We propose a novel measure valued process which models the behaviour of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species and continuous movement of molecules in space. Reactions rates at a spatial location are proportional to the mass of different species present locally and to a location specific chemical rate, which may be a function of the local or global species mass as well. We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types. In particular, when the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.
{"title":"A spatial measure-valued model for chemical reaction networks in heterogeneous systems","authors":"Lea Popovic, Amandine Veber","doi":"10.1214/22-aap1904","DOIUrl":"https://doi.org/10.1214/22-aap1904","url":null,"abstract":"We propose a novel measure valued process which models the behaviour of chemical reaction networks in spatially heterogeneous systems. It models reaction dynamics between different molecular species and continuous movement of molecules in space. Reactions rates at a spatial location are proportional to the mass of different species present locally and to a location specific chemical rate, which may be a function of the local or global species mass as well. We obtain asymptotic limits for the process, with appropriate rescaling depending on the abundance of different molecular types. In particular, when the mass of some species in the scaling limit is discrete while the mass of the others is continuous, we obtain a new type of spatial random evolution process. This process can be shown, in some situations, to correspond to a measure-valued piecewise deterministic Markov process in which the discrete mass of the process evolves stochastically, and the continuous mass evolves in a deterministic way between consecutive jump times of the discrete part.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119730","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 consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system.
{"title":"Graphon mean field systems","authors":"Erhan Bayraktar, Suman Chakraborty, Ruoyu Wu","doi":"10.1214/22-aap1901","DOIUrl":"https://doi.org/10.1214/22-aap1901","url":null,"abstract":"We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon mean field system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Well-posedness, continuity and stability of such systems are provided. We also consider a not-so-dense analogue of the finite particle system, obtained by percolation with vanishing rates and suitable scaling of interactions. A law of large numbers result is proved for the convergence of such systems to the corresponding graphon mean field system.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136168194","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 consider the stochastic system of interacting neurons introduced in (J. Stat. Phys. 158 (2015) 866–902) and in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1844–1876) and then further studied in (Electron. J. Probab. 26 (2021) 20) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1/ N. In between successive spikes, each neuron’s potential follows a deterministic flow. In our previous article (Electron. J. Probab. 26 (2021) 20) we proved the convergence of the system, as N→∞, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in (Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58) of the point process representing the small jumps of the particle system with the limit Brownian motion.
{"title":"Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling","authors":"Xavier Erny, Eva Löcherbach, Dasha Loukianova","doi":"10.1214/22-aap1900","DOIUrl":"https://doi.org/10.1214/22-aap1900","url":null,"abstract":"We consider the stochastic system of interacting neurons introduced in (J. Stat. Phys. 158 (2015) 866–902) and in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1844–1876) and then further studied in (Electron. J. Probab. 26 (2021) 20) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1/ N. In between successive spikes, each neuron’s potential follows a deterministic flow. In our previous article (Electron. J. Probab. 26 (2021) 20) we proved the convergence of the system, as N→∞, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in (Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58) of the point process representing the small jumps of the particle system with the limit Brownian motion.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996637","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}
This paper aims to systematically solve stochastic team optimization of large-scale system, in linear-quadratic-Gaussian framework. Concretely, the underlying large-scale system involves considerable weakly-coupled cooperative agents for which the individual admissible controls: ( i ) enter the diffusion terms, ( ii ) are constrained in some closed-convex subsets, and ( iii ) subject to a general partial decentralized information structure. A more im-portant but serious feature: ( iv ) all agents are heterogenous with continuum instead of finite diversity. Combination of ( i )-( iv ) yields a quite general modeling of stochastic team-optimization, but on the other hand, also fails current existing techniques of team analysis. In particular, classical team consistency with continuum heterogeneity collapses because of ( i ). As the resolution, a novel unified approach is proposed under which the intractable continuum heterogeneity can be converted to a more tractable homogeneity . As a trade-off, the underlying randomness is augmented, and all agents become (quasi) weakly-exchangeable. Such approach essentially involves a subtle balance between homogeneity v.s. heterogeneity, and left (prior-sampling)-v.s. right (posterior-sampling) information filtration. Subsequently, the consistency condition (CC) system takes a new type of forward-backward stochastic system with double-projections (due to ( ii ), ( iii )), along with spatial mean on continuum heterogenous index (due to ( iv )). Such system is new in team literature and its well-posedness is also challenging. We address this is-sue under mild conditions. Related asymptotic optimality is also established.
{"title":"A unified approach to linear-quadratic-Gaussian mean-field team: Homogeneity, heterogeneity and quasi-exchangeability","authors":"Xinwei Feng, Ying Hu, Jianhui Huang","doi":"10.1214/22-aap1878","DOIUrl":"https://doi.org/10.1214/22-aap1878","url":null,"abstract":"This paper aims to systematically solve stochastic team optimization of large-scale system, in linear-quadratic-Gaussian framework. Concretely, the underlying large-scale system involves considerable weakly-coupled cooperative agents for which the individual admissible controls: ( i ) enter the diffusion terms, ( ii ) are constrained in some closed-convex subsets, and ( iii ) subject to a general partial decentralized information structure. A more im-portant but serious feature: ( iv ) all agents are heterogenous with continuum instead of finite diversity. Combination of ( i )-( iv ) yields a quite general modeling of stochastic team-optimization, but on the other hand, also fails current existing techniques of team analysis. In particular, classical team consistency with continuum heterogeneity collapses because of ( i ). As the resolution, a novel unified approach is proposed under which the intractable continuum heterogeneity can be converted to a more tractable homogeneity . As a trade-off, the underlying randomness is augmented, and all agents become (quasi) weakly-exchangeable. Such approach essentially involves a subtle balance between homogeneity v.s. heterogeneity, and left (prior-sampling)-v.s. right (posterior-sampling) information filtration. Subsequently, the consistency condition (CC) system takes a new type of forward-backward stochastic system with double-projections (due to ( ii ), ( iii )), along with spatial mean on continuum heterogenous index (due to ( iv )). Such system is new in team literature and its well-posedness is also challenging. We address this is-sue under mild conditions. Related asymptotic optimality is also established.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44814943","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 { ξ i } i ≥ 1 be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in i -th vertical column to another in the ( i + 1) -th vertical column by an edge having length ξ i . Then declare independently each edge e in the resulting lattice open with probability p e = p | e | where p ∈ [0 , 1] and | e | is the length of e . We relate the occurrence of a nontrivial phase transition for this model to moment properties of ξ 1 . More precisely, we prove that the model undergoes a nontrivial phase transition when E ( ξ η 1 ) < ∞ , for some η > 1 . On the other hand, when E ( ξ 1 ) = ∞ , percolation never occurs for p < 1 . We also show that the probability of the one-arm event decays no faster than a polynomial in an open interval of parameters p close to the critical point.
{"title":"Phase transition for percolation on a randomly stretched square lattice","authors":"Emy, Anchis, ugusto, eixeira","doi":"10.1214/22-aap1887","DOIUrl":"https://doi.org/10.1214/22-aap1887","url":null,"abstract":"Let { ξ i } i ≥ 1 be a sequence of i.i.d. positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in i -th vertical column to another in the ( i + 1) -th vertical column by an edge having length ξ i . Then declare independently each edge e in the resulting lattice open with probability p e = p | e | where p ∈ [0 , 1] and | e | is the length of e . We relate the occurrence of a nontrivial phase transition for this model to moment properties of ξ 1 . More precisely, we prove that the model undergoes a nontrivial phase transition when E ( ξ η 1 ) < ∞ , for some η > 1 . On the other hand, when E ( ξ 1 ) = ∞ , percolation never occurs for p < 1 . We also show that the probability of the one-arm event decays no faster than a polynomial in an open interval of parameters p close to the critical point.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49635155","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}
{"title":"An SPDE approach to perturbation theory of Φ24: Asymptoticity and short distance behavior","authors":"Hao Shen, Rongchan Zhu, Xiangchan Zhu","doi":"10.1214/22-aap1873","DOIUrl":"https://doi.org/10.1214/22-aap1873","url":null,"abstract":"","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47121502","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 the global well-posedness and wave-breaking phenomenon for the stochastic rotation-two-component Camassa-Holm (R2CH) system. First, we find a Hamiltonian structure of the R2CH system and use the stochastic Hamiltonian to derive the stochastic R2CH system. Then, we establish the local well-posedness of the stochastic R2CH system using a dispersion-dissipation approximation system and the regularization method. We also show a precise blow-up criterion for the stochastic R2CH system. Moreover, we prove that the global existence of the stochastic R2CH system occurs with high probability. At the end, we consider the transport noise case and establish the local well-posedness and another blow-up criterion.
{"title":"Well-posedness and wave-breaking for the stochastic rotation-two-component Camassa–Holm system","authors":"Yong Chen, Jinqiao Duan, Hongjun Gao","doi":"10.1214/22-aap1877","DOIUrl":"https://doi.org/10.1214/22-aap1877","url":null,"abstract":"We study the global well-posedness and wave-breaking phenomenon for the stochastic rotation-two-component Camassa-Holm (R2CH) system. First, we find a Hamiltonian structure of the R2CH system and use the stochastic Hamiltonian to derive the stochastic R2CH system. Then, we establish the local well-posedness of the stochastic R2CH system using a dispersion-dissipation approximation system and the regularization method. We also show a precise blow-up criterion for the stochastic R2CH system. Moreover, we prove that the global existence of the stochastic R2CH system occurs with high probability. At the end, we consider the transport noise case and establish the local well-posedness and another blow-up criterion.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42938785","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}
{"title":"A sample-path large deviation principle for dynamic Erdős–Rényi random graphs","authors":"Peter Braunsteins, F. den Hollander, M. Mandjes","doi":"10.1214/22-aap1892","DOIUrl":"https://doi.org/10.1214/22-aap1892","url":null,"abstract":"","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43616231","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 a class of Metropolis–Hastings algorithms for target measures that are absolutely continuous with respect to a large class of non-Gaussian prior measures on Banach spaces. The algorithm is shown to have a spectral gap in a Wasserstein-like semimetric weighted by a Lyapunov function. A number of error bounds are given for computationally tractable approximations of the algorithm including bounds on the closeness of Cesáro averages and other pathwise quantities via perturbation theory. Several applications illustrate the breadth of problems to which the results apply such as various likelihood approximations and perturbations of prior measures.
{"title":"Spectral gaps and error estimates for infinite-dimensional Metropolis–Hastings with non-Gaussian priors","authors":"Bamdad Hosseini, James E Johndrow","doi":"10.1214/22-aap1854","DOIUrl":"https://doi.org/10.1214/22-aap1854","url":null,"abstract":"We study a class of Metropolis–Hastings algorithms for target measures that are absolutely continuous with respect to a large class of non-Gaussian prior measures on Banach spaces. The algorithm is shown to have a spectral gap in a Wasserstein-like semimetric weighted by a Lyapunov function. A number of error bounds are given for computationally tractable approximations of the algorithm including bounds on the closeness of Cesáro averages and other pathwise quantities via perturbation theory. Several applications illustrate the breadth of problems to which the results apply such as various likelihood approximations and perturbations of prior measures.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136177382","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 investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on (cid:90) 2 with infinitely many sources that are the points of the vertical axis I ( ∞ ) = {0} × (cid:90) . Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t.the horizontal direction) random forest spanning (cid:90) 2 , based on an IDLA protocol, which is invariant in distribution w.r.t.vertical translations.
{"title":"The bi-dimensional Directed IDLA forest","authors":"Nicolas Chenavier, D. Coupier, A. Rousselle","doi":"10.1214/22-aap1865","DOIUrl":"https://doi.org/10.1214/22-aap1865","url":null,"abstract":"We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on (cid:90) 2 with infinitely many sources that are the points of the vertical axis I ( ∞ ) = {0} × (cid:90) . Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t.the horizontal direction) random forest spanning (cid:90) 2 , based on an IDLA protocol, which is invariant in distribution w.r.t.vertical translations.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42941335","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：