Stochastics and Dynamics

We study the large time limiting properties of a Lévy process in a symmetric space of noncompact type, both pathwise and in terms of distribution.

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

Diffusion processes associated with semi-Dirichlet forms are studied in the paper. The main results are Fukushima's decomposition for the diffusions and a transformation formula for the corresponding martingale part of the decomposition. The results are applied to some concrete examples.

In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.

For the shift σ in Σ = {0, 1}^ℕ, we define the renormalization for potentials by

We show that the stable set of the Hofbauer potential, i. e. the set of potentials V such that converges to V* is characterized by the germ of these potentials close to 0^∞ = 000….

Then, we make connections with the Manneville–Pomeau map f : [0, 1]↺. In particular we show that the lift in Σ of log f′ is in the stable set of V*.

In the second part, we characterize "good" H, such that σ² ◦ H = H ◦ σ.

In the last part, we study the thermodynamic formalism for some special potentials in the stable set of V*. They are called virtual Manneville–Pomeau maps.

The theorem of Shannon–McMillan–Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). In this paper we prove that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.