Infinite Measure Research Articles

Infinite measure mixing for some mechanical systems

We show that if an infinite measure preserving system is well approximated on most of the phase space by a system satisfying the local limit theorem, then the original system enjoys mixing with respect to global observables, that is, the observables which admit an infinite volume average. The systems satisfying our conditions include the Lorentz gas with Coulomb potential, the Galton board, and piecewise smooth Fermi-Ulam pingpongs.

Exact simulation of continuous max-id processes with applications to exchangeable max-id sequences

An algorithm for the unbiased simulation of continuous max-(resp. min-) infinitely divisible stochastic processes is developed. The algorithm only requires the simulation of finite Poisson random measures on the space of continuous functions and avoids the necessity of computing conditional distributions of infinite (exponent) measures. The complexity of the algorithm is characterized in terms of the expected number of simulated atoms of the Poisson random measures on the space of continuous functions. Special emphasis is put on the simulation of exchangeable max-(or min-) infinitely divisible sequences, in particular exchangeable Sato-frailty sequences. Additionally, exact simulation schemes of exchangeable exogenous shock models and exchangeable max-stable sequences are sketched.

Exponential Fermi Acceleration in a Switching Billiard

In this paper we show an infinite measure set of exponentially escaping orbits for a resonant Fermi accelerator, which is realised as a square billiard with a periodically oscillating platform. We use normal forms to describe how the energy changes in a period and we employ techniques for hyperbolic systems with singularities to show the exponential drift of these normal forms on a divided time-energy phase.

Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic; hence these two notions are not equivalent for infinite measure. We also give type IIIλ examples of such systems, for 0<λ≤1. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence (ni) that is not measure theoretically rigid along (ni) for any nonsingular ergodic finite measure.

Superrigidity for dense subgroups of lie groups and their actions on homogeneous spaces

An essentially free group action \(\Gamma \curvearrowright (X,\mu )\) is called W\(^*\)-superrigid if the crossed product von Neumann algebra \(L^\infty (X) \rtimes \Gamma \) completely remembers the group \(\Gamma \) and its action on \((X,\mu )\). We prove W\(^*\)-superrigidity for a class of infinite measure preserving actions, in particular for natural dense subgroups of isometries of the hyperbolic plane. The main tool is a new cocycle superrigidity theorem for dense subgroups of Lie groups acting by translation. We also provide numerous countable type II\(_1\) equivalence relations that cannot be implemented by an essentially free action of a group, both of geometric nature and through a wreath product construction.

Weak compactness and representation in variable exponent Lebesgue spaces on infinite measure spaces

Relative weakly compact sets and weak convergence in variable exponent Lebesgue spaces {L^{p(cdot )}(Omega )} for infinite measure spaces (Omega ,mu ) are characterized. Criteria recently obtained in [14] for finite measures are here extended to the infinite measure case. In particular, it is showed that the inclusions between variable exponent Lebesgue spaces for infinite measures are never L-weakly compact. A lattice isometric representation of {L^{p(cdot )}(Omega )} as a variable exponent space L^{q(cdot )}(0,1) is given.

Nonsingular Poisson Suspensions

The classical Poisson functor associates to every infinite measure preserving dynamical system (X, μ, T) a probability preserving dynamical system (X*, μ*, T*) called the Poisson suspension of T. In this paper we generalize this construction: a subgroup Aut2(X, μ) of μ-nonsingular transformations T of X is specified as the largest subgroup for which T* is μ*-nonsingular. The topological structure of this subgroup is studied. We show that a generic element in Aut2(X, μ) is ergodic and of Krieger type III1. Let G be a locally compact Polish group and let A: G → Aut2(X, μ) be a G-action. We investigate dynamical properties of the Poisson suspension A* of A in terms of an affine representation of G associated naturally with A. It is shown that G has property (T), if and only if each nonsingular Poisson G-action admits an absolutely continuous invariant probability. If G does not have property (T), then for each generating probability κ on G and t > 0, a nonsingular Poisson G-action is constructed whose Furstenberg κ-entropy is t.

Acute threshold dynamics of an epidemic system with quarantine strategy driven by correlated white noises and Lévy jumps associated with infinite measure.

Several studies have previously been conducted on the dynamics of probabilistic epidemic models driven by Lévy disorder. All of these works have used the Poisson counting process with finite Lévy measures. However, this scope disregards a considerable category of correlated Lévy jump processes governed by an infinite Lévy measure. In this research, we take into consideration this general framework applied to an epidemic model with a quarantine strategy. Under an appropriate hypothetical setting, we infer the exact threshold value between the ergodicity and the disease disappearance. Our analysis completes the work presented by Privault and Wang (J Nonlinear Sci 31(1):1–28, 2021) and puts forward a novel analytical aspect to deal with other stochastic models in several areas. As a numerical application, we implement the algorithm of Rosinski (Stoch Process Appl 117:677–707, 2007) for tempered stable Lévy processes with an infinite Lévy measure.

Convergence of volume forms on a family of log Calabi–Yau varieties to a non-Archimedean measure

We study the convergence of volume forms on a degenerating holomorphic family of log Calabi–Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let (X, B) be a holomorphic family of sub log canonical, log Calabi–Yau complex varieties parameterized by the punctured unit disk. Let \(\eta \) be a meromorphic form on X with poles along B such that the restriction of \(\eta \) is a top-dimensional form on each of the fibers. We show that the (possibly infinite) measures induced by the restriction of \(|\eta \wedge \overline{\eta }|\) to a fiber converge to a measure on the Berkovich analytification of \(X \setminus B\) as we approach the puncture. The convergence takes place on a hybrid space, which is obtained by filling in the space \(X \setminus B\) with the aforementioned Berkovich space over the puncture.

Graphical construction of spatial Gibbs random graphs

We consider a random graph model on Zd that incorporates the interplay between the statistics of the graph and the underlying space where the vertices are located. Based on a graphical construction of the model as the invariant measure of a birth and death process, we prove the existence and uniqueness of a measure defined on graphs with vertices in Zd, which coincides with the limit along the measures over graphs with the finite vertex set. As a consequence, theoretical properties, such as exponential mixing of the infinite volume measure and central limit theorem for averages of a real-valued function of the graph, are obtained. Moreover, a perfect simulation algorithm based on the clan of ancestors is described in order to sample a finite window of the equilibrium measure defined on Zd.

Singular elliptic problems in general domains

In this paper, we prove the existence of solutions to nonlinear elliptic equations, which present first-order terms with natural growth with respect to the gradient and lower order terms singular in the variable that represents the solution. The problems are considered on a domain Ω, which may have infinite Lebesgue measure.

Semisolid sets and topological measures

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is defined on open and closed sets; it is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. We examine semisolid sets and give a way of constructing topological measures from solid-set functions on locally compact, Hausdorff, connected, locally connected spaces. For compact spaces we present two simpler methods than the currently available one. We give examples of finite and infinite topological measures on locally compact spaces and present an easy way to generate topological measures on spaces whose one-point compactification has genus 0. Our results are necessary for various methods for constructing topological measures, give additional properties of topological measures, and provide a tool for determining whether two topological measures or quasi-linear functionals are the same.

The symmetric coalescent and Wright–Fisher models with bottlenecks

We define a new class of Ξ-coalescents characterized by a possibly infinite measure over the nonnegative integers. We call them symmetric coalescents since they are the unique family of exchangeable coalescents satisfying a symmetry property on their coagulation rates: they are invariant under any transformation that consists of moving one element from one block to another without changing the total number of blocks. We illustrate the diversity of behaviors of this family of processes by introducing and studying a one parameter subclass, the (β,S)-coalescents. We also embed this family in a larger class of Ξ-coalescents arising as the limit genealogies of Wright–Fisher models with bottlenecks. Some convergence results rely on a new Skorokhod type metric, that induces the Meyer–Zheng topology, which allows us to study the scaling limit of non-Markovian processes using standard techniques.

Gas of sub-recoiled laser cooled atoms described by infinite ergodic theory.

The velocity distribution of a classical gas of atoms in thermal equilibrium is the normal Maxwell distribution. It is well known that for sub-recoiled laser cooled atoms, Lévy statistics and deviations from usual ergodic behavior come into play. In a recent letter, we showed how tools from infinite ergodic theory describe the cool gas. Here, using the master equation, we derive the scaling function and the infinite invariant density of a stochastic model for the momentum of laser cooled atoms, recapitulating results obtained by Bertin and Bardou [Am. J. Phys. 76, 630 (2008)] using life-time statistics. We focus on the case where the laser trapping is strong, namely, the rate of escape from the velocity trap is R(v) ∝ |v|α for v → 0 and α > 1. We construct a machinery to investigate time averages of physical observables and their relation to ensemble averages. The time averages are given in terms of functionals of the individual stochastic paths, and here we use a generalization of Lévy walks to investigate the ergodic properties of the system. Exploring the energy of the system, we show that when α = 3, it exhibits a transition between phases where it is either an integrable or a non-integrable observable with respect to the infinite invariant measure. This transition corresponds to very different properties of the mean energy and to a discontinuous behavior of fluctuations. While the integrable phase is described by universal statistics and the Darling-Kac law, the more challenging case is the exploration of statistical properties of non-integrable observables. Since previous experimental work showed that both α = 2 and α = 4 are attainable, we believe that both phases could also be explored experimentally.

Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

It is shown that a locally compact second countable group $G$ has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free $G$-action $T=(T_g)_{g\in G}$ on an infinite $\sigma $-finite standard measure space $

Rubio de Francia extrapolation results for grand Lebesgue spaces defined on sets having possibly infinite measure

Rubio de Francia extrapolation results for grand Lebesgue spaces defined on sets having possibly infinite measure

Individual ergodic theorems for infinite measure

Given a $\sigma $-finite infinite measure space $(\Omega ,\mu )$, it is shown that any Dun\-ford–Schwartz operator $T: \mathcal {L}^1(\Omega )\to \mathcal {L}^1(\Omega )$ can be uniquely extended to the space $\mathcal {L}^1(\Omega )+\mathcal {L}^\infty (

Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces

We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space . Each measure in the product is a shifted and scaled copy of a reference probability measure on that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.

Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Let (,) be a measurable space with -finite continuous measure, ()=. A linear operator T:L1()+L()L1()+L() is called the Dunford-Schwartz operator if ||T(f)||1||f||1 (respectively, ||T(f)||||f||) for all fL1() (respectively, fL()). {Tt}t0is a strongly continuous in L1() semigroup of Dunford-Schwartz operators, then each operator At(f)=1t∫0tTs(f)ds∈L1(Ω){{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}} has a unique extension to the Dunford-Schwartz operator, which is also denoted by At, t0. It is proved that in the completely symmetric space of measurable functions on (,) the means At converge strongly as t+ for each strongly continuous in L1() semigroup {Tt}t0 of Dunford-Schwartz operators if and only if the norm ||.||E() is order continuous.

Dynamic polymers: invariant measures and ordering by noise

We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force -- One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.