Stochastics and Dynamics

This paper introduces and solves a general class of fully coupled forward–backward stochastic dynamics by investigating the associated system of functional differential equations. As a consequence, we are able to solve many different types of forward–backward stochastic differential equations (FBSDEs) that do not fit in the classical setting. In our approach, the equations are running in the same time direction rather than in a forward and backward way, and the conflicting nature of the structure of FBSDEs is therefore avoided.

We study the behavior of the empirical distribution function of iterates of intermittent maps in the Hilbert space of square integrable functions with respect to Lebesgue measure. In the long-range dependent case, we prove that the empirical distribution function, suitably normalized, converges to a degenerate stable process, and we give the corresponding almost sure result. We apply the results to the convergence of the Wasserstein distance between the empirical measure and the invariant measure. We also apply it to obtain the asymptotic distribution of the corresponding Cramér–von-Mises statistic.

We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein–Kantorovich–Rubinstein distance on path space between two Lévy diffusions in terms of the coupling distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process.

As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

We establish the existence, uniqueness and approximation of the strong solutions for the stochastic 3D LANS-α model driven by a non-Gaussian Lévy noise. Moreover, we also study the stability of solutions. In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math.189 (2012) 61–110] obtained results on mixing and mixing rates for a large class of noninvertible maps preserving an infinite ergodic invariant measure.

Here, we are concerned with extending these results to the invertible setting. Mixing is established for a large class of infinite measure invertible maps. Assuming additional structure, in particular exponential contraction along stable manifolds, it is possible to obtain good results on mixing rates and higher order asymptotics.

We consider two independent Goldstein–Kac telegraph processes X₁(t) and X₂(t) on the real line ℝ, both developing with constant speed c > 0, that, at the initial time instant t = 0, simultaneously start from the origin 0 ∈ ℝ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ > 0. Closed-form expressions for the transition density φ(x, t) and the probability distribution function Φ(x, t) = Pr{S(t) < x}, x ∈ ℝ, t > 0, of the sum S(t) = X₁(t) + X₂(t) of these processes at arbitrary time instant t > 0, are obtained. It is also proved that the shifted time derivative g(x, t) = (∂/∂t + 2λ)φ(x, t) satisfies the Goldstein–Kac telegraph equation with doubled parameters 2c and 2λ. From this fact it follows that φ(x, t) solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.