Showing papers in "Journal de l'École polytechnique in 2022"
TL;DR: In this paper , Noyau et al. showed that the unique solution Xt(x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates.
Abstract: — Let α ∈ (0, 2) and d ∈ N. Consider the following stochastic differential equation (SDE) in Rd: dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, where L(α) is a d-dimensional rotationally invariant α-stable process, b : R+ × Rd → Rd and a : R+ × Rd → Rd ⊗ Rd are Hölder continuous functions in space, with respective order β, γ ∈ (0, 1) such that (β ∧ γ) + α > 1, uniformly in t. Here b may be unbounded. When a is bounded and uniformly elliptic, we show that the unique solution Xt(x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range α ∈ (0, 1). Our proof is based on ad hoc parametrix expansions and probabilistic techniques. Résumé (Noyau de la chaleur pour des EDS surcritiques à dérive non bornée) Soit α ∈ (0, 2) et d ∈ N. Considérons l’équation différentielle stochastique (EDS) suivante dans Rd : dXt = b(t,Xt) dt+ a(t,Xt−) dL (α) t , X0 = x, où L(α) est un processus α-stable isotrope de dimension d, b : R+×R → Rd et a : R+×R → Rd⊗Rd sont des fonctions Hölder continues en espace, d’indices respectifs β, γ ∈ (0, 1) tels que (β∧γ)+α > 1, uniformément en t. En particulier b peut être non bornée. Lorsque a est bornée et uniformément elliptique, nous montrons que la solution Xt(x) de l’EDS admet une densité continue, que l’on peut encadrer, à constante multiplicative près, par une même quantité. Nous obtenons également une borne supérieure précise pour la dérivée logarithmique de la densité. En particulier, nous traitons complètement le régime surcritique α ∈ (0, 1). Notre approche se base sur des développements parametrix ad hoc et des techniques probabilistes.
5 citations
TL;DR: In this paper , a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow was developed, and moment bounds for the associated correctors and optimal estimates on the homogenisation error were derived.
Abstract: This work develops a quantitative homogenization theory for random suspensions of rigid particles in a steady Stokes flow, and completes recent qualitative results. More precisely, we establish a large-scale regularity theory for this Stokes problem, and we prove moment bounds for the associated correctors and optimal estimates on the homogenization error; the latter further requires a quantitative ergodicity assumption on the random suspension. Compared to the corresponding quantitative homogenization theory for divergence-form linear elliptic equations, substantial difficulties arise from the analysis of the fluid incompressibility and the particle rigidity constraints. Our analysis further applies to the problem of stiff inclusions in (compressible or incompressible) linear elasticity and in electrostatics; it is also new in those cases, even in the periodic setting.
4 citations
TL;DR: In this article , it was shown that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs, and that they are superrigid for measure equivalence in the strongest possible sense.
Abstract: We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group G is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when G is nonabelian, the automorphism group of the universal cover of the Salvetti complex of G always contains infinitely generated (non-uniform) lattices.
4 citations
TL;DR: In this paper , the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisors.
Abstract: We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of 0-cycles over p-adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.
4 citations
TL;DR: In this paper , a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type was derived.
Abstract: We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.
4 citations
TL;DR: In this paper , the intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures was studied and proved for any Jordan type.
Abstract: We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on previous result from Mellit [Mel17b] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let13].
3 citations
TL;DR: In this article , it was shown that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Latt\`{e}s map.
Abstract: In this article, we show that every rational map whose multipliers all lie in a given number field is a power map, a Chebyshev map or a Latt\`{e}s map. This strengthens a conjecture by Milnor concerning rational maps with integer multipliers, which was recently proved by Ji and Xie.
3 citations
TL;DR: For graphs with non-negative curvature, this article showed that their conductance decreases logarithmically with the number of states and that their displacement is at least diffusive until the mixing time.
Abstract: We establish three remarkable consequences of non-negative curvature for sparse Markov chains. First, their conductance decreases logarithmically with the number of states. Second, their displacement is at least diffusive until the mixing time. Third, they never exhibit the cutoff phenomenon. The first result provides a nearly sharp quantitative answer to a classical question of Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for graphs with non-negative curvature. The third offers a striking counterpoint to the recently established cutoff for non-negatively curved chains with uniform expansion.
3 citations
TL;DR: In this article , the authors studied the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric, and determined the precise asymptotic behavior of the resolute operator when the resolver parameter tends to zero.
Abstract: We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibered boundary (aka ϕ-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibers are points. The new feature in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate ‘split’ pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on ϕ-manifolds.
3 citations
TL;DR: In this article , the authors define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof.
Abstract: Let A be an integral Banach ring, and X/A be a projective scheme of finite type, endowed with a semi-ample line bundle L. We define a class PSH(X,L) of plurisubharmonic metrics on L on the Berkovich analytification X^an and prove various basic properties thereof. We focus in particular on the case where A is a hybrid ring of complex power series and X/A is a smooth variety, so that X^an is the hybrid space associated to a degeneration X of complex varieties over the punctured disk. We then prove that when L is ample, any plurisubharmonic metric on L with logarithmic growth at zero admits a canonical plurisubharmonic extension to the hybrid space X^hyb . We also discuss the continuity of the family of Monge-Amp\`ere measures associated to a continuous plurisubharmonic hybrid metric. In the case where X is a degeneration of canonically polarized manifolds, we prove that the canonical psh extension is continuous on Xhyb and describe it explicitly in terms of the canonical model (in the sense of MMP) of the degeneration.
2 citations
TL;DR: In this article , it was shown that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios for Zoll contact forms.
Abstract: A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.
TL;DR: In this paper , the semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions, where the magnetic fields are sign changing with exactly two distinct values and are discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge.
Abstract: The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.
TL;DR: The first quantitative uniform in time propagation of chaos result for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion was given in this article .
Abstract: In this article, we prove the first quantitative uniform in time propagation of chaos for a class of systems of particles in singular repulsive interaction in dimension one that contains the Dyson Brownian motion. We start by establishing existence and uniqueness for the Riesz gases, before proving propagation of chaos with an original approach to the problem, namely coupling with a Cauchy sequence type argument. We also give a general argument to turn a result of weak propagation of chaos into a strong and uniform in time result using the long time behavior and some bounds on moments, in particular enabling us to get a uniform in time version of the result of C\'epa-L\'epingle.
TL;DR: In this article , the scaling limit of a discrete directed polymer in a heavy-tail environment with Lévy noise was constructed for α-stable noises with α∈(1,2).
Abstract: We present in this paper the construction of a continuum directed polymer model in an environment given by space-time Lévy noise. One of the main objectives of this construction is to describe the scaling limit of a discrete directed polymer in a heavy-tail environment and for this reason we put special emphasis on the case of α-stable noises with α∈(1,2). Our construction can be performed in arbitrary dimension, provided that the Lévy measure satisfies specific (and dimension dependent) conditions. We also discuss a few basic properties of the continuum polymer and the relation between this model and the stochastic heat equation with multiplicative Lévy noise.
TL;DR: In this paper , the authors consider the limit where the thickness of these rigid bodies tends to zero with a common rate ϵ, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves.
Abstract: We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely, we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate ϵ, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive ϵ, the bodies’ dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method.
TL;DR: In this paper , the authors considered variations of the spatial domain on which the solution of the linear heat equation is defined at each time, and investigated three main issues: (i) approximate desensitizing, (ii) approximate-desensitising combined with an exact desENSITizing for a finite-dimensional subspace, and (iii) exact desenitizing.
Abstract: . — This article is dedicated to desensitizing issues for a quadratic functional involving the solution of the linear heat equation with respect to domain variations. This work can be seen as a continuation of [28], insofar as we generalize several of the results it contains and investigate new related properties. In our framework, we consider variations of the spatial domain on which the solution of the PDE is defined at each time, and investigate three main issues: (i) approximate desensitizing, (ii) approximate desensitizing combined with an exact desensitizing for a finite-dimensional subspace, and (iii) exact desensitizing. We provide positive answers to questions (i) and (ii) and partial results to question (iii). pour
TL;DR: In this paper , the authors define a dynamical residue which generalizes the Guillemin-Wodzicki residue density of pseudo-differential operators, and prove that residues of Lorentzian spectral zeta functions are dynamical residues.
Abstract: We define a dynamical residue which generalizes the Guillemin–Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott–Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.
TL;DR: In this article , a cellular approximation for the diagonal of the Forcey-Loday realizations of the multiplihedra is defined, and a compatible topological cellular operadic bimodule structure over the Loday realization of the associahedra.
Abstract: We define a cellular approximation for the diagonal of the Forcey--Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A-infinity morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology.
TL;DR: In this paper , the authors studied the exponential stability in the H 2 norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end of the channel.
Abstract: We study the exponential stability in the H 2 norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single proportional-integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain of the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steady-state considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the input-to-state stability (ISS) of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.
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TL;DR: Toric Kato manifolds as mentioned in this paper generalize Tsuchihashi and Oda's manifold constructions to complex dimension 2 and obtain the properly blown-up Inoue surfaces.
Abstract: We introduce and study a special class of Kato manifolds, which we call toric Kato manifolds. Their construction stems from toric geometry, as their universal covers are open subsets of toric algebraic varieties of non-finite type. This generalizes previous constructions of Tsuchihashi and Oda, and in complex dimension 2, retrieves the properly blown-up Inoue surfaces. We study the topological and analytical properties of toric Kato manifolds and link certain invariants to natural combinatorial data coming from the toric construction. Moreover, we produce families of flat degenerations of any toric Kato manifold, which serve as an essential tool in computing their Hodge numbers. In the last part, we study the Hermitian geometry of Kato manifolds. We give a characterization result for the existence of locally conformally Kähler metrics on any Kato manifold. Finally, we prove that no Kato manifold carries balanced metrics and that a large class of toric Kato manifolds of complex dimension ≥3 do not support pluriclosed metrics.
TL;DR: In this paper , a noncommutative Bader-shalom factor theorem for lattices with dense projections in product groups was proved for higher rank semisimple algebraic groups.
Abstract: We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices $\Gamma
TL;DR: In this paper , it was shown that the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption.
Abstract: —We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics. Résumé (Une caractérisation du champ libre gaussien dans le continu en toute dimension) Nous montrons que, sous de faibles hypothèses de moment et de continuité, le champ libre gaussien dans le continu à d dimensions est le seul processus stochastique satisfaisant à la propriété habituelle de Markov sur le domaine et une propriété d’échelle. Notre preuve est basée sur une décomposition de l’espace fonctionnel sous-jacent en termes de processus radiaux et d’harmoniques sphériques.
TL;DR: In this article , the authors modify the definition of the completed Iwahori-Hecke algebra given in our previous article (J. Éc. Polytechnique 6 , and explain why the construction we gave earlier is not correct as such).
Abstract: . — We modify the definition of the completed Iwahori-Hecke algebra given in our previous article (J. Éc. Polytechnique 6 , and explain why the construction we gave earlier is not correct as such.
TL;DR: In this article , the authors derived a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation, a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions.
Abstract: We consider a damped plate equation on an open bounded subset of ℝ d , or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskiĭ-Šapiro condition. The damping term acts on an internal region without imposing any geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first through microlocal estimates, then local estimates, and finally a global estimate.
TL;DR: In this article , a probabilistic reinforcement learning model for ants searching for the shortest path(s) from their nest to a food source, represented here by two vertices of a finite graph.
Abstract: In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) from their nest to a food source, represented here by two vertices of a finite graph. In this model, the ants each take a random walk on the graph, starting from the nest vertex, until they reach the food vertex, and then reinforce the weight of the set of crossed edges. We show that if the graph is a finite tree, in which the set of leaves is identified with a single food vertex, and the root with the nest vertex, and if there is an edge between the nest and the food, then almost surely the proportion of ants that end up taking this last edge tends to 1. On the other hand we show on three other examples that in general ants do not always choose the shortest path. Our techniques use stochastic approximation methods, as well as couplings with urn processes.
TL;DR: In this paper , a lower estimate of the growth function of any sub-semi-group is given for the periodic quotient of a torsion-free hyperbolic group.
Abstract: Given a periodic quotient of a torsion-free hyperbolic group, we provide a fine lower estimate of the growth function of any sub-semi-group. This generalizes results of Razborov and Safin for free groups.
TL;DR: In this article , the authors present new families of examples of analogous phenomena when counting prime ideals in number fields of higher degree where the bias takes place for all large enough x, and their proofs are unconditional.
Abstract: Chebyshev’s bias is the phenomenon according to which for most x, the interval [2,x] contains more primes congruent to 3 modulo 4 than primes congruent to 1 modulo 4. We present new families of examples of analogous phenomena when counting prime ideals in number fields of higher degree where the bias takes place for all large enough x. Our proofs are unconditional.
TL;DR: In this article , a compact connected manifold (X,g) of negative curvature and a family of semi-classical Lagrangian states f h (x)=a(x)e iϕ(x)/h on X was considered, and it was shown that f h , when evolved by the semiclassical Schrödinger equation during a long time, resembles a random Gaussian field.
Abstract: In this paper, we consider a compact connected manifold (X,g) of negative curvature, and a family of semi-classical Lagrangian states f h (x)=a(x)e iϕ(x)/h on X. For a wide family of phases ϕ, we show that f h , when evolved by the semi-classical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry’s random waves conjecture for Lagrangian states.