A Liv\v{s}ic-type theorem and some regularity properties for nonadditive sequences of potentials

We study some notions of cohomology for asymptotically additive sequences and prove a Liv\v{s}ic-type result for almost additive sequences of potentials. As a consequence, we are able to characterize almost additive sequences based on their equilibrium measures and also show the existence of almost (and asymptotically) additive sequences of H\"older continuous functions satisfying the bounded variation condition (with a unique equilibrium measure) and which are not physically equivalent to any additive sequence generated by a H\"older continuous function. None of these examples were previously known, even in the case of full shifts of finite type. Moreover, we also use our main result to suggest a classification of almost additive sequences based on physical equivalence relations with respect to the classical additive setup.Comment: 36 page

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$\beta$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian ($1<q<3$) distribution and tend to a Gaussian ($q=1$) for longer times, as the orbits eventually enter into "large size" chaotic domains. However, in agreement with other studies, we find in certain cases that the $q$-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called "crossover" function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these $q$-Gaussian distributions to identify two energy regimes of "weak chaos"-one where the system melts and one where it transforms from liquid to a gas state-by observing where the $q$-index of the distribution increases significantly above the $q=1$ value of strong chaos.Comment: 32 pages, 13 figures, Submitted for publication to Physica

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.Comment: 23 pages, to appear in Nonlinearity; small changes made according to comments from the referee

Asymptotic analysis for the generalized langevin equation

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity

Intertemporal resource allocation and intergenerational equity: compatibility of efficiency and equity

研究成果の概要 (和文) : 異時点間資源配分の理論研究として、(1)最適経済成長経路の特徴付け、(2)異時点間選好の効用関数表現、(3)衡平分割問題の解法、という3つの観点から研究を行った。代表的消費者が時間選好率が消費経路に依存する再帰的効用関数を持つ場合に、収穫逓増をともなう経済で最適成長経路が存在するための条件を求め、最適成長経路の支持価格による特徴付けを行った。衡平分割問題において、効率性と衡平性を同時に満たす解の存在を示した。研究成果の概要 (英文) : I have investigated the following three theoretical aspects of intertemporal resource allocations : (1) a characterization of optimal paths for economic growth ; (2) representation of intertemporal preferences by utility functions ; (3) solutions to fair division problems. I have proved the existence of optimal path for economies with increasing returns in which a representative agent possesses a recursive utility function that endogenizes time preference for consumption paths. In the fair division problem, I have demonstrated the existence of solutions which satisfy both efficiency and fairness

Macroeconomic Policy Analysis using utility function with non-constant discount factor

研究成果の概要 (和文) : 割引因子が一定でないマクロモデルを用いて、財政政策と金融政策の分析を行った。特に、逐次的効用関数と呼ばれる、割引因子が効用もしくは消費や実物貨幣残高といったマクロ変数の関数に商店を当てた。いくつかの論文を書き上げ、国際学会でプレゼンテーションを実施し、国際査読付き雑誌に投稿し、いくつかはすでに出版された。また、研究代表者は、この成果を含めた論文集「選好と国際マクロ経済学」法政大学出版局、2012を編集した。この本は2012年3月に法政大学出版局から出版された。研究成果の概要 (英文) : We conduct a public/fiscal policy analysis using macroeconomic models with non-constant discount factors. In particular, we focused on recursive utility models, in which the discount factor is a function of utility or some aggregate macroeconomic variables including consumption and real balances. We wrote some research papers to give presentations at international academic conferences and to submit to international refereed journals. Some of them have been already published. In addition, Kenji Miyazaki edited a Japanese book including our research papers. This book, titled \u27Preference and International Macroeconomics,\u27was published in March, 2012

Nonadditive families of potentials: physical equivalence and some regularity relations

We show that additive and asymptotically additive families of continuous functions with respect to suspension flows are physically equivalent. In particular, the equivalence result holds for hyperbolic flows and some classes of expansive flows in general. Moreover, we show how this equivalence result can be used to extend the nonadditive thermodynamic formalism and multifractal analysis. In the second part of this work, we obtain a Liv\v{s}ic-like result for nonadditive families of potentials and also address the H\"older and Bowen regularity problem for the physical equivalence relations with respect to hyperbolic symbolic flows.Comment: This material is a substantial upgrade on the previous submission "Asymptotically additive families of functions and a physical equivalence problem for flows" (68 pages