We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.
{"title":"Stochastic evolution equations with Wick-polynomial nonlinearities","authors":"T. Levajković, S. Pilipovic, D. Seleši, M. Zigic","doi":"10.1214/18-EJP241","DOIUrl":"https://doi.org/10.1214/18-EJP241","url":null,"abstract":"We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"23 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48741498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Johannes Ruf, Martin Larsson, Wouter M. Koolen, Aaditya Ramdas
{"title":"A composite generalization of Ville’s martingale theorem using e-processes","authors":"Johannes Ruf, Martin Larsson, Wouter M. Koolen, Aaditya Ramdas","doi":"10.1214/23-ejp1019","DOIUrl":"https://doi.org/10.1214/23-ejp1019","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"109 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135213398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new class of Markov chain Monte Carlo methods, called k-polar slice sampling (k-PSS), as a technical tool that interpolates between and extrapolates beyond uniform and polar slice sampling. By examining Wasserstein contraction rates and spectral gaps of k-PSS, we obtain strong quantitative results regarding its performance for different kinds of target distributions. Because k-PSS contains uniform and polar slice sampling as special cases, our results significantly advance the theoretical understanding of both of these methods. In particular, we prove realistic estimates of the convergence rates of uniform slice sampling for arbitrary multivariate Gaussian distributions on the one hand, and near-arbitrary multivariate t-distributions on the other. Furthermore, our results suggest that for heavy-tailed distributions, polar slice sampling performs dimension-independently well, whereas uniform slice sampling suffers a rather strong curse of dimensionality.
{"title":"Wasserstein contraction and spectral gap of slice sampling revisited","authors":"Philip Schär","doi":"10.1214/23-ejp1030","DOIUrl":"https://doi.org/10.1214/23-ejp1030","url":null,"abstract":"We propose a new class of Markov chain Monte Carlo methods, called k-polar slice sampling (k-PSS), as a technical tool that interpolates between and extrapolates beyond uniform and polar slice sampling. By examining Wasserstein contraction rates and spectral gaps of k-PSS, we obtain strong quantitative results regarding its performance for different kinds of target distributions. Because k-PSS contains uniform and polar slice sampling as special cases, our results significantly advance the theoretical understanding of both of these methods. In particular, we prove realistic estimates of the convergence rates of uniform slice sampling for arbitrary multivariate Gaussian distributions on the one hand, and near-arbitrary multivariate t-distributions on the other. Furthermore, our results suggest that for heavy-tailed distributions, polar slice sampling performs dimension-independently well, whereas uniform slice sampling suffers a rather strong curse of dimensionality.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"276 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135448141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the d-dimensional integers dies out almost surely at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.
{"title":"Contact process in an evolving random environment","authors":"Marco Seiler, Anja Sturm","doi":"10.1214/23-ejp1002","DOIUrl":"https://doi.org/10.1214/23-ejp1002","url":null,"abstract":"In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the d-dimensional integers dies out almost surely at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136202544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simple random walk on Z2 perturbed on the axes (renewal case)","authors":"Pierre Andreoletti, P. Debs","doi":"10.1214/23-ejp969","DOIUrl":"https://doi.org/10.1214/23-ejp969","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48018140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sharp estimates for martingale transforms with unbounded transforming sequences","authors":"T. Gałązka, A. Osȩkowski","doi":"10.1214/23-ejp953","DOIUrl":"https://doi.org/10.1214/23-ejp953","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49192598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By introducing a new type of minimality condition, this paper gives a novel approach to the reflected backward stochastic differential equations (RBSDEs) with càdlàg obstacles. Our first step is to prove the dynamic programming principles for nonlinear optimal stopping problems with g-expectations. We then use the nonlinear Doob-Meyer decomposition theorem for g-supermartingales to get the existence of the solution. With a new type of minimality condition, we prove a representation formula of solutions to RBSDEs, in an efficient way. Finally, we derive some a priori estimates and stability results.
{"title":"Dynamic programming approach to reflected backward stochastic differential equations","authors":"Hun O, Mun-Chol Kim, Kon-Gun Kim","doi":"10.1214/23-ejp999","DOIUrl":"https://doi.org/10.1214/23-ejp999","url":null,"abstract":"By introducing a new type of minimality condition, this paper gives a novel approach to the reflected backward stochastic differential equations (RBSDEs) with càdlàg obstacles. Our first step is to prove the dynamic programming principles for nonlinear optimal stopping problems with g-expectations. We then use the nonlinear Doob-Meyer decomposition theorem for g-supermartingales to get the existence of the solution. With a new type of minimality condition, we prove a representation formula of solutions to RBSDEs, in an efficient way. Finally, we derive some a priori estimates and stability results.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose–Einstein condensation phase transition is sought for. A qualitative understanding of the free energy would be helpful, but this is currently far out of reach. In this paper, we demonstrate a path towards gaining such an understanding for a simplified version of the model with deterministic boxes instead of Brownian cycles. This model is a marked Poisson point process with unbounded marks containing particles and bounded-reach interactions between the particles. Even though it is not a quantum model, it is close to that in spirit. We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step meso-macro large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks; an application of well-known level-three principles á la Georgii/Zessin is not possible because of the appearance of macro marks. The characteristic variational formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition.
相互作用的量子玻色气体是许多不同长度的布朗桥(循环)的随机集合，其中任意一对循环腿之间存在相互作用。它是一种标准的数学模型，用来证明著名的玻色-爱因斯坦凝聚相变。对自由能的定性理解是有帮助的，但目前还远远达不到。在本文中，我们展示了一种途径，以获得这种理解的模型的简化版本与确定性框而不是布朗循环。该模型是一个带有无界标记的泊松点过程，其中包含粒子和粒子之间的有界相互作用。尽管它不是一个量子模型，但它在精神上与量子模型很接近。对于任意粒子密度值的正则系综的极限自由能，我们导出了一个明确的、可解释的热力学极限变分公式。这个公式包含了模型的所有相关物理量，如微观和宏观的粒子密度，以及它们的互能和自能以及它们的熵。证明方法包括对标记泊松点过程的两步中宏大偏差方法和对小标记和大标记的明确区分;由于宏观标记的出现，不可能应用著名的三级原理 la Georgii/Zessin。特征变分公式使我们能够证明极限自由能作为粒子密度函数的一些性质，如可微性和明确的上下边界，以及低于和高于临界阈值的定性图像(如果它是有限的)。这证明了相变的一种修正的饱和性质。然而，我们还没有成功地证明这种相变的存在。
{"title":"A micro-macro variational formula for the free energy of a many-body system with unbounded marks","authors":"Orphée Collin, Benedikt Jahnel, Wolfgang König","doi":"10.1214/23-ejp1014","DOIUrl":"https://doi.org/10.1214/23-ejp1014","url":null,"abstract":"The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose–Einstein condensation phase transition is sought for. A qualitative understanding of the free energy would be helpful, but this is currently far out of reach. In this paper, we demonstrate a path towards gaining such an understanding for a simplified version of the model with deterministic boxes instead of Brownian cycles. This model is a marked Poisson point process with unbounded marks containing particles and bounded-reach interactions between the particles. Even though it is not a quantum model, it is close to that in spirit. We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step meso-macro large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks; an application of well-known level-three principles á la Georgii/Zessin is not possible because of the appearance of macro marks. The characteristic variational formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135212403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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：