{"title":"A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus","authors":"Antônio Francisco Neto","doi":"10.4171/aihpd/173","DOIUrl":"https://doi.org/10.4171/aihpd/173","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"4 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87940903","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}
In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa– Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space H with s > 3/2. Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously. Résumé. Dans cet article, nous considérons les effets du bruit sur une classe d’équations d’évolution stochastiques y compris les équations stochastiques de Camassa–Holm avec ou sans rotation. Nous obtenons d’abord l’existence, l’unicité et un critère d’explosion de solutions pathwises dans l’espace de Sobolev H avec s > 3/2. Ensuite, nous prouvons qu’un bruit suffisamment fort peut empêcher l’explosion avec une probabilité de 1, ce qui justifie l’effet régularisant du bruit non linéaire fort dans la prévention des singularités. De plus, de telles forces de bruit sont estimées dans les examples différents. Enfin, pour l’interaction entre l’effet de régularisation induit par le bruit et la dépendance aux conditions initiales, nous introduisons et étudions la stabilité du temps de sortie et construisons un exemple pour montrer que le bruit multiplicatif ne peut pas améliorer simultanément la stabilité du temps de sortie et la continuité de la dépendance aux données initiales. MSC2020 subject classifications: Primary 60H15, 35Q51; Secondary 35A01, 35B30
本文考虑了噪声对一类随机演化方程的影响，其中包括有或无旋转的随机Camassa - Holm方程。首先得到了Sobolev空间H中s > 3/2的路径解的存在唯一性和爆破判据。然后，我们证明了足够强的噪声可以以1的概率防止爆炸，这证明了强非线性噪声在防止奇异性方面的正则化效果。此外，在不同的例子中估计了噪声的强度。最后，针对噪声引起的正则化效应与初始条件依赖性之间的相互作用，引入并研究了存在时间的稳定性，并构造了一个例子，表明乘性噪声不能同时提高存在时间的稳定性和对初始数据依赖性的连续性。的简历。在cet(中央东部东京)的文章中,鉴于les运用du散播关于一个架势等式中d以stochastiques y理解les方程stochastiques Camassa-Holm用或者无旋转。已知的存在性条件、单一性条件、爆炸解路径和Sobolev空间均大于3/2。套房,常识prouvons曲一个谣传说我们堡empecher l 'explosion用一个概率是1,ce, justifie l 'effet regularisant du散播非线性在预防des singularites堡。另外，它还可以强制计算出不同的栅格和栅格。最后，将“相互作用”和“影响”与“先决条件”和“先决条件”相结合，将“先决条件”与“先决条件”相结合，将“先决条件”与“先决条件”相结合，将“先决条件”与“先决条件”相结合，将“先决条件”与“先决条件”相结合，将“先决条件”与“先决条件”相结合，将“先决条件”与“先决条件”相结合。MSC2020学科分类:初级60H15, 35Q51;二级35A01, 35B30
{"title":"Noise effects in some stochastic evolution equations: Global existence and dependence on initial data","authors":"Hao Tang, Anita S Yang","doi":"10.1214/21-aihp1241","DOIUrl":"https://doi.org/10.1214/21-aihp1241","url":null,"abstract":"In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa– Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space H with s > 3/2. Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously. Résumé. Dans cet article, nous considérons les effets du bruit sur une classe d’équations d’évolution stochastiques y compris les équations stochastiques de Camassa–Holm avec ou sans rotation. Nous obtenons d’abord l’existence, l’unicité et un critère d’explosion de solutions pathwises dans l’espace de Sobolev H avec s > 3/2. Ensuite, nous prouvons qu’un bruit suffisamment fort peut empêcher l’explosion avec une probabilité de 1, ce qui justifie l’effet régularisant du bruit non linéaire fort dans la prévention des singularités. De plus, de telles forces de bruit sont estimées dans les examples différents. Enfin, pour l’interaction entre l’effet de régularisation induit par le bruit et la dépendance aux conditions initiales, nous introduisons et étudions la stabilité du temps de sortie et construisons un exemple pour montrer que le bruit multiplicatif ne peut pas améliorer simultanément la stabilité du temps de sortie et la continuité de la dépendance aux données initiales. MSC2020 subject classifications: Primary 60H15, 35Q51; Secondary 35A01, 35B30","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73074875","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}
{"title":"On the boundaries of the $m=2$ amplituhedron","authors":"T. Łukowski","doi":"10.4171/aihpd/124","DOIUrl":"https://doi.org/10.4171/aihpd/124","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"36 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73672700","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}
We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erdős-Rényi– Shepp strong law for the increments. 1. Definitions and main results The classical Erdős-Rényi–Shepp strong law of large numbers [4], [5], asserts as follows. Theorem 1.1 (Erdős-Rényi 1970, Shepp 1964). Consider a random walk Sn = ∑n i=1Xi with Xi i.i.d., satisfying EX1 = 0. Set φ(t) = E[e tX ] and let D φ = {t > 0 : φ(t) < ∞}. Let α > 0 be such that φ(t)e −αt achieves its minimum value for some t in the interior of D φ . Set 1/Aα := − logmin t>0 φ(t)e −αt. Then, Aα > 0 and (1.1) max 0≤j≤n−⌊Aα logn⌋ Sj+⌊Aα logn⌋ − Sj ⌊Aα log n⌋ a.s. → α, a.s. In the particular case of Xi ∈ {−1, 1}, the assumptions of the theorem are satisfied for any α ∈ (0, 1). The theorem also trivially generalizes to EX1 ̸= 0, by considering Yi = Xi − EXi. Theorem 1.1 is closely related to the large deviation principle for Sn/n given by Cramér’s theorem, see e.g. [3] for background. Indeed, with I(x) = supt(tx−log φ(t)) denoting the rate function, one observes that I(α) = 1/Aα and that (1.2) α = inf{x > 0 : I(x) > 1/Aα}. In this paper, we prove an analogous statement for standard one dimensional random walk in random environment (RWRE), in the case of positive velocity. We begin by introducing the model. Fix a realization ω = {ωi}i∈Z with ωi ∈ (0, 1) of a collection of i.i.d. random variables, which we call the environment. With p denoting the law of ω0 and σ(p) its support, denote by P = pZ the law of the environment on Σp := σ(p) Z. We make throughout the following assumption. Condition 1.2 (Uniform Ellipticity). There exists a κ ∈ (0, 1) such that σ(p) ⊂ [κ, 1− κ] almost surely. Date: May 4, 2020. Revised May 19, 2021 and July 8, 2021. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452). 1 2 DARCY CAMARGO, YURI KIFER, AND OFER ZEITOUNI Letting ρi := (1 − ωi)/ωi, we note that the ellipticity assumption gives a deterministic uniform upper and lower bounds on ρi. It will be useful for us to consider also different laws of the environment Σ = [κ, 1 − κ]Z, not necessarily product laws. Such laws will be denoted η. Equipping Σ with the standard shift operator θ, so that (θω)j := ωi+j , the spaces of probability measures (stationary/ ergodic wrt θ) on Σ are denoted M1(Σ) (M s 1 (Σ)/M e 1 (Σ)), respectively; similar definitions hold when Σ is replaced by Σp. On top of ω we consider the RWRE, which is a nearest neighbor random walk {Xt}t∈Z. Conditioned on the environment ω, {Xt} is a Markov chain with transition probabilities π(i, i+ 1) = 1− π(i, i− 1) = ωi. We denote the law of the random walk, started at i ∈ Z and conditioned on a fixed realization of the environment ω, by Pi (the so-called quenched law). For any measure η ∈ M1(Σ), the measure η(dω) ⊗ Pi is referred to as the annealed law, and denoted by P i ; with some abuse of notation, w
{"title":"The Erdős-Rényi-Shepp law of large numbers for ballistic random walk in random environment","authors":"Darcy Camargo, Y. Kifer, O. Zeitouni","doi":"10.1214/21-aihp1210","DOIUrl":"https://doi.org/10.1214/21-aihp1210","url":null,"abstract":"We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erdős-Rényi– Shepp strong law for the increments. 1. Definitions and main results The classical Erdős-Rényi–Shepp strong law of large numbers [4], [5], asserts as follows. Theorem 1.1 (Erdős-Rényi 1970, Shepp 1964). Consider a random walk Sn = ∑n i=1Xi with Xi i.i.d., satisfying EX1 = 0. Set φ(t) = E[e tX ] and let D φ = {t > 0 : φ(t) < ∞}. Let α > 0 be such that φ(t)e −αt achieves its minimum value for some t in the interior of D φ . Set 1/Aα := − logmin t>0 φ(t)e −αt. Then, Aα > 0 and (1.1) max 0≤j≤n−⌊Aα logn⌋ Sj+⌊Aα logn⌋ − Sj ⌊Aα log n⌋ a.s. → α, a.s. In the particular case of Xi ∈ {−1, 1}, the assumptions of the theorem are satisfied for any α ∈ (0, 1). The theorem also trivially generalizes to EX1 ̸= 0, by considering Yi = Xi − EXi. Theorem 1.1 is closely related to the large deviation principle for Sn/n given by Cramér’s theorem, see e.g. [3] for background. Indeed, with I(x) = supt(tx−log φ(t)) denoting the rate function, one observes that I(α) = 1/Aα and that (1.2) α = inf{x > 0 : I(x) > 1/Aα}. In this paper, we prove an analogous statement for standard one dimensional random walk in random environment (RWRE), in the case of positive velocity. We begin by introducing the model. Fix a realization ω = {ωi}i∈Z with ωi ∈ (0, 1) of a collection of i.i.d. random variables, which we call the environment. With p denoting the law of ω0 and σ(p) its support, denote by P = pZ the law of the environment on Σp := σ(p) Z. We make throughout the following assumption. Condition 1.2 (Uniform Ellipticity). There exists a κ ∈ (0, 1) such that σ(p) ⊂ [κ, 1− κ] almost surely. Date: May 4, 2020. Revised May 19, 2021 and July 8, 2021. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452). 1 2 DARCY CAMARGO, YURI KIFER, AND OFER ZEITOUNI Letting ρi := (1 − ωi)/ωi, we note that the ellipticity assumption gives a deterministic uniform upper and lower bounds on ρi. It will be useful for us to consider also different laws of the environment Σ = [κ, 1 − κ]Z, not necessarily product laws. Such laws will be denoted η. Equipping Σ with the standard shift operator θ, so that (θω)j := ωi+j , the spaces of probability measures (stationary/ ergodic wrt θ) on Σ are denoted M1(Σ) (M s 1 (Σ)/M e 1 (Σ)), respectively; similar definitions hold when Σ is replaced by Σp. On top of ω we consider the RWRE, which is a nearest neighbor random walk {Xt}t∈Z. Conditioned on the environment ω, {Xt} is a Markov chain with transition probabilities π(i, i+ 1) = 1− π(i, i− 1) = ωi. We denote the law of the random walk, started at i ∈ Z and conditioned on a fixed realization of the environment ω, by Pi (the so-called quenched law). For any measure η ∈ M1(Σ), the measure η(dω) ⊗ Pi is referred to as the annealed law, and denoted by P i ; with some abuse of notation, w","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"63 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80500925","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}
Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To contribute to the statistical understanding in more general situations, we demonstrate how combining βmixing assumptions on the process and heat kernel bounds on the transition density representing controls on the longand short-time transitional behaviour, allow to obtain sup-norm and L kernel invariant density estimation rates that match the well-understood case of reversible multidimensional diffusion processes and are faster than in a sampled discrete data scenario. Moreover, we demonstrate how, up to log-terms, optimal sup-norm adaptive invariant density estimation can be achieved within our framework, based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous-time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of popular Markov models. We highlight this point by showing how multidimensional jump SDEs with Lévy-driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive sup-norm estimation rates for this class of processes. MSC2020 subject classifications: Primary 62M05; secondary 62G05, 62G20, 60G10, 60J25
{"title":"Adaptive invariant density estimation for continuous-time mixing Markov processes under sup-norm risk","authors":"Niklas Dexheimer, C. Strauch, Lukas Trottner","doi":"10.1214/21-aihp1235","DOIUrl":"https://doi.org/10.1214/21-aihp1235","url":null,"abstract":"Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To contribute to the statistical understanding in more general situations, we demonstrate how combining βmixing assumptions on the process and heat kernel bounds on the transition density representing controls on the longand short-time transitional behaviour, allow to obtain sup-norm and L kernel invariant density estimation rates that match the well-understood case of reversible multidimensional diffusion processes and are faster than in a sampled discrete data scenario. Moreover, we demonstrate how, up to log-terms, optimal sup-norm adaptive invariant density estimation can be achieved within our framework, based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous-time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of popular Markov models. We highlight this point by showing how multidimensional jump SDEs with Lévy-driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive sup-norm estimation rates for this class of processes. MSC2020 subject classifications: Primary 62M05; secondary 62G05, 62G20, 60G10, 60J25","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"199 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80046489","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}
An inequality of Marton [Mar96] shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We generalize this inequality to Markov chains indexed by trees.
{"title":"Concentration of Markov chains indexed by trees","authors":"Christopher Shriver","doi":"10.1214/21-aihp1224","DOIUrl":"https://doi.org/10.1214/21-aihp1224","url":null,"abstract":"An inequality of Marton [Mar96] shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We generalize this inequality to Markov chains indexed by trees.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"18 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81107336","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}
{"title":"Weyl law for the Anderson Hamiltonian on a two-dimensional manifold","authors":"Mouzard Antoine","doi":"10.1214/21-aihp1216","DOIUrl":"https://doi.org/10.1214/21-aihp1216","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"11 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76883169","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}
Abstract: We consider the density Xt(x) of the critical (α, β)-superprocess in R d with α ∈ (0, 2) and β < α d . Our starting point is a recent result from PDE [2] which implies the following dichotomy: if x ∈ R is fixed and β ≤ β∗(α) := α d+α , then Xt(x) > 0 a.s. on {Xt 6= 0}; otherwise, the probability that Xt(x) is positive when conditioned on {Xt 6= 0} has power law decay. We strengthen this and prove probabilistically that if β < β∗(α) and the density is continuous, which holds if and only if d = 1 and α > 1+ β, then Xt(x) > 0 for all x ∈ R a.s. on {Xt 6= 0}. The above complements a classical superprocess result that if Xt is non-zero, then it charges every open set almost surely. We unify and extend these results by giving close to sharp conditions on a measure μ such that μ(Xt) := ∫ Xt(x)μ(dx) > 0 a.s. on {Xt 6= 0}. Our characterization is based on the size of supp(μ), in the sense of Hausdorff measure and dimension. For s ∈ [0, d], if β ≤ β∗(α, s) = α d−s+α and supp(μ) has positive x-Hausdorff measure, then μ(Xt) > 0 a.s. on {Xt 6= 0}; and when β > β ∗(α, s), if μ satisfies a uniform lower density condition which implies dim(supp(μ)) < s, then P (μ(Xt) = 0 |Xt 6= 0) > 0. Our methods also give new results for the fractional PDE which is dual to the (α, β)superprocess, i.e. ∂tu(t, x) = ∆αu(t, x)− u(t, x) 1+β with domain (t, x) ∈ (0,∞) × R, where ∆α = −(−∆) α
{"title":"The density of the (α,d,β)-superprocess and singular solutions to a fractional non-linear PDE","authors":"Thomas Hughes","doi":"10.1214/21-aihp1180","DOIUrl":"https://doi.org/10.1214/21-aihp1180","url":null,"abstract":"Abstract: We consider the density Xt(x) of the critical (α, β)-superprocess in R d with α ∈ (0, 2) and β < α d . Our starting point is a recent result from PDE [2] which implies the following dichotomy: if x ∈ R is fixed and β ≤ β∗(α) := α d+α , then Xt(x) > 0 a.s. on {Xt 6= 0}; otherwise, the probability that Xt(x) is positive when conditioned on {Xt 6= 0} has power law decay. We strengthen this and prove probabilistically that if β < β∗(α) and the density is continuous, which holds if and only if d = 1 and α > 1+ β, then Xt(x) > 0 for all x ∈ R a.s. on {Xt 6= 0}. The above complements a classical superprocess result that if Xt is non-zero, then it charges every open set almost surely. We unify and extend these results by giving close to sharp conditions on a measure μ such that μ(Xt) := ∫ Xt(x)μ(dx) > 0 a.s. on {Xt 6= 0}. Our characterization is based on the size of supp(μ), in the sense of Hausdorff measure and dimension. For s ∈ [0, d], if β ≤ β∗(α, s) = α d−s+α and supp(μ) has positive x-Hausdorff measure, then μ(Xt) > 0 a.s. on {Xt 6= 0}; and when β > β ∗(α, s), if μ satisfies a uniform lower density condition which implies dim(supp(μ)) < s, then P (μ(Xt) = 0 |Xt 6= 0) > 0. Our methods also give new results for the fractional PDE which is dual to the (α, β)superprocess, i.e. ∂tu(t, x) = ∆αu(t, x)− u(t, x) 1+β with domain (t, x) ∈ (0,∞) × R, where ∆α = −(−∆) α","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"9 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81535420","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}
{"title":"The Brownian disk viewed from a boundary point","authors":"Jean-François Le Gall","doi":"10.1214/21-aihp1179","DOIUrl":"https://doi.org/10.1214/21-aihp1179","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"4 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79952290","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}
We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on R describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gapN = 1 − cos(π/N), and that for all ǫ ∈ (0, 1), its ǫ-mixing time satisfies TN (ǫ) ∼ logN 2 gapN as N → ∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential. MSC 2010 subject classifications: Primary 60J25; Secondary 37A25, 82C22.
{"title":"Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ interfaces with convex potential","authors":"P. Caputo, Cyril Labbé, H. Lacoin","doi":"10.1214/21-aihp1174","DOIUrl":"https://doi.org/10.1214/21-aihp1174","url":null,"abstract":"We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on R describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gapN = 1 − cos(π/N), and that for all ǫ ∈ (0, 1), its ǫ-mixing time satisfies TN (ǫ) ∼ logN 2 gapN as N → ∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential. MSC 2010 subject classifications: Primary 60J25; Secondary 37A25, 82C22.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"46 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73562045","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}
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