Complexity and the Fractional Calculus

We study complex processes whose evolution in time rests on the occurrence of a large and random number of events. The mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories each of which satisfies the stochastic central limit theorem and the condition for the Mittag-Leffler universality.

ERGODIC CONDITION AND MAGNETIC MODELS

A condition for ergodicity is derived, applicable to a Hermitian many body model in both the classical and quantum domains. Using this ergodic condition, the validity of the ergodic hypothesis is examined in certain solvable 1d magnetic models. A simple but general picture has emerged which shows why the hypothesis can be valid and why it can also fail.

Ergodicity and asymptotically almost periodic solutions of some differential equations

Using ergodicity of functions, we prove the existence and uniqueness of (asymptotically) almost periodic solution for some nonlinear differential equations. As a consequence, we generalize a Massera’s result. A counterexample is given to show that the ergodic condition cannot be dropped.

A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

A uniform convergence theorem for the numerical solving of the nonlinear filtering problem

The filtering problem concerns the estimation of a stochastic process X from its noisy partial information Y. With the notable exception of the linear-Gaussian situation, general optimal filters have no finitely recursive solution. The aim of this work is the design of a Monte Carlo particle system approach to solve discrete time and nonlinear filtering problems. The main result is a uniform convergence theorem. We introduce a concept of regularity and we give a simple ergodic condition on the signal semigroup for the Monte Carlo particle filter to converge in law and uniformly with respect to time to the optimal filter, yielding what seems to be the first uniform convergence result for a particle approximation of the nonlinear filtering equation.

Zero-sum two-person semi-Markov games

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.

Zero-sum two-person semi-Markov games

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.