It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.
{"title":"An entropic proof of cutoff on Ramanujan graphs","authors":"N. Ozawa","doi":"10.1214/20-ecp358","DOIUrl":"https://doi.org/10.1214/20-ecp358","url":null,"abstract":"It is recently proved by Lubetzky and Peres that the simple random walk on a Ramanujan graph exhibits a cutoff phenomenon, that is to say, the total variation distance of the random walk distribution from the uniform distribution drops abruptly from near $1$ to near $0$. There are already a few alternative proofs of this fact. In this note, we give yet another proof based on functional analysis and entropic consideration.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87906861","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 provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.
{"title":"Couplings, generalized couplings and uniqueness of invariant measures","authors":"M. Scheutzow","doi":"10.1214/20-ecp363","DOIUrl":"https://doi.org/10.1214/20-ecp363","url":null,"abstract":"We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provide an example showing that uniqueness can fail if the state space is separable and metric (but not Polish) even though a coupling defined via a continuous and positive definite function exists.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75676697","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 recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.
{"title":"Constructive approach to limit theorems for recurrent diffusive random walks on a strip","authors":"D. Dolgopyat, I. Goldsheid","doi":"10.3233/asy-201619","DOIUrl":"https://doi.org/10.3233/asy-201619","url":null,"abstract":"We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77402413","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}
Consider the parabolic Anderson model $partial_tu=frac{1}{2}partial_x^2u+u, eta$ on the interval $[0, L]$ with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise $eta$. Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral $int_0^Lu(t,, x), mathrm{d} x$ as $L$ tends to infinity, where the limiting Gaussian distribution is independent of the choice of the boundary conditions and coincides with the Gaussian fluctuation for the spatial average of parabolic Anderson model on the whole space $mathbb{R}$.
{"title":"Gaussian fluctuation for spatial average of parabolic Anderson model with Neumann/Dirichlet/periodic boundary conditions","authors":"Fei Pu","doi":"10.1090/tran/8565","DOIUrl":"https://doi.org/10.1090/tran/8565","url":null,"abstract":"Consider the parabolic Anderson model $partial_tu=frac{1}{2}partial_x^2u+u, eta$ on the interval $[0, L]$ with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise $eta$. Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral $int_0^Lu(t,, x), mathrm{d} x$ as $L$ tends to infinity, where the limiting Gaussian distribution is independent of the choice of the boundary conditions and coincides with the Gaussian fluctuation for the spatial average of parabolic Anderson model on the whole space $mathbb{R}$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76508893","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 show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the KPZ equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of cite{wu},cite{DM20}, the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with $|h(x)|le C(1+sqrt{|x|})$ for some $C>0$, or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts. The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates. Just before posting the first version of this article, we learned that, emph{independently and at the same time and place}, Balint Virag found a completely different proof of the convergence of the KPZ equation to the KPZ fixed point. The methods invite extensions in different directions and it will be very interesting to see how this plays out.
{"title":"Convergence of exclusion processes and the KPZ equation to the KPZ fixed point","authors":"J. Quastel, S. Sarkar","doi":"10.1090/jams/999","DOIUrl":"https://doi.org/10.1090/jams/999","url":null,"abstract":"We show that under the 1:2:3 scaling, critically probing large space and time, the height function of finite range asymmetric exclusion processes and the KPZ equation converge to the KPZ fixed point, constructed earlier as a limit of the totally asymmetric simple exclusion process through exact formulas. Consequently, based on recent results of cite{wu},cite{DM20}, the KPZ line ensemble converges to the Airy line ensemble. For the KPZ equation we are able to start from a continuous function plus a finite collection of narrow wedges. For nearest neighbour exclusions, we can take (discretizations) of continuous functions with $|h(x)|le C(1+sqrt{|x|})$ for some $C>0$, or one narrow wedge. For non-nearest neighbour exclusions, we are restricted at the present time to a class of (random) initial data, dense in continuous functions in the topology of uniform convergence on compacts. The method is by comparison of the transition probabilities of finite range exclusion processes and the totally asymmetric simple exclusion processes using energy estimates. Just before posting the first version of this article, we learned that, emph{independently and at the same time and place}, Balint Virag found a completely different proof of the convergence of the KPZ equation to the KPZ fixed point. The methods invite extensions in different directions and it will be very interesting to see how this plays out.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77165693","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}
Pub Date : 2020-08-13DOI: 10.1007/978-3-030-60754-8_9
M. Cabezas, L. Rolla
{"title":"Avalanches in Critical Activated Random Walks","authors":"M. Cabezas, L. Rolla","doi":"10.1007/978-3-030-60754-8_9","DOIUrl":"https://doi.org/10.1007/978-3-030-60754-8_9","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"92 1","pages":"187-205"},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86173245","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 solve multidimensional SDEs with distributional drift driven by symmetric, $alpha$-stable Levy processes for $alphain (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2alpha)/3$, and in particular we go beyond the by now well understood "Young regime", where the drift must have better regularity than $(1-alpha)/2$. The analysis of the Kolmogorov backward equation in the low regularity regime is based on paracontrolled distributions. As an application of our results we construct a Brox diffusion with Levy noise. Keywords: Singular diffusions, stable Levy noise, distributional drift, paracontrolled distributions, Brox diffusion
{"title":"Multidimensional SDE with distributional drift and Lévy noise","authors":"Helena Kremp, Nicolas Perkowski","doi":"10.3150/21-bej1394","DOIUrl":"https://doi.org/10.3150/21-bej1394","url":null,"abstract":"We solve multidimensional SDEs with distributional drift driven by symmetric, $alpha$-stable Levy processes for $alphain (1,2]$ by studying the associated (singular) martingale problem and by solving the Kolmogorov backward equation. We allow for drifts of regularity $(2-2alpha)/3$, and in particular we go beyond the by now well understood \"Young regime\", where the drift must have better regularity than $(1-alpha)/2$. The analysis of the Kolmogorov backward equation in the low regularity regime is based on paracontrolled distributions. As an application of our results we construct a Brox diffusion with Levy noise. \u0000Keywords: Singular diffusions, stable Levy noise, distributional drift, paracontrolled distributions, Brox diffusion","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"133 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85613169","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 a Poisson equation in $mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Holder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable, and does not depend on their regularities with respect to the fast component.
{"title":"Diffusion approximation for fully coupled stochastic differential equations","authors":"M. Rockner, Longjie Xie","doi":"10.1214/20-AOP1475","DOIUrl":"https://doi.org/10.1214/20-AOP1475","url":null,"abstract":"We consider a Poisson equation in $mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Holder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with respect to the slow variable, and does not depend on their regularities with respect to the fast component.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77409457","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}
Pub Date : 2020-08-08DOI: 10.1016/J.SPL.2021.109073
A. Frolov
{"title":"On upper and lower bounds for probabilities of combinations of events","authors":"A. Frolov","doi":"10.1016/J.SPL.2021.109073","DOIUrl":"https://doi.org/10.1016/J.SPL.2021.109073","url":null,"abstract":"","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82273322","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}
D. Aldous, P. Caputo, R. Durrett, A. Holroyd, Paul Jung, Amber L. Puha
Thomas Milton Liggett was a world renowned UCLA probabilist, famous for his monograph Interacting Particle Systems. He passed away peacefully on May 12, 2020. This is a perspective article in memory of both Tom Liggett the person and Tom Liggett the mathematician.
{"title":"The Life and Mathematical Legacy of Thomas M. Liggett","authors":"D. Aldous, P. Caputo, R. Durrett, A. Holroyd, Paul Jung, Amber L. Puha","doi":"10.1090/noti2203","DOIUrl":"https://doi.org/10.1090/noti2203","url":null,"abstract":"Thomas Milton Liggett was a world renowned UCLA probabilist, famous for his monograph Interacting Particle Systems. He passed away peacefully on May 12, 2020. This is a perspective article in memory of both Tom Liggett the person and Tom Liggett the mathematician.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74521815","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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