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Kloeckner discovered that the quadratic Wasserstein space over the real line (denoted by W 2 (R)) is quite peculiar, as its isometry group contains an exotic isometry flow. His result implies that it can happen that an isometry Φ fixes all Dirac measures, but still, Φ is not the identity of W 2 (R). This is the only known example of this surprising and counterintuitive phenomenon. Kloeckner also proved that the image of each finitely supported measure under these isometries (and thus under all isometry) is a finitely supported measure. Recently we showed that the exotic isometry flow can be represented as a unitary group on L 2 ((0 , 1)). In this paper, we calculate the generator of this group, and we show that every exotic isometry (and thus every isometry) maps the set of all absolutely continuous measures belonging to W 2 (R) onto itself. [ABSTRACT FROM AUTHOR]

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov's one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call "dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the "big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to consider a discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation in the case that the observable measurements, obtained discretely in time, may be contaminated by systematic errors, which works for a general class of observable measurements, such as low Fourier modes and local spatial averages over finite volume elements. We will provide some suitable conditions to establish asymptotic in time estimates of the difference between the approximating solution and the unknown exact (reference) solution in some appropriate norms for these two different kinds of interpolation operators, which also shows that the approximation solution of the proposed discrete data assimilation algorithm will convergent to the unique unknown exact (reference) solution of the original system at an exponential rate, asymptotically in time if the observational measurements are free of error. [ABSTRACT FROM AUTHOR]

Let (X, G) be a G-system which means that X is a compact metric space and G is an amenable group continuously acting on X. In this paper, we introduce the notions of density-equicontinuity with respect to a given Følner sequence and Banach density-equicontinuity for amenable group actions and we give some relations among Banach mean equicontinuity, Banach density-t-equicontinuity and Banach density-equicontinuity. Moreover, we introduce the concept of density n-sensitive tuple with respect to a given Følner sequence for amenable group actions and we prove that every topological sequence entropy n-tuple is a density n-sensitive tuple with respect to each tempered Følner sequence for an abelian group action which admits an ergodic measure with full support. For an invariant measure μ of (X, G), we introduce the concept of μ -density n-sensitive tuple with respect to a given Følner sequence and we show that if μ is ergodic and G is abelian, then every μ -sequence entropy n-tuple is a μ -density n-sensitive tuple with respect to any given tempered Følner sequence of G. [ABSTRACT FROM AUTHOR]

Covariate shift in regression problems and the associated distribution mismatch between training and test data is a commonly encountered phenomenon in machine learning. In this paper, we extend recent results on nonparametric convergence rates for i.i.d. data to Markovian dependence structures. We demonstrate that under Hölder smoothness assumptions on the regression function, convergence rates for the generalization risk of a Nadaraya–Watson kernel estimator are determined by the similarity between the invariant distributions associated to source and target Markov chains. The similarity is explicitly captured in terms of a bandwidth-dependent similarity measure recently introduced in Pathak, Ma and Wainwright [ ICML , 2022]. Precise convergence rates are derived for the particular cases of finite Markov chains and spectral gap Markov chains for which the similarity measure between their invariant distributions grows polynomially with decreasing bandwidth. For the latter, we extend the notion of a distribution transfer exponent from Kpotufe and Martinet [ Ann. Stat. , 49(6), 2021] to kernel transfer exponents of uniformly ergodic Markov chains in order to generate a rich class of Markov kernel pairs for which convergence guarantees for the covariate shift problem can be formulated. [ABSTRACT FROM AUTHOR]

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic Φ 3 4 -model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ 3 4 -model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018). The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates. The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate. [ABSTRACT FROM AUTHOR]

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group G has only been investigated when G is a cyclic group; an alternating group; a simple factor of a Weyl group of type E 6 , E 7 , or E 8 ; or PSL (2 , F 7). In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) RD k ≤ d -versality, which we connect to the existence of "special points" on varieties. [ABSTRACT FROM AUTHOR]

We introduce a notion of a point-wise entropy of measures (i.e. local entropy) called neutralized local entropy , and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension. [ABSTRACT FROM AUTHOR]

The axiomatic theory of ordinary differential equations, owing to its simplicity, can provide a useful framework to describe various generalizations of dynamical systems. In this study, we consider how dynamical properties can be generalized to this setting. First, we study the generalizations of dynamical properties, such as invariance and limit sets and show that compact weakly invariant sets can be characterized in terms of the corresponding shift-invariant sets in the function space. This result enables us to consider weakly invariant sets in terms of shift-invariant sets. Next, we compare the axiomatic theory of ODE with other formalisms of generalized dynamical systems, namely generalized semiflow and multivalued semigroup. Finally, we introduce the notion of invariant measures and show that we can generalize classical results, such as Poincaré recurrence theorem. [ABSTRACT FROM AUTHOR]

We consider fertile models with hard interactions, four states, and an activity parameter on a Cayley tree. We show that there are three types of such models: "stick," "key," and "generalized key." For the "generalized key" model on a Cayley tree of order , the uniqueness of the translation-invariant Gibbs measure is proved, and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones are found. Moreover, in the case of a fertile graph of the "stick" type, the translation invariance of double-periodic Gibbs measures on a Cayley tree of orders is shown and conditions for the existence of double-periodic Gibbs measures other than the translation-invariant ones on a Cayley tree of order are found. [ABSTRACT FROM AUTHOR]

Background: Daydreaming may contribute to the maintenance of grandiose delusions. Repeated, pleasant and vivid daydreams about the content of grandiose delusions may keep the ideas in mind, elaborate the details, and increase the degree of conviction in the delusion. Pleasant daydreams more generally could contribute to elevated mood, which may influence the delusion content. Aims: We sought to develop a brief questionnaire, suitable for research and clinical practice, to assess daydreaming and test potential associations with grandiosity. Method: 798 patients with psychosis (375 with grandiose delusions) and 4518 non-clinical adults (1788 with high grandiosity) were recruited. Participants completed a daydreaming item pool and measures of grandiosity, time spent thinking about the grandiose belief, and grandiose belief conviction. Factor analysis was used to derive the Qualities of Daydreaming Scale (QuOD) and associations were tested using pairwise correlations and structural equation modelling. Results: The questionnaire had three factors: realism, pleasantness, and frequency of daydreams. The measure was invariant across clinical and non-clinical groups. Internal consistency was good (alpha-ordinals: realism=0.86, pleasantness=0.93, frequency=0.82) as was test–retest reliability (intra-class coefficient=0.75). Daydreaming scores were higher in patients with grandiose delusions than in patients without grandiose delusions or in the non-clinical group. Daydreaming was significantly associated with grandiosity, time spent thinking about the grandiose delusion, and grandiose delusion conviction, explaining 19.1, 7.7 and 5.2% of the variance in the clinical group data, respectively. Similar associations were found in the non-clinical group. Conclusions: The process of daydreaming may be one target in psychological interventions for grandiose delusions. [ABSTRACT FROM AUTHOR]

Similar to (asymptotic) average shadowing but regardless of the order of points in orbits, the notion of (asymptotic) quasi-shadowing is introduced and studied with the aid of mean orbital pseudo-metric. The asymptotic quasi-shadowing property is stronger than the quasi-shadowing property. The quasi-shadowing property with surjection implies chain transitivity. Each uniquely ergodic system has the asymptotic quasi-shadowing property. For systems with the asymptotic quasi-shadowing property, every invariant measure has generic points. Additionally, we give an equivalent characterization of the existence of generic points for every invariant measure. [ABSTRACT FROM AUTHOR]

We consider the (rough) stochastic differential equations driven by a linear multiplicative fractional Brownian motion with Hurst index H ∈ (1 2 , 1) (or a linear multiplicative geometric fractional Brownian rough path with Hurst index H ∈ (1 3 , 1 2 ]), we construct the pullback random attractors for these two type noises via the transformation method. Furthermore, we obtain the invariant probability measures, which are supported on the pullback random attractors. Finally, we obtain the limit behavior of the invariant probability measures. [ABSTRACT FROM AUTHOR]

We investigate the convergence, in the small mass limit, of the stationary solutions of a class of stochastic damped wave equations, where the friction coefficient depends on the state and the noisy perturbation is of multiplicative type. We show that the Smoluchowski–Kramers approximation that has been previously shown to be true in any fixed time interval, is still valid in the long time regime. Namely, we prove that the first marginals of any sequence of stationary solutions for the damped wave equation converge to the unique invariant measure of the limiting stochastic quasilinear parabolic equation. The convergence is proved with respect to the Wasserstein distance associated with the H - 1 norm. [ABSTRACT FROM AUTHOR]

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [ 0 , T ] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance. [ABSTRACT FROM AUTHOR]

We study the invariant measure of the transport correlator for a chiral Hamiltonian and analyze its properties. The Jacobian of the invariant measure is a function of random phases. Then we distinguish the invariant measure before and after the phase integration. In the former case we found quantum diffusion of fermions and a uniform zero mode that is associated with particle conservation. After the phase integration the transport correlator reveals two types of evolution processes, namely classical diffusion and back-folded random walks. Which one dominates the other depends on the details of the underlying chiral Hamiltonian and may lead either to classical diffusion or to the suppression of diffusion. [ABSTRACT FROM AUTHOR]

This work focuses on the convergence of the numerical invariant measure for a stochastic age-dependent population–toxicant model with Markov switching. Considering that Euler–Maruyama (EM) has the advantage of fast computation and low cost, explicit EM was used to discretize the time variable. With the help of the p-th moment boundedness of the analytical and numerical solutions of the model, the existence and uniqueness of the corresponding invariant measures were obtained. Under suitable assumptions, the conclusion that the numerical invariant measure converges to the invariant measure of the analytic solution was proven by defining the Wasserstein distance. A numerical simulation was performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological one induced by a cancerous tumor. A mean-field approximation for a stochastic model of angiogenesis consists of a partial differential equation (PDE) for the density of active vessel tips. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite-dimensional Lévy process and is the basis of a statistical theory of angiogenesis. The associated functional equation has been solved and the invariant measure obtained. The results of this theory are compared to direct numerical simulations of the underlying angiogenesis model. The invariant measure and the moments are functions of a Korteweg–de Vries-like soliton, which approximates the deterministic density of active vessel tips. [ABSTRACT FROM AUTHOR]

We prove a non-extinction result for Fleming–Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times. [ABSTRACT FROM AUTHOR]

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification. [ABSTRACT FROM AUTHOR]

According to the recurrent frequency, many levels of recurrent points are found such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an uncountable distributionally chaotic subset. For a wide range of dynamical systems such as expansive systems with the shadowing property, we also show the distributional chaos of the points that are recurrent but not Banach recurrent. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotics of the ODE's flow induced by a regular vector field b on the d-dimensional torus R d / Z d . First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of R 2 . Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field b = a ζ , where ζ is a constant vector in R 2 . Furthermore, several extensions of the Franks-Misiurewicz theorem are obtained in the two-dimensional ODE's context. On the one hand, we provide some interesting stability properties in the case where the Herman rotation set has a commensurable direction. On the other hand, we present new results highlighting the exceptional character of the opposite case, i.e. when the Herman rotation set is a closed line segment with 0 R 2 at one end and with an irrational slope, if it is not reduced to a single point. Besides this, given a pair (μ , ν) of invariant probability measures for the flow, we establish new Fourier relations between the determinant det (μ b ^ (j) , ν b ^ (k)) and the determinant det (j , k) for any pair (j, k) of non null integer vectors, which can be regarded as an extension of the Franks-Misiurewicz theorem. Next, in contrast with dimension two, any three-dimensional closed convex polyhedron with rational vertices is shown to be the rotation set associated with a suitable vector field b. Finally, in the case of an invariant measure μ with a regular density and a non null mass μ (b) with respect to b, we show that the homogenization of the two-dimensional transport equation with the oscillating velocity b (x / ε) as ε tends to 0, leads us to a nonlocal limit transport equation, but with the effective constant velocity μ (b) . [ABSTRACT FROM AUTHOR]

In this paper, stochastic FitzHugh–Nagumo lattice system with nonlinear noise in weighted spaces is considered. Firstly, the well-posedness of solution of such system in a weighted space L 2 (Ω , l σ 2 × l σ 2) is established, based on which we further prove the existence and uniqueness of weak pullback mean random attractor in the weighted space. Then the existence and uniqueness of invariant measure are proved in the weighted space l σ 2 × l σ 2 as well as exponentially mixing property in the sense of Wasserstein metric. Moreover, the limit behaviors of invariant measure in the weighted space l σ 2 × l σ 2 are also investigated with respect to noise intensity. [ABSTRACT FROM AUTHOR]

This work is concerned with the effective dynamics for the stochastic reaction diffusion equations with cubic nonlinearity driven by a multiplicative noise. By splitting the solution into the finite dimension kernel space and its complement space with some appropriate multi-scale, it derives the dominant solution and the effective invariant measure in the sense of the Wasserstein distance, which capture the complex dynamical behaviors of the original system as a singular parameter is enough small. Furthermore, the effective invariant measure is decomposed to a product of two invariant measures. [ABSTRACT FROM AUTHOR]

We study the homogenization of the equation A(\cdot\varepsilon): D2u\varepsilon =f posed in a bounded convex domain Ω\subset Rn subject to a Dirichlet boundary condition and the numerical approximation of the corresponding homogenized problem, where the measurable, uniformly elliptic, periodic, and symmetric diffusion matrix A is merely assumed to be essentially bounded and (if n > 2) to satisfy the Cordes condition. In the first part, we show existence and uniqueness of an invariant measure by reducing to a Lax--Milgram-type problem, we obtain L2-bounds for periodic problems in doubledivergence-form, we prove homogenization under minimal regularity assumptions, and we generalize known corrector bounds and results on optimal convergence rates from the classical case of H\"older continuous coefficients to the present case. In the second part, we suggest and rigorously analyze an approximation scheme for the effective coefficient matrix and the solution to the homogenized problem based on a finite element method for the approximation of the invariant measure, and we demonstrate the performance of the scheme through numerical experiments. [ABSTRACT FROM AUTHOR]

We study the equilibrium temperature distribution in a model for strongly magnetized plasmas in dimensions two and three. Provided the magnetic field is sufficiently structured (integrable in the sense that it is fibered by co-dimension one invariant tori, on most of which the field lines ergodically wander) and the effective thermal diffusivity transverse to the tori is small, it is proved that the temperature distribution is well approximated by a function that only varies across the invariant surfaces. The same result holds for "nearly integrable" magnetic fields up to a "critical" size. In this case, a volume of non-integrability is defined in terms of the temperature defect distribution and is related to the non-integrable structure of the magnetic field, confirming a physical conjecture of Paul et al (J Plasma Phys 88(1):905880107, 2022). Our proof crucially uses a certain quantitative ergodicity condition for the magnetic field lines on a full measure set of invariant tori, which is automatic in two dimensions for magnetic fields without null points and, in higher dimensions, is guaranteed by a Diophantine condition on the rotational transform of the magnetic field. [ABSTRACT FROM AUTHOR]

In this paper, we study discrete spectrum of invariant measures for group actions. We show that an invariant measure has discrete spectrum if and only if it has finite max-mean-measure-complexity. As an application, we show that for countable discrete amenable group actions an invariant measure has discrete spectrum if and only if it has bounded mean-measure-complexity along any Følner sequence. [ABSTRACT FROM AUTHOR]

In this paper, we studied the dynamic properties of predator-prey and scavenger three species system by using ergodic invariant measures. Pengyu Ma. find the five points of dynamical bifurcation of the stochastic model, which happened between extinction and survival of each species. Environmental noise was added and proved by the fact that driving force produced by environmental noise influence the system and it was find that system may extinct or partially extinct. Here, we have analysed the stochastic bifurcation phenomena of the prey-predator with scavenger system from the nature of dynamic bifurcation. The phase plots and time diagram plotted for the different values of parameters. We have verified all the results by numerical simulations. [ABSTRACT FROM AUTHOR]

We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain's invariant measure argument is not applicable to this problem. In order to overcome this difficulty, we implement a variant of Bourgain's argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance 1+t is an evolution system of measures for SKdV with the white noise initial data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α -model, convective Brinkman-Forchheimer equations, fractional (s , p) -Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others. [ABSTRACT FROM AUTHOR]

3D Burgers equation is an important model for turbulence. It is natural to expect the long-time behaviour for this hydrodynamics equation. However, there is no result about the long-time behaviour for this deterministic model. Surprisingly, if the system is perturbed by stochastic noise, we establish the existence and uniqueness of invariant measure for 3D stochastic Burgers equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model for chaotic diffusion with amplification on graphs associated with piecewise-linear maps of the interval [S. Lepri, Chaotic fluctuations in graphs with amplification, Chaos, Solitons & Fractals 139 (2020) 110003]. We determine the conditions for having fat-tailed invariant measures by considering approximate solution of the Perron–Frobenius equation for generic graphs. An analogy with the statistical mechanics of a directed polymer is presented that allows for a physically appealing interpretation of the statistical regimes. The connection between non-Gaussian statistics and the generalized Lyapunov exponents L (q) is illustrated. Finally, some results concerning large graphs are reported. [ABSTRACT FROM AUTHOR]

We consider a stochastic nonlinear defocusing Schrödinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear noise in the Itô form. We work at the same time on compact Riemannian manifolds without boundary and on relatively compact smooth domains with either the Dirichlet or the Neumann boundary conditions, always in dimension two. We construct a martingale solution using a modified Faedo–Galerkin's method, following Brzeźniak et al (2019 Probab. Theory Relat. Fields 174 1273–338). Then by means of the Strichartz estimates deduced from Blair et al (2008 Proc. Am. Math. Soc. 136 247–56) but modified for our stochastic setting we show the pathwise uniqueness of solutions. Finally, we prove the existence of an invariant measure by means of a version of the Krylov–Bogoliubov method, which involves the weak topology, as proposed by Maslowski and Seidler (1999 Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10 69–78). This is the first result of this type for stochastic nonlinear Schrödinger equation (NLS) on compact Riemannian manifolds without boundary and on relatively compact smooth domains even for an additive noise. Some remarks on the uniqueness in a particular case are provided as well. [ABSTRACT FROM AUTHOR]

This research aimed to provide new psychometric evidence of the Argentine Beck Anxiety Inventory (BAI) version in a sample of adults from Argentina. More specifically, the objectives were: 1) test the factor structure of the BAI; 2) assess gender, education, age, and region invariance; 3) test the latent factor mean difference across gender, education, age, and region; 4) analyze the reliability of the BAI; 5) provide population-based norms. The sample was composed of 1,410 adults ranging between 18 and 65 years (M = 32.41; SD = 10.45). Results showed that the BAI should be interpreted as unidimensional. Full configural, metric, and scalar invariance across gender, education, age, and region were obtained for the unidimensional model. The BAI presented adequate reliability values. Latent mean comparisons showed that men experienced more anxiety than women, that college-educated experienced more anxiety than non-college, that older people experienced more anxiety than younger people, and that people living in Greater Buenos Aires experienced more anxiety than those from Buenos Aires City. The BAI is an invariant measure of anxiety symptoms with good psychometric properties. [ABSTRACT FROM AUTHOR]

Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows rigorous estimation of the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection. [ABSTRACT FROM AUTHOR]

We consider Kolmogorov operators with constant diffusion matrices and linear drifts, i.e., Ornstein–Uhlenbeck operators, and show that all solutions to the corresponding stationary Fokker–Planck–Kolmogorov equations (including signed solutions) are invariant measures for the generated semigroups. This also gives a relatively explicit description of all solutions. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with limiting behaviors of invariant measures for neural field lattice models in a random environment. First of all, we consider the convergence relation of invariant measures between the stochastic neural field lattice model and the corresponding deterministic model in weighted spaces, and prove any limit of a sequence of invariant measures of such a lattice model must be an invariant measure of its limiting system as the noise intensity tends to zero. Then we are devoted to studying the numerical approximation of invariant measures of such a stochastic neural lattice model. To this end, we first consider convergence of invariant measures between such a neural lattice model and the system with neurons only interacting with its n-neighborhood; then we further prove the convergence relation of invariant measures between the system with an n-neighborhood and its finite dimensional truncated system. By this procedure, the invariant measure of the stochastic neural lattice models can be approximated by the numerical invariant measure of a finite dimensional truncated system based on the backward Euler--Maruyama (BEM) scheme. Therefore, the invariant measure of a deterministic neural field lattice model can be observed by the invariant measure of the BEM scheme when the noise is not negligible. [ABSTRACT FROM AUTHOR]

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the 2-dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author [Comm. Math. Phys. 377 (2020), pp. 1311–1347] to a situation where the invariant measure is not explicitly known. [ABSTRACT FROM AUTHOR]

We address the problem of existence and uniqueness of solutions (c,u(.)) to ergodic Hamilton--Jacobi--Bellman (HJB) equations of the form H(x,∇ u(x), D2u(x))=c in the whole space Rm with unbounded and merely measurable data and where H is a Bellman Hamiltonian. The method we use is different from classical approaches. It relies on duality theory and optimization in abstract Banach spaces together with maximal dissipativity of the diffusion operator. [ABSTRACT FROM AUTHOR]

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 (λ N − γ) in space and O (τ γ) in time, where γ ∈ (0 , 1 ] from the assumption ∥ A γ − 1 2 Q 1 2 ∥ L 2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations are established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that Hγ (Ϭ) is compactly embedded in L²(Ϭ) with γ γ (0, 1). Moreover, the uniqueness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of stochastic PDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations. [ABSTRACT FROM AUTHOR]

We study in this article a stochastic version of a coupled Allen-Cahn-Navier-Stokes model in a two-dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter, both endowed with suitable boundary conditions. We prove the existence of solutions via a semigroup approach. We also obtain the existence and uniqueness of an invariant measure via coupling methods. [ABSTRACT FROM AUTHOR]

We consider the one dimensional boundary driven harmonic model and its continuous version, both introduced in (Frassek et al. in J Stat Phys 180: 135–171, 2020). By combining duality and integrability the authors of (Frassek and Giardiná in J Math Phys 63: 103301, 2022) obtained the invariant measures in a combinatorial representation. Here we give an integral representation of the invariant measures which turns out to be a convex combination of inhomogeneous product of geometric distributions for the discrete model and a convex combination of inhomogeneous product of exponential distributions for the continuous one. The mean values of the geometric and of the exponential variables are distributed according to the order statistics of i.i.d. uniform random variables on a suitable interval fixed by the boundary sources. The result is obtained solving exactly the stationary condition written in terms of the joint generating function. The method has an interest in itself and can be generalized to study other models. We briefly discuss some applications. [ABSTRACT FROM AUTHOR]

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic p -Laplace equations and, more generally, singular stochastic Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir et al. (2012) [22] , and improve the results by Liu and Tölle (2011) [41]. [ABSTRACT FROM AUTHOR]

We analyze plankton-nutrient food chain models composed of phytoplankton, herbivorous zooplankton and a limiting nutrient. These models have played a key role in understanding the dynamics of plankton in the oceanic layer. Given the strong environmental and seasonal fluctuations that are present in the oceanic layer, we propose a stochastic model for which we are able to fully classify the longterm behavior of the dynamics. In order to achieve this we had to develop new analytical techniques, as the system does not satisfy the regular dissipativity conditions and the analysis is more subtle than in other population dynamics models. [ABSTRACT FROM AUTHOR]

We study the well-posedness of the initial value problem and the long-time behavior of almost periodic solutions to stochastic scalar conservation laws in any space dimension, under the assumption of Lipschitz continuity of the flux functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions. [ABSTRACT FROM AUTHOR]

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]