We study the empirical process arising from a multi-dimensional diffusion process with periodic drift and diffusivity. The smoothing properties of the generator of the diffusion are exploited to prove the Donsker property for certain classes of smooth functions. We partially generalise the finding from the one-dimensional case studied in [29]: that the diffusion empirical process exhibits stronger regularity than in the classical case of i.i.d. observations. As an application, precise asymptotics are deduced for the Wasserstein-1 distance between the time-T occupation measure and the invariant measure in dimensions d≤3.
{"title":"Donsker theorems for occupation measures of multi-dimensional periodic diffusions","authors":"Neil Deo","doi":"10.1214/23-ecp547","DOIUrl":"https://doi.org/10.1214/23-ecp547","url":null,"abstract":"We study the empirical process arising from a multi-dimensional diffusion process with periodic drift and diffusivity. The smoothing properties of the generator of the diffusion are exploited to prove the Donsker property for certain classes of smooth functions. We partially generalise the finding from the one-dimensional case studied in [29]: that the diffusion empirical process exhibits stronger regularity than in the classical case of i.i.d. observations. As an application, precise asymptotics are deduced for the Wasserstein-1 distance between the time-T occupation measure and the invariant measure in dimensions d≤3.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136003268","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}
We establish a Chung-type law of the iterated logarithm for the solutions of a class of stochastic heat equations driven by a multiplicative noise whose coefficient depends on the solution, and this dependence takes us away from Gaussian setting. Based on the literature on small ball probabilities and the technique of freezing coefficients, the limiting constant in Chung’s law of the iterated logarithm can be evaluated almost surely.
{"title":"Chung’s law of the iterated logarithm for a class of stochastic heat equations","authors":"Jiaming Chen","doi":"10.1214/23-ecp542","DOIUrl":"https://doi.org/10.1214/23-ecp542","url":null,"abstract":"We establish a Chung-type law of the iterated logarithm for the solutions of a class of stochastic heat equations driven by a multiplicative noise whose coefficient depends on the solution, and this dependence takes us away from Gaussian setting. Based on the literature on small ball probabilities and the technique of freezing coefficients, the limiting constant in Chung’s law of the iterated logarithm can be evaluated almost surely.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135955014","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}
{"title":"A note on the α-Sun distribution","authors":"T. Simon","doi":"10.1214/23-ecp526","DOIUrl":"https://doi.org/10.1214/23-ecp526","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44969643","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}
We find uniform lower bounds on the drift for a large family of random walks on graph products, of the form P(|Zn|≤κn)≤e−κn for κ>0. This includes the simple random walk for a right-angled Artin group with a sparse defining graph. This is done by extending an argument of Gouëzel, along with the combinatorial notion of a piling introduced by Crisp, Godelle, and Wiest. We do not use any moment conditions, instead considering random walks which alternate between one measure uniformly distributed on vertex groups, and another measure over which we make no assumptions.
{"title":"Effective drift estimates for random walks on graph products","authors":"Kunal Chawla","doi":"10.1214/23-ecp546","DOIUrl":"https://doi.org/10.1214/23-ecp546","url":null,"abstract":"We find uniform lower bounds on the drift for a large family of random walks on graph products, of the form P(|Zn|≤κn)≤e−κn for κ>0. This includes the simple random walk for a right-angled Artin group with a sparse defining graph. This is done by extending an argument of Gouëzel, along with the combinatorial notion of a piling introduced by Crisp, Godelle, and Wiest. We do not use any moment conditions, instead considering random walks which alternate between one measure uniformly distributed on vertex groups, and another measure over which we make no assumptions.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953415","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}
In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_t^q-L_x^p$ space. Contrary to the large deviation principle approach recently proposed in [2], the main ingredient of the proof here are the textit{Partial Girsanov transformations} introduced in [3] and developed in a general setting in this work.
{"title":"Propagation of chaos for stochastic particle systems with singular mean-field interaction of Lq−Lp type","authors":"Milica Tomavsevi'c","doi":"10.1214/23-ecp539","DOIUrl":"https://doi.org/10.1214/23-ecp539","url":null,"abstract":"In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_t^q-L_x^p$ space. Contrary to the large deviation principle approach recently proposed in [2], the main ingredient of the proof here are the textit{Partial Girsanov transformations} introduced in [3] and developed in a general setting in this work.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49365044","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}
Let BH be a fractional Brownian motion on R with Hurst parameter H∈(0,1) and let F be its pathwise antiderivative (so F is a differentiable random function such that F′(x)=BxH) with F(0)=0. Let B be a standard Brownian motion, independent of BH. We show that the zero energy part At=F(Bt)−∫0tF′(Bs)dBs of F(B) has positive and finite p-th variation in a special sense for p0=2 1+H. We also present some simulation results about the zero energy part of a certain median process which suggest that its 4∕3-th variation is positive and finite.
{"title":"Example of a Dirichlet process whose zero energy part has finite p-th variation","authors":"László Bondici, Vilmos Prokaj","doi":"10.1214/23-ecp558","DOIUrl":"https://doi.org/10.1214/23-ecp558","url":null,"abstract":"Let BH be a fractional Brownian motion on R with Hurst parameter H∈(0,1) and let F be its pathwise antiderivative (so F is a differentiable random function such that F′(x)=BxH) with F(0)=0. Let B be a standard Brownian motion, independent of BH. We show that the zero energy part At=F(Bt)−∫0tF′(Bs)dBs of F(B) has positive and finite p-th variation in a special sense for p0=2 1+H. We also present some simulation results about the zero energy part of a certain median process which suggest that its 4∕3-th variation is positive and finite.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704682","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}
We prove that a supercritical branching random walk on a transient Markov chain converges almost surely under rescaling to a random measure on the Martin boundary of the underlying Markov chain. Several open problems and conjectures about this limiting measure are presented.
{"title":"On the boundary at infinity for branching random walk","authors":"Elisabetta Candellero, Tom Hutchcroft","doi":"10.1214/23-ecp560","DOIUrl":"https://doi.org/10.1214/23-ecp560","url":null,"abstract":"We prove that a supercritical branching random walk on a transient Markov chain converges almost surely under rescaling to a random measure on the Martin boundary of the underlying Markov chain. Several open problems and conjectures about this limiting measure are presented.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704893","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}
In this paper, it is shown that $alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.
{"title":"A note on α-permanent and loop soup","authors":"Xiaodan Li, Yushu Zheng","doi":"10.1214/23-ECP530","DOIUrl":"https://doi.org/10.1214/23-ECP530","url":null,"abstract":"In this paper, it is shown that $alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47950495","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}
The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.
{"title":"Investigation of Airy equations with random initial conditions","authors":"L. Sakhno","doi":"10.1214/23-ecp522","DOIUrl":"https://doi.org/10.1214/23-ecp522","url":null,"abstract":"The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48209881","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}
The Hierarchical Dirichlet process is a discrete random measure serving as an im-portant prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.
{"title":"Hierarchical Dirichlet process and relative entropy","authors":"S. Feng","doi":"10.1214/23-ecp511","DOIUrl":"https://doi.org/10.1214/23-ecp511","url":null,"abstract":"The Hierarchical Dirichlet process is a discrete random measure serving as an im-portant prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43022214","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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