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The typicality of the canonical state shows that majority of the states are indistinguishable from equilibrium, and thus the nonequilibrium states are exceptionally rare in the extremely high-dimensional Hilbert space. On the contrary, we…

For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structrue of the space of invariant measures: The ergodic measures of intermediate entropies and…

We propose a definition o meta-stability and obtain sufficient conditions for a sequence of Markov processes on finite state spaces to be meta-stable. In the reversible case, these conditions reduce to estimates of the capacity and the…

In this paper we classify the pathwise asymptotic behaviour of the discretisation of a general autonomous scalar differential equation which has a unique and globally stable equilibrium. The underlying continuous equation is subjected to a…

We study the ergodic property of a continuous-state branching process with immigration and competition. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to…

We establish central limit theorems for a large class of supercritical branching Markov processes in infinite dimension with spatially dependent and non-necessarily local branching mechanisms. This result relies on a fourth moment…

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin methods. Our results apply to DDEs with multiple discrete…

The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics,…

We introduce the notion of perturbations of quantum stochastic models using the series product, and establish the asymptotic convergence of sequences of quantum stochastic models under the assumption that they are related via a right series…

We present some new nonparametric estimators of entropies and we establish almost sure consistency and central limit Theorems for some of the most important entropies in the discrete case. Our theorical results are validated by simulations.

Ensemble of initial conditions for nonlinear maps can be described in terms of entropy. This ensemble entropy shows an asymptotic linear growth with rate K. The rate K matches the logarithm of the corresponding asymptotic sensitivity to…

We derive high-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of a finite number of diametrically small inhomogeneities with conductivities different from the background conductivity. Our…

We show the entanglement entropy in certain quantum field theories to contain state-dependent divergences. Both perturbative and holographic examples are exhibited. However, quantities such as the relative entropy and the generalized…

The hypercontractivity is proved for the Markov semigroup associated to a class of finite/infinite dimensional stochastic Hamiltonian systems. Consequently, the Markov semigroup is exponentially convergent to the invariant probability…

The Lyapunov exponent characterizes an exponential growth rate of the difference of nearby orbits. A positive Lyapunov exponent is a manifestation of chaos. Here, we propose the Lyapunov pair, which is based on the generalized Lyapunov…

Continuing our investigation into the Hierarchical Reference Theory of fluids for thermodynamic states of infinite isothermal compressibility kappa[T] we now turn to the available numerical evidence to elucidate the character of the partial…

The entropy of a classical thermally isolated Hamiltonian system is given by the logarithm of the measure of phase space enclosed by the constant energy hyper-surface, also known as volume entropy. It has been shown that on average the…

We consider the problem of asymptotic convergence to invariant sets in interconnected nonlinear dynamic systems. Standard approaches often require that the invariant sets be uniformly attracting. e.g. stable in the Lyapunov sense. This,…

We introduce a new family of integrable stochastic processes, called \textit{dynamical stochastic higher spin vertex models}, arising from fused representations of Felder's elliptic quantum group $E_{\tau, \eta} (\mathfrak{sl}_2)$. These…

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,\pi)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with…