Henri Berestycki, Alexei Novikov, Jean-Michel Roquejoffre, Lenya Ryzhik
The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.
{"title":"Diffusion of knowledge and the lottery society","authors":"Henri Berestycki, Alexei Novikov, Jean-Michel Roquejoffre, Lenya Ryzhik","doi":"arxiv-2409.11479","DOIUrl":"https://doi.org/arxiv-2409.11479","url":null,"abstract":"The Lucas-Moll system is a mean-field game type model describing the growth\u0000of an economy by means of diffusion of knowledge. The individual agents in the economy advance their\u0000knowledge by learning from each other and via internal innovation. Their\u0000cumulative distribution function satisfies a forward in time nonlinear\u0000non-local reaction-diffusion type equation. On the other hand, the learning\u0000strategy of the agents is based on the solution to a backward in time nonlocal\u0000Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation\u0000for the agents density. Together, these equations form a system of the\u0000mean-field game type. When the learning rate is sufficiently large, existence\u0000of balanced growth path solutions to the Lucas-Moll system was proved\u0000in~cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the\u0000balanced growth paths do not exist. The main result is a long time convergence\u0000theorem. Namely, the solution to the initial-terminal value problem behaves in\u0000such a way that at large times an overwhelming majority of the agents spend no\u0000time producing at all and are only learning. In particular, the agents density\u0000propagates at the Fisher-KPP speed. We name this type of solutions a lottery\u0000society.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"196 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262313","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 stochastic reaction-diffusion equation in $1+1$ dimensions driven by multiplicative space-time white noise, with a distributional drift belonging to a Besov-H"older space with any regularity index larger than $-1$. We assume that the diffusion coefficient is a regular function which is bounded away from zero. By using a combination of stochastic sewing techniques and Malliavin calculus, we show that the equation admits a unique solution.
{"title":"Regularisation by multiplicative noise for reaction-diffusion equations","authors":"Konstantinos Dareiotis, Teodor Holland, Khoa Lê","doi":"arxiv-2409.11130","DOIUrl":"https://doi.org/arxiv-2409.11130","url":null,"abstract":"We consider the stochastic reaction-diffusion equation in $1+1$ dimensions\u0000driven by multiplicative space-time white noise, with a distributional drift\u0000belonging to a Besov-H\"older space with any regularity index larger than $-1$.\u0000We assume that the diffusion coefficient is a regular function which is bounded\u0000away from zero. By using a combination of stochastic sewing techniques and\u0000Malliavin calculus, we show that the equation admits a unique solution.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262315","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}
Bony's paraproduct is one of the main tools in the theory of paracontrolled calculus. The paraproduct is usually defined via Fourier analysis, so it is not a local operator. In the previous researches [7, 8], however, the author proved that the pointwise estimate like (1.2) holds for the paraproduct and its iterated versions when the sum of the regularities is smaller than 1. The aim of this article is to extend these results for higher regularities.
{"title":"A note on the Taylor estimates of iterated paraproducts","authors":"Masato Hoshino","doi":"arxiv-2409.10817","DOIUrl":"https://doi.org/arxiv-2409.10817","url":null,"abstract":"Bony's paraproduct is one of the main tools in the theory of paracontrolled\u0000calculus. The paraproduct is usually defined via Fourier analysis, so it is not\u0000a local operator. In the previous researches [7, 8], however, the author proved\u0000that the pointwise estimate like (1.2) holds for the paraproduct and its\u0000iterated versions when the sum of the regularities is smaller than 1. The aim\u0000of this article is to extend these results for higher regularities.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262321","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}
The Potts spin glass is an analogue of the Sherrington-Kirkpatrick model in which each spin can take one of $kappa$ possible values, which we interpret as colors. It was suggested in arXiv:2310.06745 that the order parameter for this model is always invariant with respect to permutations of the colors. We show here that this is false whenever $kappa ge 58$.
波茨自旋玻璃是谢林顿-柯克帕特里克（Sherrington-Kirkpatrick）模型的一个类似物，其中每个自旋都可以取$kappa$可能值中的一个，我们将其解释为颜色。有人在 arXiv:2310.06745 中提出，这个模型的阶次参数在颜色的排列上总是不变的。我们在这里证明，只要 $kappa ge 58$，这就是假的。
{"title":"Color symmetry breaking in the Potts spin glass","authors":"Jean-Christophe Mourrat","doi":"arxiv-2409.10437","DOIUrl":"https://doi.org/arxiv-2409.10437","url":null,"abstract":"The Potts spin glass is an analogue of the Sherrington-Kirkpatrick model in\u0000which each spin can take one of $kappa$ possible values, which we interpret as\u0000colors. It was suggested in arXiv:2310.06745 that the order parameter for this\u0000model is always invariant with respect to permutations of the colors. We show\u0000here that this is false whenever $kappa ge 58$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262323","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 study the statistics of the Mallows measure on permutations in the limit pioneered by Starr (2009). Our main result is the local central limit theorem for its height function. We also re-derive versions of the law of large numbers and the large deviation principle, obtain the standard central limit theorem from the local one, and establish a multi-point version of the local central limit theorem.
{"title":"Local central limit theorem for Mallows measure","authors":"Alexey Bufetov, Kailun Chen","doi":"arxiv-2409.10415","DOIUrl":"https://doi.org/arxiv-2409.10415","url":null,"abstract":"We study the statistics of the Mallows measure on permutations in the limit\u0000pioneered by Starr (2009). Our main result is the local central limit theorem\u0000for its height function. We also re-derive versions of the law of large numbers\u0000and the large deviation principle, obtain the standard central limit theorem\u0000from the local one, and establish a multi-point version of the local central\u0000limit theorem.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262325","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}
If one seeks to estimate the total variation between two product measures $||P^otimes_{1:n}-Q^otimes_{1:n}||$ in terms of their marginal TV sequence $delta=(||P_1-Q_1||,||P_2-Q_2||,ldots,||P_n-Q_n||)$, then trivial upper and lower bounds are provided by$ ||delta||_infty le ||P^otimes_{1:n}-Q^otimes_{1:n}||le||delta||_1$. We improve the lower bound to $||delta||_2lesssim||P^otimes_{1:n}-Q^otimes_{1:n}||$, thereby reducing the gap between the upper and lower bounds from $sim n$ to $simsqrt $. Furthermore, we show that {em any} estimate on $||P^otimes_{1:n}-Q^otimes_{1:n}||$ expressed in terms of $delta$ must necessarily exhibit a gap of $simsqrt n$ between the upper and lower bounds in the worst case, establishing a sense in which our estimate is optimal. Finally, we identify a natural class of distributions for which $||delta||_2$ approximates the TV distance up to absolute multiplicative constants.
{"title":"On the tensorization of the variational distance","authors":"Aryeh Kontorovich","doi":"arxiv-2409.10368","DOIUrl":"https://doi.org/arxiv-2409.10368","url":null,"abstract":"If one seeks to estimate the total variation between two product measures\u0000$||P^otimes_{1:n}-Q^otimes_{1:n}||$ in terms of their marginal TV sequence\u0000$delta=(||P_1-Q_1||,||P_2-Q_2||,ldots,||P_n-Q_n||)$, then trivial upper and\u0000lower bounds are provided by$ ||delta||_infty le\u0000||P^otimes_{1:n}-Q^otimes_{1:n}||le||delta||_1$. We improve the lower bound\u0000to $||delta||_2lesssim||P^otimes_{1:n}-Q^otimes_{1:n}||$, thereby reducing\u0000the gap between the upper and lower bounds from $sim n$ to $simsqrt $.\u0000Furthermore, we show that {em any} estimate on\u0000$||P^otimes_{1:n}-Q^otimes_{1:n}||$ expressed in terms of $delta$ must\u0000necessarily exhibit a gap of $simsqrt n$ between the upper and lower bounds\u0000in the worst case, establishing a sense in which our estimate is optimal.\u0000Finally, we identify a natural class of distributions for which $||delta||_2$\u0000approximates the TV distance up to absolute multiplicative constants.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262326","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}
For the Moran model with strong or moderately strong selection we prove that the fluctuations around the deterministic limit of the line counting process of the ancestral selection graph converges to an Ornstein-Uhlenbeck process. To this purpose we provide an extension of a functional limit theorem by Ethier and Kurtz 1986. This result and a small adaptation of our arguments can also be used to obtain the scaling limit for the fluctuations of certain logistic branching processes.
{"title":"Ornstein-Uhlenbeck fluctuations for the line counting process of the ancestral selection graph","authors":"Florin Boenkost, Anna-Lena Weinel","doi":"arxiv-2409.10360","DOIUrl":"https://doi.org/arxiv-2409.10360","url":null,"abstract":"For the Moran model with strong or moderately strong selection we prove that\u0000the fluctuations around the deterministic limit of the line counting process of\u0000the ancestral selection graph converges to an Ornstein-Uhlenbeck process. To\u0000this purpose we provide an extension of a functional limit theorem by Ethier\u0000and Kurtz 1986. This result and a small adaptation of our arguments can also be\u0000used to obtain the scaling limit for the fluctuations of certain logistic\u0000branching processes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262493","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 study the half-space TASEP with a reservoir at the origin. We solve the model for a general deterministic initial condition. Taking the 1:2:3 KPZ scaling, we derive the transition probability for the half-space KPZ fixed point.
{"title":"TASEP in half-space","authors":"Xincheng Zhang","doi":"arxiv-2409.09974","DOIUrl":"https://doi.org/arxiv-2409.09974","url":null,"abstract":"We study the half-space TASEP with a reservoir at the origin. We solve the\u0000model for a general deterministic initial condition. Taking the 1:2:3 KPZ\u0000scaling, we derive the transition probability for the half-space KPZ fixed\u0000point.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262497","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 establish a log-Sobolev inequality for the stationary distribution of mean-field Langevin dynamics with a constant that is independent of the number of particles $N$. Our proof proceeds by establishing the existence of a Lipschitz transport map from the standard Gaussian measure via the reverse heat flow of Kim and Milman.
我们为平均场朗格文动力学的静态分布建立了一个对数-索博列夫不等式，其常数与粒子数 $N$ 无关。我们的证明是通过 Kim 和 Milman 的反向热流，从标准高斯量度建立利普希兹传输映射的存在性。
{"title":"Uniform-in-$N$ log-Sobolev inequality for the mean-field Langevin dynamics with convex energy","authors":"Sinho Chewi, Atsushi Nitanda, Matthew S. Zhang","doi":"arxiv-2409.10440","DOIUrl":"https://doi.org/arxiv-2409.10440","url":null,"abstract":"We establish a log-Sobolev inequality for the stationary distribution of\u0000mean-field Langevin dynamics with a constant that is independent of the number\u0000of particles $N$. Our proof proceeds by establishing the existence of a\u0000Lipschitz transport map from the standard Gaussian measure via the reverse heat\u0000flow of Kim and Milman.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262324","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}
Subordinators are infinitely divisible distributions on the positive half-line. They are often used as mixing distributions in Poisson mixtures. We show that appropriately scaled Poisson mixtures can approximate the mixing subordinator and we derive a rate of convergence in $L^p$ for each $pin[1,infty]$. This includes the Kolmogorov and Wasserstein metrics as special cases. As an application, we develop an approach for approximate simulation of the underlying subordinator. In the interest of generality, we present our results in the context of more general mixtures, specifically those that can be represented as differences of randomly stopped L'evy processes. Particular focus is given to the case where the subordinator belongs to the class of tempered stable distributions.
{"title":"On Approximations of Subordinators in $L^p$ and the Simulation of Tempered Stable Distributions","authors":"Michael Grabchak, Sina Saba","doi":"arxiv-2409.09909","DOIUrl":"https://doi.org/arxiv-2409.09909","url":null,"abstract":"Subordinators are infinitely divisible distributions on the positive\u0000half-line. They are often used as mixing distributions in Poisson mixtures. We\u0000show that appropriately scaled Poisson mixtures can approximate the mixing\u0000subordinator and we derive a rate of convergence in $L^p$ for each\u0000$pin[1,infty]$. This includes the Kolmogorov and Wasserstein metrics as\u0000special cases. As an application, we develop an approach for approximate\u0000simulation of the underlying subordinator. In the interest of generality, we\u0000present our results in the context of more general mixtures, specifically those\u0000that can be represented as differences of randomly stopped L'evy processes.\u0000Particular focus is given to the case where the subordinator belongs to the\u0000class of tempered stable distributions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262499","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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