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For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular, we generalise a basic result by Bhattacharya and Majumdar (2007) where a certain form of mixing, or swap condition was assumed uniformly over the state space. We do not rely on continuity properties of transition probabilities., Comment: 20 pages

In the article 'Criteria for Strong and Weak Random Attractors' necessary and sufficient conditions for strong attractors and weak attractors are studied. In this note we correct two of its theorems on strong attractors., Comment: 4 pages

We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of the SDE can be split into two terms, one of which is singular and the radial component of the other term has a radial component of sufficient strength in the direction of the origin, then the random dynamical system generated by the SDE admits a pullback attractor., Comment: 35 pages

We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + \sigma(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{{p}}(\mathbb{R}^d)$, $p>d$ and $p\geq 2$, and show that such SDEs can be approximated by random ordinary differential equations by smoothing the noise and the singular drift at the same time. We further prove a support theorem for this class of SDEs in a rather simple way using the Girsanov theorem., Comment: 19 pages

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. 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) random dynamical system or not.

We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previous works, we do not assume a monotonicity condition to hold. Further, our conditions are even weaker than some results in the literature on invariant probability measures for functional SDEs without dependence on the law of the solution., Comment: 16 pages

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

We consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the random dynamical system corresponding to the stochastic equation is not only strongly complete but even admits a random attractor., Comment: updated version. Small corrections

We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists., Comment: 9 pages

In this work, we address the occurrence of infinite pinning in a random medium. We suppose that an initially flat interface starts to move through the medium due to some constant driving force. The medium is assumed to contain random obstacles. We model their positions by a Poisson point process and their strengths are not bounded. We determine a necessary condition on its distribution so that regardless of the driving force the interface gets pinned., Comment: 12 pages, 4 figures

Consider a discrete time Markov chain with rather general state space which has an invariant probability measure $\mu$. There are several sufficient conditions in the literature which guarantee convergence of all or $\mu$-almost all transition probabilities to $\mu$ in the total variation (TV) metric: irreducibility plus aperiodicity, equivalence properties of transition probabilities, or coupling properties. In this work, we review and improve some of these criteria in such a way that they become necessary and sufficient for TV convergence of all respectively $\mu$-almost all transition probabilities. In addition, we discuss so-called generalized couplings.

We consider a discrete and a continuum model for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inhibitory or an acceleratory force. We show that the interface remains bounded for all times even when a small constant external driving force is applied. This phenomenon has already been known when only inhibitory obstacles are present. In this work we extend this result to the case of, for example, a random medium of random zero mean forcing., Comment: 18 pages, 10 figures

We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the supremum norm. In this set-up, the usual proof using the ordinary Gronwall lemma together with the Burkholder-Davis-Gundy inequality seems impossible. In order to solve this problem, we prove a new and quite general stochastic Gronwall lemma for cadlag martingales using Lenglart's inequality., Comment: 18 pages

In this paper we consider a class of discrete time prey-predator models with three interacting species defined on the two-dimensional simplex. For some choices of parameters of the operator describing the evolution of the relative frequencies, we show that the ergodic hypothesis does not hold. Moreover, we prove that any order Ces\`aro mean of the trajectories diverges. For another class of parameters, we show that all orbits starting from the interior of the simplex converge to the unique fixed point of the operator while for the remaining choices of parameters all orbits converge to one of the vertices of the simplex. Contrary to many authors we study discrete time models but we include a speed function $f$ in the dynamics which allows us to approximate the continuous-time case arbitrarily well when $f$ is small., Comment: 14 pages

We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction-diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer, Mattingly (2011).

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single maps, starting with a generic initial condition where in particular all genetic types occur with positive frequency, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur. We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particular its random point attractors. The long term dynamics of the random composition of maps is asymptotic, almost surely, to equilibria. In contrast to the deterministic system, for generic initial conditions these can be equilibria with one or two or three types present (depending only on the distribution)., Comment: 15 pages

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in recent work by Hans Crauel and the author that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward {\em point} attractors but no minimal one remained open. Here we provide such an example., Comment: 4 pages

We develop a new generalized coupling approach to the study of stochastic delay equations with H\"older continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solutions, and establish weak ergodic rates for the corresponding segment processes. We also prove, under additional smoothness assumptions on the coefficients, stabilization rates for the sensitivities in the initial value of the corresponding semigroups

We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.

Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean-Vlasov equation and a corresponding propagation of chaos result in the infinite population limit are proven. Our existence result for the McKean-Vlasov equation is based on the Euler approximation, that is applied to this type of equation for the first time., Comment: In this version, a shorter title has been chosen. The manuscript has been accepted for publication in Annals of Applied Probability

It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary., Comment: 19 pages

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving connectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the compact set, these nullsets can be chosen independently of the compact set (which is clear for $\sigma$-compact spaces but not at all clear for spaces which are not $\sigma$-compact)., Comment: 7 pages

Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: $\Delta$-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most $\Delta$, where $\Delta$ is a non-negative number, while cc-attractors attract all countable compact sets. We provide two examples showing that a given random dynamical system may have various different $\Delta$-attractors for different values of $\Delta$. It seems that both concepts are new even in the context of deterministic dynamical systems., Comment: v1: 20 pages v2: 20 pages, corrected typos, streamlined proofs

We establish new general sufficient conditions for the existence of an invariant measure for stochastic functional differential equations and for exponential or subexponential convergence to the equilibrium. The obtained conditions extend Veretennikov--Khasminskii conditions for SDEs and are optimal in a certain sense., Comment: 25 pages

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

We analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems., Comment: 27 pages

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply "strong completeness" and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin-Wentzell-type large deviation result., Comment: 25 pages

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong $\Delta$-completeness which means that there exists a modification of the solution which if restricted to any set of dimension $\Delta$ is almost surely continuous with respect to the initial condition., Comment: 18 pages

We derive a discrete version of the stochastic Gronwall Lemma found in [Scheutzow, IDAQP, 2013]. The proof is based on a corresponding deterministic version of the discrete Gronwall Lemma and an inequality bounding the supremum in terms of the infimum for time discrete martingales. As an application the proof of an a priori estimate for the backward Euler-Maruyama method is included., Comment: 9 pages

We provide sufficient conditions for the uniqueness of an invariant measure of a Markov process as well as for the weak convergence of transition probabilities to the invariant measure. Our conditions are formulated in terms of generalized couplings. We apply our results to several SPDEs for which unique ergodicity has been proven in a recent paper by Glatt-Holtz, Mattingly, and Richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true.

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 $\mathbb{R}^d$., Comment: 14 pages

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $\gamma$ 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 $\gamma \to \pi/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $\gamma \to \pi/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor., Comment: 23 pages, 1 figure

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

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., Comment: 12 pages

We provide sufficient conditions for synchronization by noise, i.e. under these conditions we prove that weak random attractors for random dynamical systems consist of single random points. In the case of SDE with additive noise, these conditions are also essentially necessary. In addition, we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point. As a result, synchronization by noise is proven for a large class of SDE with additive noise. In particular, we prove that the random attractor for an SDE with drift given by a (multidimensional) double-well potential and additive noise consists of a single random point. All examples treated in [Tearne, PTRF; 2008] are also included., Comment: 45 pages

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., Comment: 9 pages

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 $\mu$ converge to $\mu$ 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 $\mu$-almost all initial conditions.

We consider a discretized version of the quenched Edwards-Wilkinson model for the propagation of a driven interface through a random field of obstacles. Our model consists of a system of ordinary differential equations on a $d$-dimensional lattice coupled by the discrete Laplacian. At each lattice point, the system is subject to a constant driving force and a random obstacle force impeding free propagation. The obstacle force depends on the current state of the solution and thus renders the problem non-linear. For independent and identically distributed obstacle strengths with exponential moment we prove ballistic propagation (i.e., propagation with a positive velocity) of the interface if the driving force is large enough. For a specific case of dependent obstacles, we show that no stationary solution exists, but still the propagation of the front is not ballistic., Comment: 13 pages, 1 figure

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

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., Comment: Technical Report, School of Electrical, Computer, and Energy Eng., Arizona State University, Tempe, AZ

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 L\'{e}vy measure., Comment: Published at http://dx.doi.org/10.3150/14-BEJ651 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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 $\kappa_p$ (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., Comment: To appear in {\em Infin. Dimens. Anal. Quantum Probab. Relat. Top.}