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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 and Mattingly (Electron J Probab 16:658–738, 2011). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 (Probab Theory Relat Fields 141(1-2):1-18, 2008) are also included. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations. [ABSTRACT FROM AUTHOR]

We study a “new kind” of backward doubly stochastic differential equations, where the nonlinear noise term is given by Itô–Kunita's stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolev's solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise. [ABSTRACT FROM AUTHOR]

We consider a network of N nodes with given distances in which customers arrive at one of the nodes and request transportation to one of the other nodes. There is a finite number of servers (or transporters) with possibly different capacities and speed available. We show that there exists a critical arrival rate κc, such that if the arrival rate is larger than κc then the number of customers in the system increases to ∞ with linear speed no matter which routing strategy for the transporters is chosen (even if we allow the strategy to depend on the future arrival process). If, on the other hand, κ < c then there exists even a fixed periodic routing strategy which keeps the system stable (in a sense to be made precise). We prefer to start with a deterministic set-up which allows very general arrival processes and look at the stochastic case only in the final section where we get more refined results by assuming that the arrival process has integrable stationary ergodic increments (which still allows batch arrivals and long-range dependence). Potential applications include the control of elevators in highrise buildings. [ABSTRACT FROM AUTHOR]

We prove that if ϕ is a random dynamical system (cocycle) for which t→ϕ (t, ω)x is a semimartingale, then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment (helix), and conversely. This relation is succinctly expressed as 'semimartingale cocycle=exp(semimartingale helix)'. To implement it we lift stochastic calculus from the traditional one-sided time ℝ to two-sided time T=ℝ and make this consistent with ergodic theory. We also prove a general theorem on the perfection of a crude cocycle, thus solving a problem which was open for more than ten years. [ABSTRACT FROM AUTHOR]

We prove a 'propagation of chaos' result for the mean-field limit of a model for a trimolecular chemical reaction called 'Brusselator'. Then we show that the pair of nonlinear (i.e. law-dependent) stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law (in t). Finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]