Isomorphism and Embedding of Borel Systems on Full Sets

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them.

In this work we consider several aspects of the thermodynamic formalism in a randomized setting. Let $X$ be a non-trivial mixing shift of finite type, and let $f : X \to \mathbb{R}$ be a H\older continuous potential with associated Gibbs measure $\mu$. Further, fix a parameter $\alpha \in (0,1)$. For each $n \geq 1$, let $\mathcal{F}_n$ be a random subset of words of length $n$, where each word of length $n$ that appears in $X$ is included in $\mathcal{F}_n$ with probability $1-\alpha$ (and excluded with probability $\alpha$), independently of all other words. Then let $Y_n = Y(\mathcal{F}_n)$ be the random subshift of finite type obtained by forbidding the words in $\mathcal{F}_n$ from $X$. In our first main result, for $\alpha$ sufficiently close to $1$ and $n$ tending to infinity, we show that the pressure of $f$ on $Y_n$ converges in probability to the value $P_X(f) + \log(\alpha)$, where $P_X(f)$ is the pressure of $f$ on $X$. Additionally, let $H_n = H(\mathcal{F}_n)$ be the random hole in $X$ consisting of the union of the cylinder sets of the words in $\mathcal{F}_n$. For our second main result, for $\alpha$ sufficiently close to one and $n$ tending to infinity, we show that the escape rate of $\mu$-mass through $H_n$ converges in probability to the value $-\log(\alpha)$ as $n$ tends to infinity.

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.

Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group,and $\beta:\Sigma\to G$ is a continuous function, denote by $\sigma_\beta$ the skew-product of $\sigma$ by$\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ for which the skew-products $\sigma_\beta$ are topologicallytransitive for sets of parameters of full measure. If $G$ isa connected semisimple matrix Lie group, we show examples of functions $\beta$ for which theskew-products $\sigma_beta$ are topologically transitive.

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we construct a countable partition $${\cal A}$$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $${{\cal A}^\infty}$$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows, provided that the limiting measure satisfies μ(Sing(X)) = 0.

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps$T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$together with a$T$-invariant measure whose base map$\unicode[STIX]{x1D703}$satisfies certain topological and mixing conditions and the maps$T_{\unicode[STIX]{x1D714}}$on the fibers are certain non-singular distance-expanding maps. Our results hold true when$\unicode[STIX]{x1D703}$is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by$T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems.Probab. Theory Related Fields162(2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of$T$(with respect to the invariant measure) does not seem to have a spectral gap.

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another, i.e., heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

We study the hyperbolicity of a class of horseshoes exhibiting an <i>internal</i> tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly <i>not</i> uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.

Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations

The set of stationary measures of an infinite Hamiltonian system with noise is investigated. The model consists of particles moving in R 3 with bounded velocities and subject to a noise that does not violate the classical laws of conservation, see (OVY). Following (LO) we assume that the noise has also a finite radius of interaction, and prove that translation invariant stationary states of finite specific entropy are reversible with respect to the stochastic component of the evolution. Therefore the results of (LO) imply that such invariant measures are superpositions of Gibbs states.

This paper establishes a fundamental difference between $\mathbb{Z}$ subshifts of finite type and $\mathbb{Z}^{2}$ subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type $X$ as a subset of a full shift $F$. We then introduce a natural penalty function $f$, defined on $F$, which is 0 if the local configuration near the origin is legal and $-1$ otherwise. We show that in the case of $\mathbb{Z}$ subshifts, for all sufficiently small perturbations, $g$, of $f$, the $g$-maximizing invariant probability measures are supported on $X$ (that is, the set $X$ is stably maximized by $f$). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations, $g$, of $f$ for which the $g$-maximizing invariant probability measures are supported on $F\setminus X$.

We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical o singular points for the random maps.

In this paper, we consider a stochastic SIR model with the perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is usedto classify the extinction and permanence of the disease. Precisely, $\lambda0$ the epidemic takes place. In this case, we derive that the Markov process $(S(t), I(t))$ has a unique invariant probability measure. We also characterize the support of a unique invariant probability measure and prove that the transition probability converges to this invariant measures in total variation norm. Our result is considered as an significant improvement over the results in [6,7,11,18].