Journal de l’École polytechnique — Mathématiques最新文献_第3页

Heat kernel of supercritical nonlocal operators with unbounded drifts 具有无界漂移的超临界非局部算子的热核

Journal de l’École polytechnique — Mathématiques Pub Date : 2022-03-03 DOI: 10.5802/jep.189

S. Menozzi, Xicheng Zhang

{"title":"Heat kernel of supercritical nonlocal operators with unbounded drifts","authors":"S. Menozzi, Xicheng Zhang","doi":"10.5802/jep.189","DOIUrl":"https://doi.org/10.5802/jep.189","url":null,"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.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130672617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 6

Intersection cohomology of character varieties for punctured Riemann surfaces 刺破Riemann曲面特征变异的交点上同调

Journal de l’École polytechnique — Mathématiques Pub Date : 2022-01-21 DOI: 10.5802/jep.215

Mathieu Ballandras

引用次数: 4

Random walks on hyperbolic spaces: second order expansion of the rate function at the drift 双曲空间上的随机漫步:在漂移处速率函数的二阶展开

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-12-29 DOI: 10.5802/jep.225

Richard Aoun, P. Mathieu, Cagri Sert

引用次数: 1

Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters 秩一局部对称空间的谱对应:异常参数的情况

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-12-21 DOI: 10.5802/jep.220

Christian Arends, J. Hilgert

引用次数: 1

The mesoscopic geometry of sparse random maps 稀疏随机映射的介观几何

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-12-20 DOI: 10.5802/jep.207

N. Curien, I. Kortchemski, Cyril Marzouk

{"title":"The mesoscopic geometry of sparse random maps","authors":"N. Curien, I. Kortchemski, Cyril Marzouk","doi":"10.5802/jep.207","DOIUrl":"https://doi.org/10.5802/jep.207","url":null,"abstract":"We investigate the structure of large uniform random maps with $n$ edges, $mathrm{f}_n$ faces, and with genus $mathrm{g}_n$ in the so-called sparse case, where the ratio between the number vertices and edges tends to $1$. We focus on two regimes: the planar case $(mathrm{f}_n, 2mathrm{g}_n) = (mathrm{s}_n, 0)$ and the unicellular case with moderate genus $(mathrm{f}_n, 2 mathrm{g}_n) = (1, mathrm{s}_n-1)$, both when $1 ll mathrm{s}_n ll n$. Albeit different at first sight, these two models can be treated in a unified way using a probabilistic version of the classical core-kernel decomposition. In particular, we show that the number of edges of the core of such maps, obtained by iteratively removing degree $1$ vertices, is concentrated around $sqrt{n mathrm{s}_{n}}$. Further, their kernel, obtained by contracting the vertices of the core with degree $2$, is such that the sum of the degree of its vertices exceeds that of a trivalent map by a term of order $sqrt{mathrm{s}_{n}^{3}/n}$; in particular they are trivalent with high probability when $mathrm{s}_{n} ll n^{1/3}$. This enables us to identify a mesoscopic scale $sqrt{n/mathrm{s}_n}$ at which the scaling limits of these random maps can be seen as the local limit of their kernels, which is the dual of the UIPT in the planar case and the infinite three-regular tree in the unicellular case, where each edge is replaced by an independent (biased) Brownian tree with two marked points.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129269090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 4

Lattices with skew-Hermitian forms over division algebras and unlikely intersections 在除法代数和不可能相交上具有斜厄米形式的格

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-11-25 DOI: 10.5802/jep.240

Christopher Daw, M. Orr

引用次数: 5

Stochastic homogenization of degenerate integral functionals and their Euler-Lagrange equations 退化积分泛函的随机均匀化及其欧拉-拉格朗日方程

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-09-27 DOI: 10.5802/jep.218

M. Ruf, T. Ruf

引用次数: 5

Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires 几乎每个K3曲面都包含无限个线性李维平面超曲面

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-09-16 DOI: 10.5802/jep.233

F'elix Lequen

引用次数: 0

The core of the Levi distribution 李维分布的核心

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-09-10 DOI: 10.5802/jep.239

G. Dall’Ara, Samuele Mongodi

引用次数: 4

Twisted cotangent bundle of Hyperkähler manifolds (with an appendix by Simone Diverio) Hyperkähler流形的扭转余切束(附Simone Diverio的附录)

Journal de l’École polytechnique — Mathématiques Pub Date : 2021-09-03 DOI: 10.5802/jep.175

F. Anella, A. Höring

引用次数: 4