We consider large time and infinite particle limit for a system of particles living in random potentials. The randomness enters the potential through an external ergodic Markov process, modeling oscillating environment with good statistical averaging properties. From each individual particle’s point of view, both law of large number and central limit theorem type of averaging are possible. Prob...

A language L over a finite alphabet Σ is called growth-sensitive if forbidding any set of subwords F yields a sublanguage LF whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear context-free language is growth-sensitive. “Ergodic” means for a context-free grammar and language that its dependency di-graph is strongly connected. The same r...

Dynamical systems of a billiard type are a fundamental notion relevant for the understanding of numerous phenomena observed in statistical mechanics, Hamiltonian dynamics, nonlinear physics, and others. By means of billiard models, the principal ideas as Boltzmann ergodic hypothesis, related to the foundation of statistical physics, and deterministic diffusion gained more deep insight. However,...

Almost all of the technical methods, which are used in the theory of nonlinear dynamical systems, originated in the theory of one dimensional dynamical systems. In this talk, a relation between nonlinear ergodic theorems, which are developed by Prof. Takahashi and other researchers, and the asymptotic behavior of one dimensional dynamical systems will be delivered, and moreover, application of ...

Abstract. The ergodic control problem for a non-degenerate controlled diffusion controlled through its drift is considered under a uniform stability condition that ensures the well-posedness of the associated Hamilton–Jacobi– Bellman (HJB) equation. A nonlinear parabolic evolution equation is then proposed as a continuous time continuous state space analog of White’s ‘relative value iteration’ ...

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection a...

Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that the equation in general has multiple solutions. We shall specify the set of all the classical solu...

In this paper, we first introduce a class of nonlinear mappings called extended hybrid in a Hilbert space containing the class of generalized hybrid mappings. The class is different from the class of super hybrid mappings which was defined by Kocourek, Takahashi and Yao [12]. We prove a fixed point theorem for generalized hybrid nonself-mapping in a Hilbert space. Next, we prove a nonlinear erg...