The moment Lyapunov exponent is computed for the solution of the parabolic Anderson equation with an (1 + 1)dimensional time–space white noise. Our main result positively confirms an open problem posted in (Ann. Probab. (2015) to appear) and originated from the observations made in the physical literature (J. Statist. Phys. 78 (1995) 1377–1401) and (Nuclear Physics B 290 (1987) 582–602). By a link through the Feynman–Kac’s formula, our theorem leads to the evaluation of the ground state energy for the n-body problem with Dirac pair interaction. Résumé. Nous calculons les moments de l’exposant de Lyapunov de la solution de l’équation d’Anderson parabolique avec un bruit blanc en espace–temps en dimension (1 + 1). Notre résultat principal confirme un problème ouvert posé dans (Ann. Probab. (2015) à paraître) et basé sur des observations faites dans la littérature physique (J. Statist. Phys. 78 (1995) 1377–1401) et (Nuclear Physics B 290 (1987) 582–602). À travers la formule de Feynman–Kac, notre théorème permet l’évaluation de l’état fondamental pour le problème à n-corps avec interaction de Dirac par paires. MSC: 60F10; 60H15; 60H40; 60J65; 81U10
{"title":"Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise","authors":"Xia Chen","doi":"10.1214/15-AIHP673","DOIUrl":"https://doi.org/10.1214/15-AIHP673","url":null,"abstract":"The moment Lyapunov exponent is computed for the solution of the parabolic Anderson equation with an (1 + 1)dimensional time–space white noise. Our main result positively confirms an open problem posted in (Ann. Probab. (2015) to appear) and originated from the observations made in the physical literature (J. Statist. Phys. 78 (1995) 1377–1401) and (Nuclear Physics B 290 (1987) 582–602). By a link through the Feynman–Kac’s formula, our theorem leads to the evaluation of the ground state energy for the n-body problem with Dirac pair interaction. Résumé. Nous calculons les moments de l’exposant de Lyapunov de la solution de l’équation d’Anderson parabolique avec un bruit blanc en espace–temps en dimension (1 + 1). Notre résultat principal confirme un problème ouvert posé dans (Ann. Probab. (2015) à paraître) et basé sur des observations faites dans la littérature physique (J. Statist. Phys. 78 (1995) 1377–1401) et (Nuclear Physics B 290 (1987) 582–602). À travers la formule de Feynman–Kac, notre théorème permet l’évaluation de l’état fondamental pour le problème à n-corps avec interaction de Dirac par paires. MSC: 60F10; 60H15; 60H40; 60J65; 81U10","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"24 1","pages":"1486-1499"},"PeriodicalIF":1.5,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78981691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the long time asymptotics of the exponential moment for the following time-space Hamiltonian ∫ t
本文研究了下列时空哈密顿量∫t的指数矩的长时间渐近性
{"title":"Exponential asymptotics for time–space Hamiltonians","authors":"Xia Chen, Yaozhong Hu, Jiancheng Song, Fei Xing","doi":"10.1214/13-AIHP588","DOIUrl":"https://doi.org/10.1214/13-AIHP588","url":null,"abstract":"In this paper, we investigate the long time asymptotics of the exponential moment for the following time-space Hamiltonian ∫ t","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"25 1","pages":"1529-1561"},"PeriodicalIF":1.5,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78003845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(X_t)_{t = 0 }^{infty}$ be an irreducible reversible discrete time Markov chain on a finite state space $Omega $. Denote its transition matrix by $P$. To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain $(X_t^{mathrm{c}})_{t ge 0} $ whose kernel is given by $H_t:=e^{-t}sum_k (tP)^k/k! $. Another possibility is to consider the associated averaged chain $(X_t^{mathrm{ave}})_{t = 0}^{infty}$, whose distribution at time $t$ is obtained by replacing $P$ by $A_t:=(P^t+P^{t+1})/2$. A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let $(X_t^{(n)})_{t = 0 }^{infty}$ be a sequence of irreducible reversible discrete time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time $t_n$ iff the sequence of the associated averaged chains exhibits total-variation cutoff around time $t_n$. Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by Aldous and Fill.
{"title":"The power of averaging at two consecutive time steps: Proof of a mixing conjecture by Aldous and Fill","authors":"J. Hermon, Y. Peres","doi":"10.1214/16-AIHP782","DOIUrl":"https://doi.org/10.1214/16-AIHP782","url":null,"abstract":"Let $(X_t)_{t = 0 }^{infty}$ be an irreducible reversible discrete time Markov chain on a finite state space $Omega $. Denote its transition matrix by $P$. To avoid periodicity issues (and thus ensuring convergence to equilibrium) one often considers the continuous-time version of the chain $(X_t^{mathrm{c}})_{t ge 0} $ whose kernel is given by $H_t:=e^{-t}sum_k (tP)^k/k! $. Another possibility is to consider the associated averaged chain $(X_t^{mathrm{ave}})_{t = 0}^{infty}$, whose distribution at time $t$ is obtained by replacing $P$ by $A_t:=(P^t+P^{t+1})/2$. \u0000A sequence of Markov chains is said to exhibit (total-variation) cutoff if the convergence to stationarity in total-variation distance is abrupt. Let $(X_t^{(n)})_{t = 0 }^{infty}$ be a sequence of irreducible reversible discrete time Markov chains. In this work we prove that the sequence of associated continuous-time chains exhibits total-variation cutoff around time $t_n$ iff the sequence of the associated averaged chains exhibits total-variation cutoff around time $t_n$. Moreover, we show that the width of the cutoff window for the sequence of associated averaged chains is at most that of the sequence of associated continuous-time chains. In fact, we establish more precise quantitative relations between the mixing-times of the continuous-time and the averaged versions of a reversible Markov chain, which provide an affirmative answer to a problem raised by Aldous and Fill.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"78 1 1","pages":"2030-2042"},"PeriodicalIF":1.5,"publicationDate":"2015-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78290058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. Bouchard, Dylan Possamai, Xiaolu Tan, Chao Zhou
We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.
{"title":"A unified approach to a priori estimates for supersolutions of BSDEs in general filtrations","authors":"B. Bouchard, Dylan Possamai, Xiaolu Tan, Chao Zhou","doi":"10.1214/16-AIHP798","DOIUrl":"https://doi.org/10.1214/16-AIHP798","url":null,"abstract":"We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"451 1","pages":"154-172"},"PeriodicalIF":1.5,"publicationDate":"2015-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77024136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.
{"title":"Non-fixation for biased Activated Random Walks","authors":"L. Rolla, L. Tournier","doi":"10.1214/17-AIHP827","DOIUrl":"https://doi.org/10.1214/17-AIHP827","url":null,"abstract":"We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"91 1","pages":"938-951"},"PeriodicalIF":1.5,"publicationDate":"2015-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81359541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDE, and show in particular that existence of a solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.
{"title":"BSDEs with diffusion constraint and viscous Hamilton–Jacobi equations with unbounded data","authors":"Andrea Cosso, H. Pham, Hao Xing","doi":"10.1214/16-AIHP762","DOIUrl":"https://doi.org/10.1214/16-AIHP762","url":null,"abstract":"We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with constraint in the martingale part. We compare our result with the classical representation in terms of (super)quadratic BSDE, and show in particular that existence of a solution to the viscous HJ equation can be obtained under more general growth assumptions on the coefficients, including both unbounded diffusion coefficient and terminal data.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"8 1","pages":"1528-1547"},"PeriodicalIF":1.5,"publicationDate":"2015-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85623592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:Xrightarrow X$ with exponential specification property and a H$ddot{text{o}}$lder continuous matrix cocycle $A:Xrightarrow G (m,mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_delta$ set).
由Oseledec乘法遍历定理可知，给定连续循环的Oseledec平均值发散的lyapunov -不规则点集对于任意不变概率测度具有零测度。与此相反，对于任何具有指数规范性质的动力系统$f:Xrightarrow X$和一个H $ddot{text{o}}$年长的连续矩阵共循环$A:Xrightarrow G (m,mathbb{R})$，我们证明了如果存在具有不同Lyapunov谱的遍历测度，那么$A$的Lyapunov-不规则集是残差的(即包含一个稠密的$G_delta$集)。
{"title":"Nonexistence of Lyapunov exponents for matrix cocycles","authors":"Xueting Tian","doi":"10.1214/15-AIHP733","DOIUrl":"https://doi.org/10.1214/15-AIHP733","url":null,"abstract":"It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:Xrightarrow X$ with exponential specification property and a H$ddot{text{o}}$lder continuous matrix cocycle $A:Xrightarrow G (m,mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_delta$ set).","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"3 1","pages":"493-502"},"PeriodicalIF":1.5,"publicationDate":"2015-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80988205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily non-monotone. On fixed finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity
{"title":"Monotonicity and condensation in homogeneous stochastic particle systems","authors":"T. Rafferty, P. Chleboun, S. Grosskinsky","doi":"10.1214/17-AIHP821","DOIUrl":"https://doi.org/10.1214/17-AIHP821","url":null,"abstract":"We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on fixed finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing homogeneous particle systems with finite critical density are necessarily \u0000non-monotone. On fixed finite lattices condensation can occur even when the critical density is infinite, in this case we give an example of a condensing process that numerical evidence suggests is monotone, and give a partial proof of its monotonicity","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"2005 1","pages":"790-818"},"PeriodicalIF":1.5,"publicationDate":"2015-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88348652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a spectral approach to prove a parametric first-order Edgeworth expansion for bivariate additive functionals of strongly ergodic Markov chains. In particular, given any V -geometrically ergodic Markov chain (Xn)n∈N whose distribution depends on a parameter θ , we prove that {ξp(Xn−1,Xn);p ∈P, n ≥ 1} satisfies a uniform (in (θ,p)) first-order Edgeworth expansion provided that {ξp(·, ·);p ∈ P} satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence (Xn)n∈N need not be stationary. This result is applied to M-estimators of Markov chains and in particular of V -geometrically ergodic Markov chains. The M-estimators of some autoregressive processes are studied. Résumé. Grâce à une approche spectrale, nous donnons des conditions assurant la validité du développement d’Edgeworth d’ordre 1 paramétrique, dans le cadre général des fonctionnelles bivariées et additives de chaînes de Markov fortement ergodiques. En particulier, soit (Xn)n∈N une chaîne de Markov V -géométriquement ergodique dont la loi dépend d’un paramètre θ . Nous montrons alors que {ξp(Xn−1,Xn);p ∈P, n ≥ 1} satisfait un développement d’Edgeworth d’ordre 1 uniforme (en (θ,p)) si {ξp(·, ·);p ∈P} satisfait une condition de type non-lattice ainsi qu’une condition quasi-optimale de moment-domination. De plus, ce résultat est établi dans le cas où les données (Xn)n∈N ne sont pas nécessairement stationnaires. Ce résultat est appliqué en particulier aux M-estimateurs associés à des chaînes de Markov V -géométriquement ergodiques. Les M-estimateurs de processus autorégressifs sont étudiés. MSC: 60F05; 60J05; 62F12; 62M05
给出了证明强遍历马尔可夫链二元加性泛函的参数一阶Edgeworth展开式的谱方法。特别地，给定任意V -几何遍历马尔可夫链(Xn)n∈n，其分布依赖于参数θ，我们证明了{ξp(Xn−1,Xn);p∈p, n≥1}满足一致(In (θ，p))一阶Edgeworth展开，条件是{ξp(·，·);p∈p}满足非格条件和几乎最优矩支配条件。此外，序列(Xn)n∈n不必是平稳的。这一结果应用于马尔可夫链的m -估计，特别是V -几何遍历马尔可夫链。研究了一些自回归过程的m估计量。的简历。恩有一个approche spectrale,常识donnons des条件assurant la validite du开发署d 'Edgeworth词1 parametrique,在干部一般des fonctionnelles bivariees等添加剂德德马尔可夫链fortement ergodiques。特别地，soit (Xn)n∈nne cha ne de Markov V - gsamomactriquement ergodique don ' la loi dsamdpend 'un paramtre θ。Nous montrons alors que {ξp(Xn−1,Xn);p∈p, n≥1}满足edgeworth d 'ordre 1一致(en (θ，p)) si {ξp(·，·);p∈p}满足一类非格ainsi qu ' one条件准最优矩控制。此外，我们还将所有的 (Xn))和所有的 (Xn) (n)和所有的 (Xn) (n) (n)发送给所有的 (Xn))。这个结果是贴花en particulier辅助M-estimateurs过渡群系des德马尔可夫链V -geometriquement ergodiques。m -估计器处理自动的和自动的。MSC: 60 f05;60 j05;62 f12;62年m05
{"title":"Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations","authors":"D. Ferre","doi":"10.1214/13-AIHP592","DOIUrl":"https://doi.org/10.1214/13-AIHP592","url":null,"abstract":"We give a spectral approach to prove a parametric first-order Edgeworth expansion for bivariate additive functionals of strongly ergodic Markov chains. In particular, given any V -geometrically ergodic Markov chain (Xn)n∈N whose distribution depends on a parameter θ , we prove that {ξp(Xn−1,Xn);p ∈P, n ≥ 1} satisfies a uniform (in (θ,p)) first-order Edgeworth expansion provided that {ξp(·, ·);p ∈ P} satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence (Xn)n∈N need not be stationary. This result is applied to M-estimators of Markov chains and in particular of V -geometrically ergodic Markov chains. The M-estimators of some autoregressive processes are studied. Résumé. Grâce à une approche spectrale, nous donnons des conditions assurant la validité du développement d’Edgeworth d’ordre 1 paramétrique, dans le cadre général des fonctionnelles bivariées et additives de chaînes de Markov fortement ergodiques. En particulier, soit (Xn)n∈N une chaîne de Markov V -géométriquement ergodique dont la loi dépend d’un paramètre θ . Nous montrons alors que {ξp(Xn−1,Xn);p ∈P, n ≥ 1} satisfait un développement d’Edgeworth d’ordre 1 uniforme (en (θ,p)) si {ξp(·, ·);p ∈P} satisfait une condition de type non-lattice ainsi qu’une condition quasi-optimale de moment-domination. De plus, ce résultat est établi dans le cas où les données (Xn)n∈N ne sont pas nécessairement stationnaires. Ce résultat est appliqué en particulier aux M-estimateurs associés à des chaînes de Markov V -géométriquement ergodiques. Les M-estimateurs de processus autorégressifs sont étudiés. MSC: 60F05; 60J05; 62F12; 62M05","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"82 1","pages":"781-808"},"PeriodicalIF":1.5,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86844498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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