Quasi-invariant Measures Research Articles

Quasi-invariant measures for generalized approximately proper equivalence relations

We introduce a generalization of the notion of approximately proper equivalence relations studied by Renault and with it, we build an étale groupoid. Choosing a suitable set of continuous functions to play the role of a potential, we construct a cocycle in that groupoid and discuss the corresponding Radon-Nikodym problem.

Almost Sure Scattering for the One Dimensional Nonlinear Schrödinger Equation

We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree p > 1 p>1 . On compact manifolds many probability measures are invariant by the flow of the linear Schrödinger equation (e.g. Wiener measures), and it is sometimes possible to modify them suitably and get invariant (Gibbs measures) or quasi-invariant measures for the non linear problem. On R d \mathbb {R}^d , the large time dispersion shows that the only invariant measure is the δ \delta measure on the trivial solution u = 0 u =0 , and the good notion to track is whether the non linear evolution of the initial measure is well described by the linear (nontrivial) evolution. This is precisely what we achieve in this work. We exhibit measures on the space of initial data for which we describe the nontrivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. Actually, we give precise (and optimal) bounds on the Radon–Nikodym derivatives of these measures with respect to each other and we characterise their L p L^p regularity. We deduce from this precise description the global well-posedness of the equation for p > 1 p>1 and scattering for p > 3 p>3 (actually even for 1 > p ≤ 3 1>p \leq 3 , we get a dispersive property of the solutions and exhibit an almost sure polynomial decay in time of their L p + 1 L^{p+1} norm). To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties or decay) for the nonlinear evolution.

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

Quasi-invariant measures on topological groups and ω-powers

Abstract Under GCH, there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card ⁢ ( G ) {{\rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G. Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group 𝐒 1 {{\bf S}_{1}} .

Quasi-Invariant and Invariant Functionals and Measures on Systems of Topological Loops and Quasigroups

Quasi-Invariant and Invariant Functionals and Measures on Systems of Topological Loops and Quasigroups

A Mixing Property for the Action of SL(3,ℤ) × SL(3,ℤ) on the Stone–Čech Boundary of SL(3,ℤ)

Abstract By analogy with the construction of the Furstenberg boundary, the Stone–Čech boundary of $\textrm {SL}(3,\mathbb {Z})$ is a fibered space over products of projective matrices. The proximal behaviour on this space is exploited to show that the preimages of certain sequences have accumulation points that belong to specific regions, defined in terms of flags. We show that the $\textrm {SL}(3,\mathbb {Z})\times \textrm {SL}(3,\mathbb {Z})$-quasi-invariant Radon measures supported on these regions are tempered. Thus, every quasi-invariant Radon boundary measure for $\textrm {SL}(3,\mathbb {Z})$ is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal $c^{\prime}_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$, slightly larger than $c_0 ((\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$. Hence, the left–right representation of $C^*(\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ in the Calkin algebra of $\textrm {SL}(3,\mathbb {Z})$ factors through $C^*_{c^{\prime}_0} (\textrm {SL}(3,\mathbb {Z}) \times \textrm {SL}(3,\mathbb {Z}))$ and the centralizer of every infinite subgroup of $\textrm {SL}(3,\mathbb {Z})$ is amenable.

Strongly recurrent transformation groups

We define the notions of strong and strict recurrency for actions of countable ordered groups on σ-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

Invariant Measures for Horospherical Actions and Anosov Groups

AbstractLet $\Gamma $ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to ${\mathbb {R}}^{\text {rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup that normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures.

Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.

Integral operators for nonlocally compact group modules

The article is devoted to integral operators for nonlocally compact group modules. Operator valued functions for nonlocally compact groups and their normed spaces of different types are investigated. For this purpose quasi-invariant measures on nonlocally compact groups are used. Estimates of norms of integral operators are given. Moreover, convolutions of functions having operator values and values in Banach spaces are scrutinized. Examples are given.

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation in negative Sobolev spaces

We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [29,26] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [34] with the normal form approach in [29,26].

On a Non-Periodic Modified Euler Equation: Well-Posedness and Quasi-Invariant Measures

We consider a modified Euler equation on $\mathbb {R}^{2}$ . We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be globally defined.

Quasi-invariant Gaussian measures for the two-dimensional defocusing cubic nonlinear wave equation

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this by introducing a simultaneous renormalization of the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third-order dispersion

In this paper, we consider the cubic nonlinear Schrödinger equation with third-order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces Hs(T), s>34, are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.

Symmetries of the KMS Simplex

A continuous groupoid homomorphism $c$ on a locally compact second countable Hausdorff \'etale groupoid $\mathcal{G}$ gives rise to a $C^{*}$-dynamical system in which every $\beta$-KMS state can be associated to a $e^{-\beta c}$-quasi-invariant measure $\mu$ on $\mathcal{G}^{(0)}$. Letting $\Delta_{\mu}$ denote the set of KMS states associated to such a $\mu$, we will prove that $\Delta_{\mu}$ is a simplex for a large class of groupoids, and we will show that there is an abelian group that acts transitively and freely on the extremal points of $\Delta_{\mu}$. This group can be described using the support of $\mu$, so our theory of symmetries can be used to obtain a description of all KMS states by describing the $e^{-\beta c}$-quasi-invariant measures. To illustrate this we will describe the KMS states for the Cuntz-Krieger algebras of all finite higher rank graphs without sources and a large class of continuous one-parameter groups.

Quasi-invariant measures for some amenable groups acting on the line

In this note we show that if $G$ is a solvable group acting on the line, and if there is $T\in G$ having no fixed points, then there is a Radon measure $\mu$ on the line quasi-invariant under $G$. In fact, our method allows for the same conclusion for $G$ inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.

Current algebra, statistical mechanics and quantum models

Results obtained in the past for free boson systems at zero and nonzero temperatures are revisited to clarify the physical meaning of current algebra reducible functionals which are associated to systems with density fluctuations, leading to observable effects on phase transitions.To use current algebra as a tool for the formulation of quantum statistical mechanics amounts to the construction of unitary representations of diffeomorphism groups. Two mathematical equivalent procedures exist for this purpose. One searches for quasi-invariant measures on configuration spaces, the other for a cyclic vector in Hilbert space. Here, one argues that the second approach is closer to the physical intuition when modelling complex systems. An example of application of the current algebra methodology to the pairing phenomenon in two-dimensional fermion systems is discussed.

On some methods of extending invariant and quasi-invariant measures

In the present paper an approach to some questions in the theory of invariant (quasi-invariant) measures is discussed. It is useful in certain situations, where given topological groups or topological vector spaces are equipped with various nonzero σ-finite left invariant (left quasi-invariant) measures.

Equidistribution, ergodicity and irreducibility in CAT(–1) spaces

We prove an equidistribution theorem à la Bader–Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(–1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a von Neumann’s mean ergodic theorem for quasi-invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.

The size of the last merger and time reversal in $\Lambda$-coalescents

Nous considérons le nombre de blocs impliqués dans le dernier regroupement d’un $\Lambda$-coalescent issu de $n$ blocs. Nous donnons des conditions sous lesquelles, quand $n$ tend vers l’infini, la suite de variables aléatoires (a) est tendue (b) converge en loi vers une variable aléatoire finie ou (c) converge vers l’infini en probabilité. Nos conditions sont optimales pour les $\Lambda$-coalescents qui ont une composante de poussière. Pour un $\Lambda$ général, nous associons ces trois cas à l’existence, l’unicité et la non-existence d’une mesure invariante pour la dynamique du processus de comptage des blocs. Dans le cas (b), nous étudions le retourné en temps du processus de comptage des blocs depuis de le temps de dernier regroupement.