This paper develops a non-asymptotic approach to mean field approximations for systems of $n$ diffusive particles interacting pairwise. The interaction strengths are not identical, making the particle system non-exchangeable. The marginal law of any subset of particles is compared to a suitably chosen product measure, and we find sharp relative entropy estimates between the two. Building upon prior work of the first author in the exchangeable setting, we use a generalized form of the BBGKY hierarchy to derive a hierarchy of differential inequalities for the relative entropies. Our analysis of this complicated hierarchy exploits an unexpected but crucial connection with first-passage percolation, which lets us bound the marginal entropies in terms of expectations of functionals of this percolation process.
{"title":"Quantitative propagation of chaos for non-exchangeable diffusions via first-passage percolation","authors":"Daniel Lacker, Lane Chun Yeung, Fuzhong Zhou","doi":"arxiv-2409.08882","DOIUrl":"https://doi.org/arxiv-2409.08882","url":null,"abstract":"This paper develops a non-asymptotic approach to mean field approximations\u0000for systems of $n$ diffusive particles interacting pairwise. The interaction\u0000strengths are not identical, making the particle system non-exchangeable. The\u0000marginal law of any subset of particles is compared to a suitably chosen\u0000product measure, and we find sharp relative entropy estimates between the two.\u0000Building upon prior work of the first author in the exchangeable setting, we\u0000use a generalized form of the BBGKY hierarchy to derive a hierarchy of\u0000differential inequalities for the relative entropies. Our analysis of this\u0000complicated hierarchy exploits an unexpected but crucial connection with\u0000first-passage percolation, which lets us bound the marginal entropies in terms\u0000of expectations of functionals of this percolation process.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262514","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}
Using Stein's method and a Gaussian integration by parts, we provide a direct proof of the known fact that drifted Brownian motions are invariant measures (modulo height) for the KPZ equation.
{"title":"Integration by parts and invariant measure for KPZ","authors":"Yu Gu, Jeremy Quastel","doi":"arxiv-2409.08465","DOIUrl":"https://doi.org/arxiv-2409.08465","url":null,"abstract":"Using Stein's method and a Gaussian integration by parts, we provide a direct\u0000proof of the known fact that drifted Brownian motions are invariant measures\u0000(modulo height) for the KPZ equation.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262705","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 $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an $Ntimes N$ torus with the height at the origin rooted at $0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large inverse-temperature $beta$, vs. $asymp log |x|$ at small $beta$ (as in the Gaussian free field (GFF)). The former--rigidity at large $beta$--is known for a wide class of $|nablaphi|^p$ models ($p=1$ being SOS) yet is believed to fail once the surface is on a slope (tilted boundary conditions). It is conjectured that the slope would destabilize the rigidity and induce the GFF-type behavior of the surface at small $beta$. The only rigorous result on this is by Sheffield (2005): for these models of integer height functions, if the slope $theta$ is irrational, then Var$(h(x))toinfty$ with $|x|$ (with no known quantitative bound). We study a family of SOS surfaces at a large enough fixed $beta$, on an $Ntimes N$ torus with a nonzero boundary condition slope $theta$, perturbed by a potential $V$ of strength $epsilon_beta$ per site (arbitrarily small). Our main result is (a) the measure on the height gradients $nabla h$ has a weak limit $mu_infty$ as $Ntoinfty$; and (b) the scaling limit of a sample from $mu_infty$ converges to a full plane GFF. In particular, we recover the asymptotics Var$(h(x))sim clog|x|$. To our knowledge, this is the first example of a tilted $|nablaphi|^p$ model, or a perturbation thereof, where the limit is recovered at large $beta$. The proof looks at random monotone surfaces that approximate the SOS surface, and shows that (i) these form a weakly interacting dimer model, and (ii) the renormalization framework of Giuliani, Mastropietro and Toninelli (2017) leads to the GFF limit. New ingredients are needed in both parts, including a nontrivial extension of [GMT17] from finite interactions to any long range summable interactions.
{"title":"Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope","authors":"Benoît Laslier, Eyal Lubetzky","doi":"arxiv-2409.08745","DOIUrl":"https://doi.org/arxiv-2409.08745","url":null,"abstract":"The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening\u0000transition: on an $Ntimes N$ torus with the height at the origin rooted at\u0000$0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large\u0000inverse-temperature $beta$, vs. $asymp log |x|$ at small $beta$ (as in the\u0000Gaussian free field (GFF)). The former--rigidity at large $beta$--is known for\u0000a wide class of $|nablaphi|^p$ models ($p=1$ being SOS) yet is believed to\u0000fail once the surface is on a slope (tilted boundary conditions). It is\u0000conjectured that the slope would destabilize the rigidity and induce the\u0000GFF-type behavior of the surface at small $beta$. The only rigorous result on\u0000this is by Sheffield (2005): for these models of integer height functions, if\u0000the slope $theta$ is irrational, then Var$(h(x))toinfty$ with $|x|$ (with no\u0000known quantitative bound). We study a family of SOS surfaces at a large enough fixed $beta$, on an\u0000$Ntimes N$ torus with a nonzero boundary condition slope $theta$, perturbed\u0000by a potential $V$ of strength $epsilon_beta$ per site (arbitrarily small).\u0000Our main result is (a) the measure on the height gradients $nabla h$ has a\u0000weak limit $mu_infty$ as $Ntoinfty$; and (b) the scaling limit of a sample\u0000from $mu_infty$ converges to a full plane GFF. In particular, we recover the\u0000asymptotics Var$(h(x))sim clog|x|$. To our knowledge, this is the first\u0000example of a tilted $|nablaphi|^p$ model, or a perturbation thereof, where\u0000the limit is recovered at large $beta$. The proof looks at random monotone\u0000surfaces that approximate the SOS surface, and shows that (i) these form a\u0000weakly interacting dimer model, and (ii) the renormalization framework of\u0000Giuliani, Mastropietro and Toninelli (2017) leads to the GFF limit. New\u0000ingredients are needed in both parts, including a nontrivial extension of\u0000[GMT17] from finite interactions to any long range summable interactions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262516","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}
Stationary measures of last passage percolation with geometric weights and the log-gamma polymer in a strip of the $mathbb Z^2$ lattice are characterized in arXiv:2306.05983 using variants of Schur and Whittaker processes, called two-layer Gibbs measures. In this article, we prove contour integral formulas characterizing the multipoint joint distribution of two-layer Schur and Whittaker processes. We also express them as Doob transformed Markov processes with explicit transition kernels. As an example of application of our formulas, we compute the growth rate of the KPZ equation on $[0,L]$ with arbitrary boundary parameters.
{"title":"Integral formulas for two-layer Schur and Whittaker processes","authors":"Guillaume Barraquand","doi":"arxiv-2409.08927","DOIUrl":"https://doi.org/arxiv-2409.08927","url":null,"abstract":"Stationary measures of last passage percolation with geometric weights and\u0000the log-gamma polymer in a strip of the $mathbb Z^2$ lattice are characterized\u0000in arXiv:2306.05983 using variants of Schur and Whittaker processes, called\u0000two-layer Gibbs measures. In this article, we prove contour integral formulas\u0000characterizing the multipoint joint distribution of two-layer Schur and\u0000Whittaker processes. We also express them as Doob transformed Markov processes\u0000with explicit transition kernels. As an example of application of our formulas,\u0000we compute the growth rate of the KPZ equation on $[0,L]$ with arbitrary\u0000boundary parameters.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262513","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}
Alekos Cecchin, Giovanni Conforti, Alain Durmus, Katharina Eichinger
This article aims at quantifying the long time behavior of solutions of mean field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov control. Our main contribution is to show well-posedness of the ergodic problem and the exponential turnpike property of dynamic optimizers, which implies exponential convergence to equilibrium for both optimal states and controls to their ergodic counterparts. In contrast with previous works that require some version of the Lasry-Lions monotonicity condition, our main assumption is a weak form of asymptotic monotonicity on the drift of the controlled dynamics and some basic regularity and smallness conditions on the interaction terms. Our proof strategy is probabilistic and based on the construction of contractive couplings between controlled processes and forward-backward stochastic differential equations. The flexibility of the coupling approach allows us to cover several interesting situations. For example, we do not need to restrict ourselves to compact domains and can work on the whole space $mathbb{R}^d$, we can cover the case of non-constant diffusion coefficients and we can sometimes show turnpike estimates for the hessians of solutions to the backward equation.
{"title":"The exponential turnpike phenomenon for mean field game systems: weakly monotone drifts and small interactions","authors":"Alekos Cecchin, Giovanni Conforti, Alain Durmus, Katharina Eichinger","doi":"arxiv-2409.09193","DOIUrl":"https://doi.org/arxiv-2409.09193","url":null,"abstract":"This article aims at quantifying the long time behavior of solutions of mean\u0000field PDE systems arising in the theory of Mean Field Games and McKean-Vlasov\u0000control. Our main contribution is to show well-posedness of the ergodic problem\u0000and the exponential turnpike property of dynamic optimizers, which implies\u0000exponential convergence to equilibrium for both optimal states and controls to\u0000their ergodic counterparts. In contrast with previous works that require some\u0000version of the Lasry-Lions monotonicity condition, our main assumption is a\u0000weak form of asymptotic monotonicity on the drift of the controlled dynamics\u0000and some basic regularity and smallness conditions on the interaction terms.\u0000Our proof strategy is probabilistic and based on the construction of\u0000contractive couplings between controlled processes and forward-backward\u0000stochastic differential equations. The flexibility of the coupling approach\u0000allows us to cover several interesting situations. For example, we do not need\u0000to restrict ourselves to compact domains and can work on the whole space\u0000$mathbb{R}^d$, we can cover the case of non-constant diffusion coefficients\u0000and we can sometimes show turnpike estimates for the hessians of solutions to\u0000the backward equation.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262506","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}
Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg
Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-rho$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $rho geq sqrt{2}$. We show that in the subcritical case when $rho > sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t rightarrow infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $rho downarrow sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $epsilon^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations.
{"title":"The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift","authors":"Julien Berestycki, Jiaqi Liu, Bastien Mallein, Jason Schweinsberg","doi":"arxiv-2409.08789","DOIUrl":"https://doi.org/arxiv-2409.08789","url":null,"abstract":"Consider branching Brownian motion with absorption in which particles move\u0000independently as one-dimensional Brownian motions with drift $-rho$, each\u0000particle splits into two particles at rate one, and particles are killed when\u0000they reach the origin. Kesten (1978) showed that this process dies out with\u0000probability one if and only if $rho geq sqrt{2}$. We show that in the\u0000subcritical case when $rho > sqrt{2}$, the law of the process conditioned on\u0000survival until time $t$ converges as $t rightarrow infty$ to a\u0000quasi-stationary distribution, which we call the Yaglom limit. We give a\u0000construction of this quasi-stationary distribution. We also study the\u0000asymptotic behavior as $rho downarrow sqrt{2}$ of this quasi-stationary\u0000distribution. We show that the logarithm of the number of particles and the\u0000location of the highest particle are of order $epsilon^{-1/3}$, and we obtain\u0000a limit result for the empirical distribution of the particle locations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262515","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}
Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu
Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981, 183-208) showed that the autocorrelation polynomial $phi_w(t)$ of $w$, which records the set of self-overlaps of $w$, explicitly determines for each $n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $X_{{w}}$, the space of all bi-infinite sequences that avoid $w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981, 183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to several invariants of $X_{{w}}$. We then give a finite-state labeled graphical representation $L_w$ of $X_{{w}}$ and show that $w$ can be recovered from the graph isomorphism class of the unlabeled version of $L_w$. Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $X_{{w}}$ for different $w$. Next, we give partial results on the problem of classifying the spaces $X_{{w}}$ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.
给定一个有限词 $w$，Guibas 和 Odlyzko (J. Combin. Theory Ser. A, 30,1981, 183-208) 发现，记录了 $w$ 的自重叠集合的 $w$ 的自相关多项式 $phi_w(t)$，明确地决定了每个 $n$ 的长度为 $n$ 的词中避开 $w$ 的词的数目 $|B_n(w)|$。我们从符号动力学的角度来考虑这个问题及相关问题，重点是 $X_{{w}}$，即所有避开 $w$ 的双无限序列的空间。我们首先总结并阐述了 (J. Combin. Theory Ser. A, 30, 1981,183-208)和其他工作，以证明序列 $|B_n(w)|$ 等价于 $X_{{w}}$ 的几个不变式。然后，我们给出了$X_{{w}}$的有限状态标注图表示$L_w$，并证明$w$可以从未标明版本的$L_w$的图同构类中得到。利用$L_w$，我们应用概率论和佩伦-弗罗贝尼斯理论的思想，得到了比较不同$w$下$X_{{w}}$特征的结果。接下来，我们给出了对直到共轭的空间 $X_{{w}$ 的分类问题的部分结果。最后，我们将部分结果扩展到避免给定有限模式的多维阵列空间。
{"title":"Shifts of Finite Type Obtained by Forbidding a Single Pattern","authors":"Nishant Chandgotia, Brian Marcus, Jacob Richey, Chengyu Wu","doi":"arxiv-2409.09024","DOIUrl":"https://doi.org/arxiv-2409.09024","url":null,"abstract":"Given a finite word $w$, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30,\u00001981, 183-208) showed that the autocorrelation polynomial $phi_w(t)$ of $w$,\u0000which records the set of self-overlaps of $w$, explicitly determines for each\u0000$n$, the number $|B_n(w)|$ of words of length $n$ that avoid $w$. We consider\u0000this and related problems from the viewpoint of symbolic dynamics, focusing on\u0000the setting of $X_{{w}}$, the space of all bi-infinite sequences that avoid\u0000$w$. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981,\u0000183-208) and other work to show that the sequence $|B_n(w)|$ is equivalent to\u0000several invariants of $X_{{w}}$. We then give a finite-state labeled\u0000graphical representation $L_w$ of $X_{{w}}$ and show that $w$ can be\u0000recovered from the graph isomorphism class of the unlabeled version of $L_w$.\u0000Using $L_w$, we apply ideas from probability and Perron-Frobenius theory to\u0000obtain results comparing features of $X_{{w}}$ for different $w$. Next, we\u0000give partial results on the problem of classifying the spaces $X_{{w}}$ up to\u0000conjugacy. Finally, we extend some of our results to spaces of\u0000multi-dimensional arrays that avoid a given finite pattern.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262708","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}
Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko
The paper studies solutions of stochastic partial differential equations with random initial conditions. First, it overviews some of the known results on scaled solutions of such equations and provides several explicit motivating examples. Then, it proves multiscaling limit theorems for renormalized solutions for the case of initial conditions subordinated to the random processes with cyclic long-range dependence. Two cases of stochastic partial differential equations are examined. The spectral and covariance representations for the corresponding limit random fields are derived. Additionally, it is discussed why analogous results are not valid for subordinated cases with Hermite ranks greater than 1. Numerical examples that illustrate the obtained theoretical results are presented.
{"title":"Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence","authors":"Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko","doi":"arxiv-2409.09215","DOIUrl":"https://doi.org/arxiv-2409.09215","url":null,"abstract":"The paper studies solutions of stochastic partial differential equations with\u0000random initial conditions. First, it overviews some of the known results on\u0000scaled solutions of such equations and provides several explicit motivating\u0000examples. Then, it proves multiscaling limit theorems for renormalized\u0000solutions for the case of initial conditions subordinated to the random\u0000processes with cyclic long-range dependence. Two cases of stochastic partial\u0000differential equations are examined. The spectral and covariance\u0000representations for the corresponding limit random fields are derived.\u0000Additionally, it is discussed why analogous results are not valid for\u0000subordinated cases with Hermite ranks greater than 1. Numerical examples that\u0000illustrate the obtained theoretical results are presented.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262505","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}
Pietro Caputo, Zongchen Chen, Yuzhou Gu, Yury Polyanskiy
This paper considers the speed of convergence (mixing) of a finite Markov kernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given a Markov kernel one defines either a discrete-time Markov chain (with the $n$-step transition kernel given by the matrix power $P^n$) or a continuous-time Markov process (with the time-$t$ transition kernel given by $e^{t(P-mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ are characterized by the famous functional inequalities, the strong data processing inequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively. When $P=KK^*$ is written as the product of a kernel and its adjoint, one could also consider the ``half-step'' contraction, which is the SDPI for $K$, while the ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03] claimed that these contraction coefficients (half-step, full-step, and continuous-time) are generally within a constant factor of each other. We disprove this and related conjectures by working out a number of different counterexamples. In particular, we construct (a) a continuous-time Markov process that contracts arbitrarily faster than its discrete-time counterpart; and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than $P^m$. Hence, our main conclusion is that the four standard inequalities comparing five common notions of entropy and variance contraction are generally not improvable. In the process of analyzing the counterexamples, we survey and sharpen the tools for bounding the contraction coefficients and characterize properties of extremizers of the respective functional inequalities. As our examples range from Bernoulli-Laplace model, random walks on graphs, to birth-death chains, the paper is also intended as a tutorial on computing MLSI, SDPI and other constants for these types of commonly occurring Markov chains.
{"title":"Entropy Contractions in Markov Chains: Half-Step, Full-Step and Continuous-Time","authors":"Pietro Caputo, Zongchen Chen, Yuzhou Gu, Yury Polyanskiy","doi":"arxiv-2409.07689","DOIUrl":"https://doi.org/arxiv-2409.07689","url":null,"abstract":"This paper considers the speed of convergence (mixing) of a finite Markov\u0000kernel $P$ with respect to the Kullback-Leibler divergence (entropy). Given a\u0000Markov kernel one defines either a discrete-time Markov chain (with the\u0000$n$-step transition kernel given by the matrix power $P^n$) or a\u0000continuous-time Markov process (with the time-$t$ transition kernel given by\u0000$e^{t(P-mathrm{Id})}$). The contraction of entropy for $n=1$ or $t=0+$ are\u0000characterized by the famous functional inequalities, the strong data processing\u0000inequality (SDPI) and the modified log-Sobolev inequality (MLSI), respectively.\u0000When $P=KK^*$ is written as the product of a kernel and its adjoint, one could\u0000also consider the ``half-step'' contraction, which is the SDPI for $K$, while\u0000the ``full-step'' contraction refers to the SDPI for $P$. The work [DMLM03]\u0000claimed that these contraction coefficients (half-step, full-step, and\u0000continuous-time) are generally within a constant factor of each other. We\u0000disprove this and related conjectures by working out a number of different\u0000counterexamples. In particular, we construct (a) a continuous-time Markov\u0000process that contracts arbitrarily faster than its discrete-time counterpart;\u0000and (b) a kernel $P$ such that $P^{m+1}$ contracts arbitrarily better than\u0000$P^m$. Hence, our main conclusion is that the four standard inequalities\u0000comparing five common notions of entropy and variance contraction are generally\u0000not improvable. In the process of analyzing the counterexamples, we survey and sharpen the\u0000tools for bounding the contraction coefficients and characterize properties of\u0000extremizers of the respective functional inequalities. As our examples range\u0000from Bernoulli-Laplace model, random walks on graphs, to birth-death chains,\u0000the paper is also intended as a tutorial on computing MLSI, SDPI and other\u0000constants for these types of commonly occurring Markov chains.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212024","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 small-mass limit, also known as the Smoluchowski-Kramers diffusion approximation (see cite{kra} and cite{smolu}), for a system of stochastic damped wave equations, whose solution is constrained to live in the unitary sphere of the space of square-integrable functions on the interval $(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative Gaussian noise, where the stochastic differential is understood in Stratonovich sense. Due to its particular structure, such noise not only conserves $mathbb{P}$-a.s. the constraint, but also preserves a suitable energy functional. In the limit, we derive a deterministic system, that remains confined to the unit sphere of $L^2$, but includes additional terms. These terms depend on the reproducing kernel of the noise and account for the interaction between the constraint and the particular conservative noise we choose.
{"title":"The small-mass limit for some constrained wave equations with nonlinear conservative noise","authors":"Sandra Cerrai, Mengzi Xie","doi":"arxiv-2409.08021","DOIUrl":"https://doi.org/arxiv-2409.08021","url":null,"abstract":"We study the small-mass limit, also known as the Smoluchowski-Kramers\u0000diffusion approximation (see cite{kra} and cite{smolu}), for a system of\u0000stochastic damped wave equations, whose solution is constrained to live in the\u0000unitary sphere of the space of square-integrable functions on the interval\u0000$(0,L)$. The stochastic perturbation is given by a nonlinear multiplicative\u0000Gaussian noise, where the stochastic differential is understood in Stratonovich\u0000sense. Due to its particular structure, such noise not only conserves\u0000$mathbb{P}$-a.s. the constraint, but also preserves a suitable energy\u0000functional. In the limit, we derive a deterministic system, that remains\u0000confined to the unit sphere of $L^2$, but includes additional terms. These\u0000terms depend on the reproducing kernel of the noise and account for the\u0000interaction between the constraint and the particular conservative noise we\u0000choose.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142211955","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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