Invariant Probability Measure Research Articles

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

On the amenable subalgebras of group von Neumann algebras

We approach the study of sub-von Neumann algebras of the group von Neumann algebra L(Γ) for countable groups Γ from a dynamical perspective. It is shown that L(Γ) admits a maximal invariant amenable subalgebra. The notion of invariant probability measures (IRAs) on the space of subalgebras is introduced, analogous to the concept of Invariant Random Subgroups. And it is shown that amenable IRAs are supported on the maximal amenable invariant subalgebra.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Bauer simplices and the small boundary property

We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.

Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pagès and Panloup (2022) we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in Pagès and Panloup (2022) and Chen et al. (2023). As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularization properties — this is done under an ellipticity condition, using Malliavin calculus for jump processes.

Transport-entropy and functional forms of Blaschke–Santaló inequalities

We explore alternative functional or transport-entropy formulations of the Blaschke–Santaló inequality. In particular, we obtain new Blaschke–Santaló inequalities for s -concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions.

Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada α-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

Asymptotic Analysis for the Generalized Langevin Equation with Singular Potentials

We consider a system of interacting particles governed by the generalized Langevin equation (GLE) in the presence of external confining potentials, singular repulsive forces, as well as memory kernels. Using a Mori–Zwanzig approach, we represent the system by a class of Markovian dynamics. Under a general set of conditions on the nonlinearities, we study the large-time asymptotics of the multi-particle Markovian GLEs. We show that the system is always exponentially attractive toward the unique invariant Gibbs probability measure. The proof relies on a novel construction of Lyapunov functions. We then establish the validity of the small-mass approximation for the solutions by an appropriate equation on any finite-time window. Important examples of singular potentials in our results include the Lennard–Jones and Coulomb functions.

Wolbachia invasion dynamics of a random mosquito population model with imperfect maternal transmission and incomplete CI.

In this work, we formulate a random Wolbachia invasion model incorporating the effects of imperfect maternal transmission and incomplete cytoplasmic incompatibility (CI). Under constant environments, we obtain the following results: Firstly, the complete invasion equilibrium of Wolbachia does not exist, and thus the population replacement is not achievable in the case of imperfect maternal transmission; Secondly, imperfect maternal transmission or incomplete CI may obliterate bistability and backward bifurcation, which leads to the failure of Wolbachia invasion, no matter how many infected mosquitoes would be released; Thirdly, the threshold number of the infected mosquitoes to be released would increase with the decrease of the maternal transmission rate or the intensity of CI effect. In random environments, we investigate in detail the Wolbachia invasion dynamics of the random mosquito population model and establish the initial release threshold of infected mosquitoes for successful invasion of Wolbachia into the wild mosquito population. In particular, the existence and stability of invariant probability measures for the establishment and extinction of Wolbachia are determined.

Weak solution and invariant probability measure for McKean-Vlasov SDEs with integrable drifts

In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.

Constrained ergodic optimization for generic continuous functions

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function f every invariant probability measure that maximizes the space average of f must have zero entropy. We establish the analogical result in the context of constrained ergodic optimization, which is introduced by [Garibaldi and Lopes, Functions for relative maximization, Dyn. Syst. 22(4) (2007), pp. 511-528].

Dynamics of a stochastic epidemic model with information intervention and vertical transmission

<abstract><p>The dynamic behavior of a stochastic epidemic model with information intervention and vertical transmission was the concern of this paper. The threshold to judge the extinction and persistence of the disease was obtained. Specifically, when $ \Delta < 0 $ ($ \Delta $ appears in Section 3), the three classes $ I_t $, $ M_t $, and $ R_t $ appearing in the model go extinct at an exponential rate, and the susceptible class $ S_t $ almost surely converges to the solution of the boundary equation exponentially. When $ \Delta > 0 $, the result that the disease in the model is persistent in the mean and the existence of invariant probability measure are proved by constructing a new form of Lyapunov functions, which results in getting sufficient and nearly necessary conditions for different properties. Moreover, one of the main characteristics of this article was the study of the critical case of $ \Delta = 0 $ under some conditions. Some examples were listed to confirm the obtained results.</p></abstract>

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Geometric interpretation of the entropy of so c systems

We consider a geometric approach to the notion of metric entropy. We justify the possibility of this approach for the class of Borel invariant ergodic probability measures on so c systems, which is the rst result of such generality for non-Markovian systems.

Random and mean Lyapunov exponents for

AbstractWe consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ . Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

Existence and degenerate regularity of statistical solution for the 2D non-autonomous tropical climate model

In this paper, the authors investigate the probability distribution of solutions within the phase space for the non-autonomous tropical climate model in two-dimensional bounded domains. They first prove that the associated process possesses a pullback attractor and a family of invariant Borel probability measures. Then they establish that this family of invariant Borel probability measures satisfies Liouville’s theorem and is a statistical solution of the tropical climate model. Afterwards, they prove that the statistical solution possesses degenerate Lusin’s type regularity provided that the associated Grashof number is small enough.

Stochastic generalized Kolmogorov systems with small diffusion: I. Explicit approximations for invariant probability density function

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology.

Amenable wreath products with non almost finite actions of mean dimension zero

Almost finiteness was introduced in the seminal work of Kerr [J. Eur. Math. Soc. (JEMS) 22 (2020), pp. 3697–3745] as an dynamical analogue of Z \mathcal {Z} -stability in the Toms-Winter conjecture. In this article, we provide the first examples of minimal, topologically free actions of amenable groups that have mean dimension zero but are not almost finite. More precisely, we prove that there exists an infinite family of amenable wreath products that admit topologically free, minimal profinite actions on the Cantor space which fail to be almost finite. Furthermore, these actions have dynamical comparison. This intriguing new phenomenon shows that Kerr’s dynamical analogue of Toms-Winter conjecture fails for minimal, topologically free actions of amenable groups. The notion of allosteric group holds a significant position in our study. A group is allosteric if it admits a minimal action on a compact space with an invariant ergodic probability measure that is topologically free but not essentially free. We study allostery of wreath products and provide the first examples of allosteric amenable groups.

Sticky nonlinear SDEs and convergence of McKean–Vlasov equations without confinement

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

Evolution systems of probability measures for nonautonomous Klein-Gordon Itô equations on [formula omitted

Invariant probability measures of time-homogeneous transition semigroups for autonomous stochastic ODEs/PDEs/lattice systems have been widely discussed in the literature. In this work we investigate evolution systems of probability measures of time inhomogeneous transition operators for a class of nonautonomous Klein-Gordon Itô equation defined on an integer set ZN. The existence, periodicity, pointwise ergodicity and tightness of evolution systems of probability measures are established in an infinite-dimensional phase space by using a generalized Krylov-Bogolyubov method developed by Da Prato and Röckner [21]. The asymptotic stability of every periodic evolution system of probability measures with respect to the noise intensity is discussed by using a recent result on this topic established by Wang, Caraballo and Tuan [49]. The uniqueness, global exponential mixing, forward strong mixing and backward strong mixing of every evolution system of probability measures are also discussed under some globally monotone conditions. The idea of uniform tail-estimate due to Wang [41] is used to overcome several difficulties caused by the lack of compactness in infinite lattices. It seems that this is the first time to study evolution systems of probability measures for nonautonomous hyperbolic stochastic equations.