The specific relative entropy, introduced in the Wiener space setting by N. Gantert, allows to quantify the discrepancy between the laws of potentially mutually singular measures. It appears naturally as the large deviations rate function in a randomized version of Donsker’s invariance principle, as well as in a novel transport-information inequality recently derived by H. Föllmer. A conjecture, put forward by the aforementioned authors, concerns a closed form expression for the specific relative entropy between continuous martingale laws in terms of their quadratic variations. We provide a first partial result in this direction, by establishing this conjecture in the case of well-behaved martingale diffusions on the line.
{"title":"On the specific relative entropy between martingale diffusions on the line","authors":"Julio Backhoff-Veraguas, Clara Unterberger","doi":"10.1214/23-ecp548","DOIUrl":"https://doi.org/10.1214/23-ecp548","url":null,"abstract":"The specific relative entropy, introduced in the Wiener space setting by N. Gantert, allows to quantify the discrepancy between the laws of potentially mutually singular measures. It appears naturally as the large deviations rate function in a randomized version of Donsker’s invariance principle, as well as in a novel transport-information inequality recently derived by H. Föllmer. A conjecture, put forward by the aforementioned authors, concerns a closed form expression for the specific relative entropy between continuous martingale laws in terms of their quadratic variations. We provide a first partial result in this direction, by establishing this conjecture in the case of well-behaved martingale diffusions on the line.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"1 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":"135955011","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":"Competition on Zd driven by branching random walk","authors":"Maria Deijfen, Timo Vilkas","doi":"10.1214/23-ecp521","DOIUrl":"https://doi.org/10.1214/23-ecp521","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49209909","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 existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.
{"title":"On the existence of weak solutions to stochastic Volterra equations","authors":"David J. Prömel, David Scheffels","doi":"10.1214/23-ecp554","DOIUrl":"https://doi.org/10.1214/23-ecp554","url":null,"abstract":"The existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"20 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":"135704896","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":"14 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":"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}
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":"28 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":"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}
{"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":" ","pages":""},"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":"165 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":"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":" ","pages":""},"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":"30 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":"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":"3 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":"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}
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