Ergodic BSDEs under weak dissipative assumptions

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in Fuhrman et al. (2009) [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non-degenerate.We show the existence of solutions by the use of coupling estimates for a non-degenerate forward stochastic differential equation with bounded measurable nonlinearity. Moreover we prove the uniqueness of “Markovian” solutions by exploiting the recurrence of the same class of forward equations.Applications are then given for the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.     

Global classical solution to partially dissipative quasilinear hyperbolic systems

For 1-D quasilinear hyperbolic systems, the strict dissipation or the weak linear degeneracy can prevent the formation of singularity. More precisely, if all the inhomogeneous sources are strictly dissipative, or all the characteristics are weakly linearly degenerate and the system is homogeneous, then the Cauchy problem with small and decaying initial data admits a unique global classical solution. In this paper, under some suitable hypotheses on the interaction, new kinds of weighted formulas of wave decomposition are developed to show the same result for a general class of combined systems, in which a part of equations possesses the strict dissipation and the others are weakly linearly degenerate.     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

Ergodic properties of crystallization processes

We consider a birth and growth process with germs which are born according to a Poisson point process whose intensity rneasure is invariant under trunslations of the space. The germs can be born in the unoccupied space; then they grow until they occupy the available space. In this general framework, the crystallization process can be characterized by a random field, which assigns to any point of the state space the first time at which this point is reached by a crigstal. Under general conditions on the growth speed and geometric shape of free crystals, we prone that the random field is mixing in the sense of ergodic theory. This result is illustrated by applications to the problem of parameter estimation. Bibliography: 7 titles.     

Some properties of invariant measures of non symmetric dissipative stochastic systems

 We consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms. We prove an integration by part formula and a Logarithmic Sobolev inequality for the invariant measure. No symmetry or reversibility assumptions are made. Furthemore we prove some compactness results on the transition semigroup and on the embedding of the Sobolev spaces based on the invariant measure. We use these results to derive asymptotic properties for a stochastic reaction–diffusion equation. Received: 29 September 2000 / Revised version: 30 May 2001 / Published online: 14 June 2002     

Ergodic properties of triangle partitions

We study the ergodic properties of a map called the Triangle Sequence. We prove that the algorithm is weakly convergent almost surely, and ergodic. As far as we know, it is the first example of a 2-dimensional algorithm where a surprising diophantine phenomenon happens: there are sequences of nested cells whose intersection is a segment, although no vertex is fixed. Examples of n-dimensional algorithms presenting this behavior were known for n ≥ 3.      

Ergodic properties of triangle partitions

We study the ergodic properties of a map called the Triangle Sequence. We prove that the algorithm is weakly convergent almost surely, and ergodic. As far as we know, it is the first example of a 2-dimensional algorithm where a surprising diophantine phenomenon happens: there are sequences of nested cells whose intersection is a segment, although no vertex is fixed. Examples of n-dimensional algorithms presenting this behavior were known for n ≥ 3.     

Global classical solutions to partially dissipative quasilinear hyperbolic systems with weaker restrictions on wave interactions

A kind of partially dissipative quasilinear hyperbolic systems in which a part of equations possesses the strict dissipation and the others are weakly linearly degenerate are discussed. Under some weaker hypotheses on the interactions between two parts of equations, it is proved that for any given initial data with small W^1,1 and C¹ norms, the corresponding Cauchy problem admits a unique C¹ global classical solution. Copyright © 2012 John Wiley & Sons, Ltd.     

Generalized almost periodic solutions and ergodic properties of quasi-autonomous dissipative systems

Let H be a real Hilbert space, A a maximal monotone operator in H and $\text{? R → H}$ a measurable function which is S¹-almost periodic. Assuming that the positive trajectories of the equation $\text{du}\text{dt}\text{+ Au(t)? f(t)}$ are bounded (in H) for t ? 0, we construct a kind of generalized almost periodic “solution” of the equation, and we show how to deduce information of ergodic type on the asymptotic behavior of the trajectories as t → +∞.     

Ergodic theory of differentiable dynamical systems

Iff is a C^{1 + ɛ} diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products. Dedicated to the memory of Rufus Bowen     

Ergodic theorems for noncommutative dynamical systems

Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ₊ ^d and for general Lie groups.     

Ergodic properties of fibered rational maps

We study the ergodic properties of fibered rational maps of the Riemann sphere. In particular we compute the topological entropy of such mappings and construct a measure of maximal relative entropy. The measure is shown to be the unique one with this property. We apply the results to selfmaps of ruled surfaces and to certain holomorphic mapping of the complex projective planeP ². Supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT).     

Ergodic properties of stationary stable processes

We derive spectral necessary and sufficient conditions for stationary symmetric stable processes to be metrically transitive and mixing. We then consider some important classes of stationary stable processes: Sub-Gaussian stationary processes and stationary stable processes with a harmonic spectral representation are never metrically transitive, the latter in sharp contrast with the Gaussian case. Stable processes with a harmonic spectral representation satisfy a strong law of large numbers even though they are not generally stationary. For doubly stationary stable processes, sufficient conditions are derived for metric transitivity and mixing, and necessary and sufficient conditions for a strong law of large numbers.     

Ergodic properties of max-infinitely divisible processes

We prove that a stationary max-infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesàro summable to 0). These criteria are applied to some classes of max-infinitely divisible processes.