Z$ is bounded by a constant $κ_p$ (which does not depend on $M$) times the $p$-th 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 $c_p$ appearing in the inequality which is at most four times as large as the optimal constant.

math.PR↗

arXiv2013

Forward Brownian Motion

We consider processes which have the distribution of standard Brownian motion (in the forward direction of time) starting from random points on the trajectory which accumulate at $-\infty$. We show that these processes do not have to have the distribution of standard Brownian motion in the backward direction of time, no matter which random time we take as the origin. We study the maximum and minimum rates of growth for these processes in the backward direction. We also address the question of which extra assumptions make one of these processes a two-sided Brownian motion.

math.PR↗

arXiv2012

Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation

For a model of a driven interface in an elastic medium with random obstacles we prove existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate independent hysteresis through the interaction of the interface with the obstacles, despite a linear (force=velocity) microscopic kinetic relation. We also prove a percolation result, namely the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbor site-percolation cluster at a non-trivial percolation threshold.

math.AP↗

arXiv2012

Exponential growth rate for a singular linear stochastic delay differential equation

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation dX(t) = X(t-1) dW(t) which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).

math.PR↗

arXiv2012

Invariance and Monotonicity for Stochastic Delay Differential Equations

We study invariance and monotonicity properties of Kunita-type stochastic differential equations in $\RR^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\RR^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.

math.PR↗

arXiv2011

Constructive quantization: approximation by empirical measures

In this article, we study the approximation of a probability measure $μ$ on $\mathbb{R}^{d}$ by its empirical measure $\hatμ_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

math.PR↗

arXiv2011

Moments of recurrence times for Markov chains

We consider moments of the return times (or first hitting times) in a discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K. L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially -- neither globally nor locally -- fulfill the above statement.

math.PR↗

arXiv2011

Exponential growth rate for derivatives of stochastic flows

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.

math.PR↗

arXiv2011

Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients

We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation and a supermartingale estimate akin to the study of branching random walks.

math.PR↗