This paper concerns the ergodic theory of a class of nonlinear dissipative PDEs of parabolic type. Leaving precise statements for later, we first give an indication of the nature of our results. We view the equation in question as a semi-group or dynamical system St on a suitable function space H , and assume the existence of a compact attracting set (as in Temam [15], Chapter 1). To this deter...

This paper denes and investigates the ergodic proper-ties of the entropy of a countable partition of a fuzzy dynamical sys-tem at different points of the state space. It ultimately introducesthe local fuzzy entropy of a fuzzy dynamical system and proves itto be an isomorphism invariant.

Harnessing the quantum computation power of present noisy-intermediate-size-quantum devices has received tremendous interest in last few years. Here we study learning a one-dimensional long-range randomly-coupled spin chain, within framework reservoir computing. In time sequence tasks, find system many-body localized (MBL) phase holds long-term memory, which can be attributed to emergent local ...

In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the erg...

The connection between ergodic theory and the theory of von Neumann algebras goes back to the very beginning of the theory of “rings of operators”. Maximal inequalities in ergodic theory provide an important tool in classical analysis. In this paper we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic theorem, thereby connecting these different aspects of ergod...

We review a wide variety of applications in different branches of sciences arising from the study of dynamical systems. The emergence of chaos and fractals from iterations of simple difference equations is discussed. Notions of phase gpace, contractive mapping, attractor, invariant density and the relevance of ergodic theory for studying dynamical systems are reviewed. Various concepts of dimen...

In this paper, we prove a convergence theorem for singular perturbations problems class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to degenerate PDEs. Our approach mainly based on G-stochastic analysis argument. As byproduct, also establish averaging principle stochastic driven by G-Brow...