We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients. Regularity assumptions on the coefficients are formulated in terms of horizontal and vertical Dupire derivatives.

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Following the 1984 seminal work of Belavin, Polyakov and Zamolodchikov on two-dimensional conformal field theories, Toda conformal field theories were introduced in the physics literature as a family of two-dimensional conformal field theories that enjoy, in addition to conformal symmetry, an extended level of symmetry usually referred to as W-symmetry or higher-spin symmetry. More precisely Toda conformal field theories provide a natural way to associate to a finite-dimensional simple and complex Lie algebra a conformal field theory for which the algebra of symmetry contains the Virasoro algebra. In this document we use the path integral formulation of these models to provide a rigorous mathematical construction of Toda conformal field theories based on probability theory. By doing so we recover expected properties of the theory such as the Weyl anomaly formula with respect to the change of background metric by a conformal factor and the existence of Seiberg bounds for the correlation functions.

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S. Gersten announced an algorithm that takes as input two finite sequences $\overrightarrow{K}=(K_1,\cdots ,K_N)$ and $\overrightarrow{K^{\prime }}=(K_1^{\prime },\cdots ,K_N^{\prime })$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs:

• (1) YES or NO depending on whether or not there is an element $\theta \in \mathsf{Out}\left(F_n\right)$ such that $\theta \left(\overrightarrow{K}\right)={\overrightarrow{K}}^{\prime }$ together with one such $\theta$ if it exists and
• (2) a finite presentation for the subgroup of $\mathsf{Out}\left(F_n\right)$ fixing $\overrightarrow{K}$.

S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler–Vogtmann’s Outer space. New results include that the subgroup of $\mathsf{Out}\left(F_n\right)$ fixing $\overrightarrow{K}$ is of type $\mathsf{VF}$, an equivariant version of these results, an application, and a unified approach to such questions.

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We give two explicit sets of generators of the group of invertible regular functions over $\mathbf{Q}$ on the modular curve $Y^1\left(N\right)$.

The first set of generators is very surprising. It is essentially the set of defining equations of $Y^1\left(k\right)$ for $k\le N/2$ when all these modular curves are simultaneously embedded into the affine plane, and this proves a conjecture of Derickx and Van Hoeij [DvH14]. This set of generators is an elliptic divisibility sequence in the sense that it satisfies the same recurrence relation as the elliptic division polynomials.

The second set of generators is explicit in terms of classical analytic functions known as Siegel functions. This is both a generalization and a converse of a result of Yang [Yan04, Yan09].

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The second author has shown that existence of extremal Kähler metrics on semisimple principal toric fibrations is equivalent to a notion of weighted uniform K-stability, read off from the moment polytope. The purpose of this article is to prove various sufficient conditions of weighted uniform K-stability which can be checked effectively and explore the low dimensional new examples of extremal Kähler metrics it provides.

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We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.

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We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven by additive space-time white noise, and with initial data distributed according to the Gibbs measure. By introducing a suitable space-dependent renormalization, we prove local well-posedness of the renormalized equation. Bourgain’s invariant measure argument then allows us to establish almost sure global well-posedness and invariance of the Gibbs measure for the renormalized stochastic damped NLW. (ii) Similarly, we study the random data defocusing NLW (without stochastic forcing or damping), and establish the same results as in the previous setting. (iii) Lastly, we study the stochastic NLW without damping. By introducing a space-time dependent renormalization, we prove its local well-posedness with deterministic initial data in all subcritical spaces.

These results extend the corresponding recent results on the two-dimensional torus obtained by (i) Gubinelli–Koch–Oh–Tolomeo (2021), (ii) Oh–Thomann (2020), and (iii) Gubinelli–Koch–Oh (2018), to a general class of compact manifolds. The main ingredient is the Green’s function estimate for the Laplace–Beltrami operator in this setting to study regularity properties of stochastic terms appearing in each of the problems.

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It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a large class of groups, including amenable groups, right-angled Artin groups, relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold groups.

Finally, we show that Property QITB holds for every finitely presented group if a conjecture by Gromov on bounded primitives of differential forms holds as well.

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We give a new proof of a theorem of Giordano, Putnam and Skau characterizing orbit equivalence of minimal homeomorphisms of the Cantor space in terms of their sets of invariant Borel probability measures. The proof is based on a strengthening of a theorem of Krieger concerning minimal actions of certain locally finite groups of homeomorphisms, and we also give a new proof of the Giordano–Putnam–Skau characterization of orbit equivalence for these actions.

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We complete the study of characters on higher rank semisimple lattices initiated in [BH21, BBHP22], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo–Zimmer structure theorem obtained in [BH21] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis’ factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.

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On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty }(M;ℝ)$ be a smooth function called potential. It is known that for any $C>0$, there exists some anisotropic Sobolev space ${\mathscr{H}}_C$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle resonances. In this paper, we show a “Fractal Weyl law”: the density of resonances is bounded by $O\left({\langle \omega \rangle }^{\frac{n}{1+{\beta }_0}}\right)$ where $\omega =\mathrm{Im}\left(z\right)$, $n=dimM-1$ and $0<{\beta }_0\le 1$ is the Hölder exponent of the distribution $E_u\oplus E_s$ (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances and the invertibility of the transfer operator. Since the dynamical distributions $E_u,E_s$ are non smooth, we use some semi-classical analysis based on wave packet transform associated to an adapted metric $g$ on $T^{*}M$ and construct some specific anisotropic Sobolev spaces.

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Let $|·|$ be the standard Euclidean norm on ${ℝ}^n$ and let $X=({ℝ}^n,\parallel ·\parallel )$ be a normed space. A subspace $Y\subset X$ is strongly $\alpha$-Euclidean if there is a constant $t$ such that $t\left|y\right|\le \parallel y\parallel \le \alpha t\left|y\right|$ for every $y\in Y$, and say that it is strongly $\alpha$-complemented if $\parallel P_Y\parallel \le \alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\parallel P_Y\parallel$ denotes the operator norm of $P_Y$ with respect to the norm on $X$. We give an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly $(1+ϵ)$-Euclidean and strongly $(1+ϵ)$-complemented, where $ϵ>0$ is an absolute constant. This property is closely related to an old question of Vitali Milman. The example is probabilistic in nature.

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Using semi-classical analysis in ${ℝ}^n$ we present a quite general model for which the topological index formula of Atiyah–Singer predicts a spectral flow with the transition of a finite number of eigenvalues between clusters (energy bands). This model corresponds to physical phenomena that are well observed for quantum energy levels of small molecules [FZ00, FZ01], also in geophysics for the oceanic or atmospheric equatorial waves [DMV17, Mat66] and expected to be observed in plasma physics [QF22].

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We construct extensions of the pure-jump $\Lambda$-Wright–Fisher processes with frequency-dependent selection ($\Lambda$-WF with selection) with different behaviors at their boundary $1$. Those processes satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coagulation processes (EFC processes). One-to-one correspondences are established between the nature of the boundaries $1$ and $\infty$ of the processes involved. They provide new information on these two classes of processes. Sufficient conditions are provided for boundary $1$ to be an exit boundary or an entrance boundary. When the coalescence measure $\Lambda$ and the selection mechanism verify some regular variation properties, conditions are found in order that the extended $\Lambda$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In this case, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele, which can be carried by the whole population for a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

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We study the Dirac equation coupled to scalar and vector Klein–Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac–Fock equation on the space of Hilbert–Schmidt operators.

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In this paper, we establish an exponential inequality for random fields, which is applied in the context of convergence rates in the law of large numbers and weak invariance principle in some Hölder spaces.

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This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}\left(F_N\right)$, the outer automorphism group of a nonabelian free group $F_N$. We prove that, for every subgroup $H$ of $\mathrm{Out}\left(F_N\right)$, there exists an element $\varphi \in H$ such that, for every element $g$ of $F_N$, the conjugacy class $\left[g\right]$ has polynomial growth under iteration of $\varphi$ if and only if $\left[g\right]$ has polynomial growth under iteration of every element of $H$.

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We prove distality of quantifier-free relations on valued fields with finite residue field. By a result of Chernikov–Galvin–Starchenko, this yields Szemerédi–Trotter-like incidence bounds for function fields over finite fields. We deduce a version of the Elekes–Szabó theorem for such fields.

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We prove a radial source estimate in Hölder–Zygmund spaces for uniformly hyperbolic dynamics (also known as Anosov flows), in the spirit of Dyatlov–Zworski [DZ16]. The main consequence is a new linear stability estimate for the marked length spectrum rigidity conjecture, also known as the Burns–Katok [BK85] conjecture. We show in particular that in any dimension $\ge 2$, in the space of negatively-curved metrics, $C^{3+\epsilon }$-close metrics with same marked length spectrum are isometric. This improves recent works of Guillarmou–Knieper and the second author [GKL22, GL19]. As a byproduct, this approach also allows to retrieve various regularity statements known in hyperbolic dynamics and usually based on Journé’s lemma: the smooth Livšic Theorem of de La Llave–Marco–Moriyón [LMM86], the smooth Livšic cocycle theorem of Niticā–Török [NT98] for general (finite-dimensional) Lie groups, the rigidity of the regularity of the foliation obtained by Hasselblatt [Has92] and others.

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In this paper, we give estimates of the quadratic transportation cost in the conditional central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains, dynamical systems generated by intermittent maps and $\tau$-mixing sequences are given.

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We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2\times S^3$.

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We are interested in the construction of a smooth branch of travelling waves to the Nonlinear Schrödinger Equation and the Euler–Korteweg system for capillary fluids with nonzero condition at infinity. This branch is defined for speeds close to the speed of sound and looks qualitatively, after rescaling, as a rarefaction pulse described by the Kadomtsev–Petviashvili equation. The proof relies on a fixed point theorem based on the nondegeneracy of the lump solitary wave of the Kadomtsev–Petviashvili equation.

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Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence ${\left({ℰ}_n\right)}_{n\ge 2}$ of Liapunov functionals such that ${ℰ}_n$ is equivalent to the $L_n$-norm for each $n\ge 2$ and ${ℰ}_n^{1/n}$ controls the $L_{\infty }$-norm in the limit $n\to \infty$. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

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Fix a container polygon $P$ in the plane and consider the convex hull $P_n$ of $n\ge 3$ independent and uniformly distributed in $P$ random points. In the focus of this paper is the vertex number of the random polygon $P_n$. The precise variance expansion for the vertex number is determined up to the constant-order term, a result which can be considered as a second-order analogue of the classical expansion for the expectation of Rényi and Sulanke (1963). Moreover, a sharp Berry–Esseen bound is derived for the vertex number of the random polygon $P_n$, which is of the same order as one over the square-root of the variance. The latter is optimal and improves the earlier result of Bárány and Reitzner (2006) by removing the factor ${(loglogn)}^{60}$ in the planar case. The main idea behind the proof of both results is a decomposition of the boundary of the random polygon $P_n$ into random convex chains and a careful merging of the variance expansions and Berry–Esseen bounds for the vertex numbers of the individual chains. In the course of the proof, we derive similar results for the Poissonized model.

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The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time $T_{mix}$. In the $(d+1)$-dimensional setting, $d\ge 2$, this is to a large extent mathematically unexplored territory, especially for discrete interfaces. On the other hand, on the basis of a mean-curvature motion heuristics [Hen97, Spo93] and simulations (see [Des02] and the references in [Hen97, Wil04]), one expects convergence to equilibrium to occur on time-scales of order $\approx {\delta }^{-2}$ in any dimension, with $\delta \to 0$ the lattice mesh.

We study the single-flip Glauber dynamics for lozenge tilings of a finite domain of the plane, viewed as $(2+1)$-dimensional surfaces. The stationary measure is the uniform measure on admissible tilings. At equilibrium, by the limit shape theorem [CKP01], the height function concentrates as $\delta \to 0$ around a deterministic profile $\varphi$, the unique minimizer of a surface tension functional. Despite some partial mathematical results [LT15a, LT15b, Wil04], the conjecture $T_{mix}={\delta }^{-2+o\left(1\right)}$ had been proven, so far, only in the situation where $\varphi$ is an affine function [CMT12]. In this work, we prove the conjecture under the sole assumption that the limit shape $\varphi$ contains no frozen regions (facets).

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We consider elliptic diffusion processes on ${ℝ}^d$. Assuming that the drift contracts distances outside a compact set, we prove that, when the diffusion coefficient is sufficiently large, the Markov semi-group associated to the process is a contraction of the ${𝒲}_2$ Wasserstein distance, which implies a Poincaré inequality for its invariant measure. The result doesn’t require neither reversibility nor an explicit expression of the invariant measure, and the estimates have a sharp dependency on the dimension. Some variations of the arguments are then used to study, first, the stability of the invariant measure of the process with respect to its drift and, second, systems of interacting particles, yielding a criterion for dimension-free Poincaré inequalities and quantitative long-time convergence for non-linear McKean–Vlasov type processes.

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We study generalized Hermite polynomials with rectangular matrix arguments arising in multivariate statistical analysis. We argue that these are well-suited for expressing the Wiener–Itô chaos expansion of functionals of the spectral measure associated with Gaussian matrices. More specifically, we obtain the Wiener chaos expansion of Gaussian determinants of the form $det{\left(X{X}^T\right)}^{1/2}$ and prove that, in the setting where the rows of $X$ are i.i.d. Gaussian vectors, its projection coefficients admit a geometric interpretation in terms of intrinsic volumes of ellipsoids, thus extending the framework of Kabluchko and Zaporozhets (2012). Our proofs rely on a crucial relation between Hermite polynomials and Laguerre polynomials. We introduce the matrix analog of the classical Mehler’s formula for the Ornstein-Uhlenbeck semigroup and prove that matrix Hermite polynomials are eigenfunctions of these operators. We apply our results to the asymptotic study of a total variation associated with vectors of Arithmetic Random Waves on the full three-torus.

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One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the gauge transform method. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe–Sommerfeld property, that the spectrum contains a semi-axis — or indeed two semi-axes in the case of operators that are not semi-bounded.

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We consider non-self-adjoint operators in Hilbert spaces of the form $H=H_0+CWC$, where $H_0$ is self-adjoint, $W$ is bounded and $C$ is bounded and relatively compact with respect to $H_0$. We suppose that $C$ is a metric operator and that $C{(H_0-z)}^{-1}C$ is uniformly bounded in $z\in ℂ\setminus ℝ$. We define the spectral singularities of $H$ as the points of the essential spectrum $\lambda \in {\sigma }_{\mathrm{ess}}\left(H\right)$ such that $C{(H-\lambda \pm i\epsilon )}^{-1}CW$ does not have a limit as $\epsilon \to 0^{+}$. We prove that the spectral singularities of $H$ are in one-to-one correspondence with the eigenvalues, associated to resonant states, of an extension of $H$ to a larger Hilbert space. Next, we show that the asymptotically disappearing states for $H$, i.e. the vectors $\phi$ such that $e^{\pm itH}\phi \to 0$ as $t\to \infty$, coincide with the finite linear combinaisons of generalized eigenstates of $H$ corresponding to eigenvalues $\lambda \in ℂ$, $\mp Im\left(\lambda \right)>0$. Finally, we define the absolutely continuous spectral subspace of $H$ and show that it satisfies ${\mathscr{H}}_{\mathrm{ac}}\left(H\right)={\mathscr{H}}_{\mathrm{p}}{\left(H^{*}\right)}^{\perp }$, where ${\mathscr{H}}_{\mathrm{p}}\left(H^{*}\right)$ stands for the point spectral subspace of $H^{*}$. We thus obtain a direct sum decomposition of the Hilbert spaces in terms of spectral subspaces of $H$. One of the main ingredients of our proofs is a spectral resolution formula for a bounded operator $r\left(H\right)$ regularizing the identity at spectral singularities. Our results apply to Schrödinger operators with complex potentials.

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We study the asymptotic behaviour of a version of the one-dimensional Mott random walk in a regime that exhibits severe blocking. We establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, we give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it.

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In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini–Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in $\mathrm{PLS}(2,ℝ)\times ℝ$. As a by-product of our proofs, we obtain analogous limit theorems for high index random subgroups of non-uniform Fuchsian lattices with torsion.

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We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in a Euclidean space which is a finite topological extension of a translation on a torus. This generalizes the pioneer works of Rauzy and Arnoux–Ito providing geometric realizations to any unimodular Pisot substitution without any additional combinatorial condition.

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We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.

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We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing literature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to $1/2+\mathbb{N}$, as it involves nonlocal compatibility conditions.

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The Crump–Young model consists of two fully coupled stochastic processes modeling the substrate and micro-organisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of micro-organisms number. Micro-organisms are modeled though a pure jump process whose jump rates depend on the substrate concentration.

It goes to extinction almost-surely in the sense that micro-organism population vanishes. In this work, we show that, conditionally on the non-extinction, its distribution converges exponentially fast to a quasi-stationary distribution.

Due to the deterministic part, the dynamics of the Crump–Young model are highly degenerated. The proof is therefore original and consists of technically precise estimates and new approaches for quasi-stationary convergence.

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We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending the investigation of the quadratic case in one or more potentials by Leplaideur and Watbled.

We prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures.

In this non-linear thermodynamical formalism, as for mean-field theories of statistical mechanics, several kind of phase transitions appear, some of which cannot happen in the linear case. Our techniques can deal with known cases (Curie–Weiss and Potts models) as well as with new examples (metastable phase transition).

Finally, we apply some of these ideas to the classical, linear setting proving that freezing phase transitions can occur over any zero-entropy invariant compact subset of the phase space.

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We consider the null-controllability problem for the generalized Baouendi–Grushin equation $({\partial }_t-{\partial }_x^2-q{\left(x\right)}^2{\partial }_y^2)f={\mathbb{1}}_{\omega }u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q\left(x\right)=x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability.

Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-{\partial }_x^2+{\nu }^{2}q{\left(x\right)}^2$ when $Re\left(\nu \right)>0$, pseudo-differential-type operators on polynomials and Runge’s theorem for the lower bound.

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