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In the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

In this paper, we prove the convexity of trace functionals (A , B , C) ↦ Tr | B p A C q | s , for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems. [ABSTRACT FROM AUTHOR]

The ergodic properties of the randomly forced Navier–Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier–Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020) and Shirikyan (J Eur Math Soc, 2020) and using the asymptotic compactness of the dynamics. [ABSTRACT FROM AUTHOR]

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler. [ABSTRACT FROM AUTHOR]

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξ g , a and bi-Laplacian fields Ξ b , a . They can be characterized as follows: for f = δ the solution u of ∇ · a ∇ u = f , a is a uniformly elliptic random environment, is the covariance of Ξ g , a . When f is the white noise, the field Ξ b , a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D ⊂ R d , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ , we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator a ¯ Δ , with constant a ¯ depending on the law of the environment a . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014). [ABSTRACT FROM AUTHOR]

We prove a finite smooth version of the entropic continuity of Lyapunov exponents proved recently by Buzzi, Crovisier, and Sarig for C ∞ surface diffeomorphisms (Buzzi et al., Invent Math 230(2):767–849, 2022). As a consequence, we show that any C r , r > 1 , smooth surface diffeomorphism f with h top (f) > 1 r lim sup n 1 n log + ‖ d f n ‖ ∞ admits a measure of maximal entropy. We also prove the C r continuity of the topological entropy at f. [ABSTRACT FROM AUTHOR]

We present new exact results for the complex generalized integral means spectrum (in the sense of Duplantier et al. (Commun Math Phys 359(3):823–868, 2018)) for two kinds of whole-plane Loewner evolutions driven by a Lévy process: The case of a Lévy process with continuous trajectories, which corresponds to Schramm–Loewner evolution SLE κ with a drift term in the Brownian driving function. There is no known result for its standard integral means spectrum, and we show that a natural path to access it goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. The case of symmetric Lévy processes for which we generalize results by Loutsenko and Yermolayeva (J Phys A Math Theor 47(16):165202, 2014, J Phys A Math Theor 52(43):435202, 2019). [ABSTRACT FROM AUTHOR]

We prove global existence for Einstein's equations with a charged scalar field for initial conditions sufficiently close to the Minkowski spacetime without matter. The proof relies on generalized wave coordinates adapted to the outgoing Schwarzschild light cones and the estimates for the massless Maxwell–Klein–Gordon system, on the background of metrics asymptotically approaching Schwarzschild at null infinity in such coordinates, by Kauffman (Global stability for charged scalar fields in an asymptotically flat metric in harmonic gauge, preprint, 2018). The generalized wave coordinates are obtained from a change of variables, introduced in Lindblad (Commun Math Phys 353(1):135–184, 2017), to asymptotically Schwarzschild coordinates at null infinity. The main technical advances are that the change of coordinates makes critical components of the metric decay faster, making the quasilinear wave operator closer to the flat wave operator, and that commuting with modified Lie derivatives preserves the geometric null structure, improving the error terms. This improved decay of the metric is essential for proving the estimates in Kauffman (2018) and will likely be useful in other contexts as well. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded open domain on the Euclidean space R n and Q + be the set of all positive rational numbers. Chen and Zeng (Cal Var Part Differ Equ 56:131, 2017) investigated the eigenvalues with higher order of the fractional Laplacian (- Δ) s Ω for s > 0 and s ∈ Q + , and they obtained a universal inequality of Yang type. In the spirit of Chen and Zeng's work, we study the eigenvalues of fractional Laplacian and establish an inequality of eigenvalues with lower order under the same condition. Also, our eigenvalue inequality is universal and generalizes the eigenvalue inequality for the poly-harmonic operators given by Jost et al. (Trans Am Math Soc 363(4):1821–1854, 2011). [ABSTRACT FROM AUTHOR]

We prove results on intrinsic flat convergence of points—a concept first explored by Sormani (Commun Anal Geom 26(6):1317–1373, 2018). In particular, we discuss compatibility with Gromov–Hausdorff convergence of points—a concept first described by Gromov (Inst Hautes Études Sci Publ Math 53:53–73, 1981). We apply these results to the problem of stability of the positive mass theorem in mathematical relativity. Specifically, we revisit the article (Huang et al. in J Reine Angew Math 727:269–299, 2017) on intrinsic flat stability for the case of graphical hypersurfaces of Euclidean space: We are able to fill in some details in the proofs of Theorems 1.4 and Lemma 5.1 of Huang et al. (2017) and strengthen some statements. Moreover, in light of an acknowledged error in the proof of Theorem 1.3 of Huang et al. (2017), we provide an alternative proof that extends recent work of Allen and Perales (Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2020. arXiv:2006.13030). [ABSTRACT FROM AUTHOR]

The higher rank Racah algebra R(n) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using the results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer Z n (s l 2) of the diagonal embedding of U (s l 2) in U (s l 2) ⊗ n . This leads to a first and novel presentation of the centralizer Z n (s l 2) in terms of generators and defining relations. An explicit formula of its Hilbert–Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey–Wilson algebra and its higher rank generalizations is discussed. [ABSTRACT FROM AUTHOR]

For a pair potential Φ in a general underlying space X satisfying some natural and sufficiently general conditions in the sense of Penrose (J Math Phys 4:1312, 1963) and Poghosyan and Ueltschi (J Math Phys 50:053509, 2009) together with a locally finite measure ϱ on X we define by means of the so-called Ursell kernel a function r which is shown to be the correlation function of a unique process G , the limiting Gibbs process for (Φ , ϱ) with empty boundary conditions. This process is exhibited as a Gibbs process in the sense of Dobrushin, Lanford and Ruelle for a class of pair potentials, which contains classical stable and hard-core potentials that are called Penrose potentials here. Particularly, a class of positive potentials is included. Finally, for some class of Penrose potentials, we show that G is the unique Gibbs process for Φ . We use the classical method of Kirkwood–Salsburg equations. A decisive role is played by a generalization of Ruelle's estimate for correlation functions. [ABSTRACT FROM AUTHOR]

We consider a system of M particles in contact with a heat reservoir of N ≫ M particles. The evolution in the system and the reservoir, together with their interaction, is modeled via the Kac's master equation. We chose the initial distribution with total energy N + M and show that if the reservoir is initially in equilibrium, that is, if the initial distribution depends only on the energy of the particle in the reservoir, then the entropy of the system decays exponentially to a very small value. We base our proof on a similar property for the Information. A similar argument allows us to greatly simplify the proof of the main result in Bonetto et al. (Commun Math Phys 363(3):847–875, 2018). [ABSTRACT FROM AUTHOR]