Weighted L 2-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2(dμ) and L 2(W* dμ), where μ denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

Food Security Sustainability: A Synthesis of the Current Concepts and Empirical Approaches for Meeting SDGs

Currently, food security is becoming a fundamental problem in the global macroeconomic dynamics for policymakers and governments in developing countries. Globally, food security offers challenges both from achieving Sustainable Development Goals (SDGs) targets and the welfare perspective of many poor households. As a result, this study is guided by Neo Malthusian and Access theories to investigate Food Security Sustainability: a Synthesis of the Current Concepts and Empirical Approaches for Meeting SDGs in Nigeria using ARDL and ECM techniques. The ARDL revealed that agricultural value-added and GDP positively affect food security for commercial agrarian investments in Nigeria. However, internal displacement, population growth, food inflation, and exchange rate volatility negatively affect sustainable food security in Nigeria. The model’s coefficient of ECMt−1 also shows negative (−0.0130 approximately) and statistically significant (0.0000) at 1%. Thus, the speed of adjustment requires 1.3% annually for the long-run equilibrium convergence to be restored. The study concludes that the SDGs targets for poverty and hunger reduction, mainly for food security sustainability alongside small producers by the year 2030, can be rarely achieved because the convergence to equilibrium is more than nine years. An active value-addition strategy for sustainable food security and the provision of humanitarian interventions are recommended.

Block Factorization of the Relative Entropy via Spatial Mixing

We consider spin systems in the d-dimensional lattice $${\mathbb Z} ^d$$ Z d satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region $$V\subset {\mathbb Z} ^d$$ V ⊂ Z d in terms of a weighted sum of the entropies on blocks $$A\subset V$$ A ⊂ V when each A is given an arbitrary nonnegative weight $$\alpha _A$$ α A . These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.

Testing Real Convergence as a Prerequisite for Long Run Sustainability

The main objective of this research is to estimate the degree of real convergence of the countries that joined the European Union between 2004–2013 as an essential precondition for sustainable accession to the Euro Area. Through this study, we tried to create a clear, real and comparative image for the downward trend in the dispersion of the GDP/capita and the speed by which countries with different integration stages achieve the real economic convergence to equilibrium level. In this respect, we tested real convergence by regression models. Further, in order to verify the robustness of the results we applied a cluster analysis. The main results show that non-Euro Area countries have a tendency to individually reduce income disparities with the Euro Area average, but do not register a convergent economic growth and do not form a homogeneous convergence cluster, unlike the newer Euro Area Member Countries. Another representative result is that the Czech Republic seems to be the best prepared country to adopt the single currency in a sustainable way, while Bulgaria is at the opposite pole.

A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases

When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations, derived in \cite{tran2020boltzmann}.

Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.