Pub Date : 2023-11-07DOI: 10.1142/s0219493723400038
Zhongyang Li, Fei Lu
In the inference for systems of interacting particles or agents, a coercivity condition ensures the identifiability of the interaction kernels, providing the foundation of learning. We prove the coercivity condition for stochastic systems with an arbitrary number of particles and a class of kernels such that the system of relative positions is ergodic. When the system of relative positions is stationary, we prove the coercivity condition by showing the strictly positive definiteness of an integral kernel arising in the learning. For the non-stationary case, we show that the coercivity condition holds when the time is large based on a perturbation argument.
{"title":"On the coercivity condition in the learning of interacting particle systems","authors":"Zhongyang Li, Fei Lu","doi":"10.1142/s0219493723400038","DOIUrl":"https://doi.org/10.1142/s0219493723400038","url":null,"abstract":"In the inference for systems of interacting particles or agents, a coercivity condition ensures the identifiability of the interaction kernels, providing the foundation of learning. We prove the coercivity condition for stochastic systems with an arbitrary number of particles and a class of kernels such that the system of relative positions is ergodic. When the system of relative positions is stationary, we prove the coercivity condition by showing the strictly positive definiteness of an integral kernel arising in the learning. For the non-stationary case, we show that the coercivity condition holds when the time is large based on a perturbation argument.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"23 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1142/s0219493723400063
Ting Li, Hongbo Fu, Xianming Liu
This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical [Formula: see text]-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as [Formula: see text]. Also, the rate of weak convergence, which depends on [Formula: see text], for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal.
{"title":"On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric α-stable Levy processes","authors":"Ting Li, Hongbo Fu, Xianming Liu","doi":"10.1142/s0219493723400063","DOIUrl":"https://doi.org/10.1142/s0219493723400063","url":null,"abstract":"This paper deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical [Formula: see text]-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly to that of a stochastic differential equation driven by a Brownian motion in the Skorohod space as [Formula: see text]. Also, the rate of weak convergence, which depends on [Formula: see text], for the solution towards the solution of the limit equation is obtained. For illustration, the results are applied to a simple one-dimensional stochastic differential equation, which implies the rate of weak convergence is optimal.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"23 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s021949372350051x
Ran Wang, Beibei Zhang
In this paper, we study a large deviation principle for a reflected stochastic partial differential equation on infinite spatial domain. A new sufficient condition for the weak convergence criterion proposed by Matoussi, Sabbagh and Zhang [A. Matoussi, W. Sabbagh and T.-S. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim. 83(2) (2021) 849–879] plays an important role in the proof.
{"title":"A Large Deviation Principle for Reflected Spdes on Infinite Spatial Domain","authors":"Ran Wang, Beibei Zhang","doi":"10.1142/s021949372350051x","DOIUrl":"https://doi.org/10.1142/s021949372350051x","url":null,"abstract":"In this paper, we study a large deviation principle for a reflected stochastic partial differential equation on infinite spatial domain. A new sufficient condition for the weak convergence criterion proposed by Matoussi, Sabbagh and Zhang [A. Matoussi, W. Sabbagh and T.-S. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim. 83(2) (2021) 849–879] plays an important role in the proof.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"178 S437","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1142/s021949372340004x
Xiujun Cheng, Wenzhuo Xiong, Huiru Wang
In this paper, we apply classical non-uniform L1 formula and the compact difference scheme for solving linear fractional systems with Neumann boundary conditions. A novelty and simple demonstration strategy is presented on the convergence analysis in the discrete maximum norm. Moreover, based on the special properties of the resulting coefficient matrix, diagonalization technique and discrete cosine transform (DCT) are adopted to speed up the convergence rate of the proposed method. In addition, the numerical scheme is also extended to the three-dimensional (3D) case. Several numerical experiments are given to support our findings.
{"title":"A Optimal Estimate for Linear Reaction Subdiffusion Equations with Neumann Boundary Conditions","authors":"Xiujun Cheng, Wenzhuo Xiong, Huiru Wang","doi":"10.1142/s021949372340004x","DOIUrl":"https://doi.org/10.1142/s021949372340004x","url":null,"abstract":"In this paper, we apply classical non-uniform L1 formula and the compact difference scheme for solving linear fractional systems with Neumann boundary conditions. A novelty and simple demonstration strategy is presented on the convergence analysis in the discrete maximum norm. Moreover, based on the special properties of the resulting coefficient matrix, diagonalization technique and discrete cosine transform (DCT) are adopted to speed up the convergence rate of the proposed method. In addition, the numerical scheme is also extended to the three-dimensional (3D) case. Several numerical experiments are given to support our findings.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"178 S436","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-27DOI: 10.1142/s0219493723500557
Jinlong Wei, Guangying Lv
{"title":"The Impact of noise on Burgers equations","authors":"Jinlong Wei, Guangying Lv","doi":"10.1142/s0219493723500557","DOIUrl":"https://doi.org/10.1142/s0219493723500557","url":null,"abstract":"","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"15 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136312425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1142/s0219493723400026
Ting Gao, Jinqiao Duan
{"title":"Stochastic Dynamics and Data Science","authors":"Ting Gao, Jinqiao Duan","doi":"10.1142/s0219493723400026","DOIUrl":"https://doi.org/10.1142/s0219493723400026","url":null,"abstract":"","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1142/s0219493723400087
Wei Wei, Jianyu Hu, Jinqiao Duan
Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, we find that the rate function (or action functional) does not involve with the solution of the skeleton equation, which describes unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution-dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.
{"title":"The Most Likely Transition Path for a Class of Distribution-Dependent Stochastic Systems","authors":"Wei Wei, Jianyu Hu, Jinqiao Duan","doi":"10.1142/s0219493723400087","DOIUrl":"https://doi.org/10.1142/s0219493723400087","url":null,"abstract":"Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, we find that the rate function (or action functional) does not involve with the solution of the skeleton equation, which describes unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution-dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1142/s0219493723400105
Jian Ren, Jinqiao Duan
Extracting governing stochastic differential equation models from elusive data is crucial to understand and forecast dynamics for complex systems. We devise a method to extract the drift term and estimate the diffusion coefficient of a governing stochastic dynamical system, from its time-series data of the most probable transition trajectory. By the Onsager-Machlup theory, the most probable transition trajectory satisfies the corresponding Euler-Lagrange equation, which is a second order deterministic ordinary differential equation involving the drift term and diffusion coefficient. We first estimate the coefficients of the Euler-Lagrange equation based on the data of the most probable trajectory, and then we calculate the drift and diffusion coefficients of the governing stochastic dynamical system. These two steps involve sparse regression and optimization. Finally, we illustrate our method with an example and some discussions.
{"title":"Identifying stochastic governing equations from data of the most probable transition trajectories","authors":"Jian Ren, Jinqiao Duan","doi":"10.1142/s0219493723400105","DOIUrl":"https://doi.org/10.1142/s0219493723400105","url":null,"abstract":"Extracting governing stochastic differential equation models from elusive data is crucial to understand and forecast dynamics for complex systems. We devise a method to extract the drift term and estimate the diffusion coefficient of a governing stochastic dynamical system, from its time-series data of the most probable transition trajectory. By the Onsager-Machlup theory, the most probable transition trajectory satisfies the corresponding Euler-Lagrange equation, which is a second order deterministic ordinary differential equation involving the drift term and diffusion coefficient. We first estimate the coefficients of the Euler-Lagrange equation based on the data of the most probable trajectory, and then we calculate the drift and diffusion coefficients of the governing stochastic dynamical system. These two steps involve sparse regression and optimization. Finally, we illustrate our method with an example and some discussions.","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1142/s0219493723400099
Yi Wang, Qing Guo, Min Zhao, Chuanjun Dai, He Liu
{"title":"Dynamics of a Stochastic Phytoplankton-Zooplankton System with Defensive and Offensive Effects","authors":"Yi Wang, Qing Guo, Min Zhao, Chuanjun Dai, He Liu","doi":"10.1142/s0219493723400099","DOIUrl":"https://doi.org/10.1142/s0219493723400099","url":null,"abstract":"","PeriodicalId":51170,"journal":{"name":"Stochastics and Dynamics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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