Invariant Probability Measure Research Articles

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].

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.

Random and mean Lyapunov exponents for

Abstract We 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

AbstractWe 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.

A stochastic variant of replicator dynamics in zero-sum games and its invariant measures

We study the behavior of a stochastic variant of replicator dynamics in two-agent zero-sum games. Inspired by the well studied deterministic replicator equation, this model gives arguably a more realistic description of real world settings and, as we demonstrate here, exhibits totally distinct behavior when compared to its deterministic counterpart which is known to be recurrent. In more detail, we characterize the statistics of such systems by their invariant measures which can be shown to be entirely supported on the boundary of the space of mixed strategies. Depending on the noise strength we can furthermore characterize these invariant measures by finding accumulation of mass at specific parts of the boundary. In particular, regardless of the magnitude of noise, we show that any invariant probability measure is a convex combination of Dirac measures on pure strategy profiles, which correspond to vertices/corners of the agents’ simplices. Thus, in the presence of stochastic perturbations, even in the most classic zero-sum settings, such as Matching Pennies, we observe a stark disagreement between the axiomatic prediction of Nash equilibrium and the evolutionary emergent behavior derived by an assumption of stochastically adaptive, learning agents.

An Asymptotically Optimal Two-Part Fixed-Rate Coding Scheme for Networked Control With Unbounded Noise

It is known that under fixed-rate information constraints, adaptive quantizers can be used to stabilize an open-loop-unstable linear system on Rn driven by unbounded noise. These adaptive schemes can be designed so that they have near-optimal rate, and the resulting system will be stable in the sense of having an invariant probability measure, or ergodicity, as well as boundedness of the state second moment. Although structural results and information theoretic bounds of encoders have been studied, the performance of such adaptive fixed-rate quantizers beyond stabilization has not been addressed. In this paper, we propose a two-part adaptive (fixed-rate) coding scheme that achieves state second moment convergence to the classical optimum (i.e., for the fully observed setting) under mild moment conditions on the noise process. The first part, as in prior work, leads to ergodicity (via positive Harris recurrence) and the second part ensures that the state second moment converges to the classical optimum at high rates. These results are established using an intricate analysis which uses random-time state-dependent Lyapunov stochastic drift criteria as a core tool.

Statistical solutions and Kolmogorov entropy for the lattice long-wave–short-wave resonance equations in weighted space

This article studies the lattice long-wave–short-wave resonance equations in weighted spaces. The authors first prove the global well-posedness of the initial value problem and the existence of the pullback attractor for the process generated by the solution mappings in the weighted space. Then they establish that the process possesses a family of invariant Borel probability measures supported by the pullback attractor. Afterwards, they verify that this family of Borel probability measures satisfies the Liouville theorem and is a statistical solution of the lattice long-wave–short-wave resonance equations. Finally, they prove an upper bound of the Kolmogorov entropy of the statistical solution.

Random dynamics on real and complex projective surfaces

Abstract We initiate the study of random iteration of automorphisms of real and complex projective surfaces, as well as compact Kähler surfaces, focusing on the classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.

Empirical approximation to invariant measures for McKean–Vlasov processes: Mean-field interaction vs self-interaction

This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean–Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean–Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distances to the invariant measure. Numerical simulations of the mean-field Ornstein–Uhlenbeck process are implemented to demonstrate the theoretical results.

Decay of correlations for critically intermittent systems

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs for the existence of an absolutely continuous invariant probability measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay of correlations. This contrasts other known results, where a system maintains exponential decay of correlations under stochastic perturbations of a deterministic map with exponential rate of mixing, see e.g. Baladi and Viana (1996 Ann. Sci. l’Ecole Norm. Sup. 29 483–517).

Stability of the Poincaré constant

We study stability of the sharp Poincaré constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincaré inequalities and Stein’s method. In particular, these results are applied to gamma distributions, to the Brownian motion on spheres and to the Brascamp-Lieb inequality for one-dimensional log-concave measures.

Free energy subadditivity for symmetric random Hamiltonians

We consider a random Hamiltonian H:Σ→R defined on a compact space Σ that admits a transitive action by a compact group G. When the law of H is G-invariant, we show its expected free energy relative to the unique G-invariant probability measure on Σ, which obeys a subadditivity property in the law of H itself. The bound is often tight for weak disorder and relates free energies at different temperatures when H is a Gaussian process. Many examples are discussed, including branching random walks, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum Sherrington–Kirkpatrick and Sachdev–Ye–Kitaev models.

Characters of the group [formula omitted] for a commutative Noetherian ring [formula omitted

Let R be a commutative Noetherian ring with unit. We classify the characters of the group ELd(R) provided that d is greater than the stable range of the ring R. It follows that every character of ELd(R) is induced from a finite dimensional representation. Towards our main result we classify ELd(R)-invariant probability measures on the Pontryagin dual group of Rd.