log-Sobolev Inequality Research Articles

Log-Sobolev inequalities and hypercontractivity for Ornstein – Uhlenbeck evolution operators in infinite dimension

Log-Sobolev inequalities and hypercontractivity for Ornstein – Uhlenbeck evolution operators in infinite dimension

Arbeitsgemeinschaft: QFT and Stochastic PDEs

Quantum field theory (QFT) is a fundamental framework for a wide range of phenomena is physics. The link between QFT and SPDE was first observed by the physicists Parisi and Wu (1981), known as Stochastic Quantisation. The study of solution theories and properties of solutions to these SPDEs derived from the Stochastic Quantisation procedure has stimulated substantial progress of the solution theory of singular SPDE, especially the invention of the theories of regularity structures and paracontrolled distributions in the last decade. Moreover, Stochastic Quantisation allows us to bring in more tools including PDE and stochastic analysis to study QFT.This Arbeitsgemeinschaft starts by covering some background material and then explores some of the advances made in recent years. The focus of this Arbeitsgemeinschaft is QFT models such as the \Phi^{4} , sine-Gordon and Yang–Mills models as examples to discuss stochastic quantisation and SPDE methods and their applications in these models. We introduce the key ideas, results and applications of regularity structure and paracontrolled distributions, construction of solutions of the SPDEs corresponding to these models, and use the PDE method to study some qualitative behaviors of these QFTs, and connections with the corresponding lattice or statistical physical models. We also discuss some other topics of QFT, such as Wilsonian renormalisation group, log-Sobolev inequalities and their implications, and various connections between these topics and SPDEs.

Regularized modified log-Sobolev inequalities and comparison of Markov chains

Regularized modified log-Sobolev inequalities and comparison of Markov chains

Logarithmic Sobolev-Type Inequalities on Lie Groups

In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo–Nirenberg and log-Caffarelli–Kohn–Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Gross-type log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

Sharp weighted log-Sobolev inequalities: Characterization of equality cases and applications

By using optimal mass transport theory, we provide a direct proof to the sharp L p L^p -log-Sobolev inequality ( p ≥ 1 ) (p\geq 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n E\subseteq \mathbb R^n . The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the L p L^p -log-Sobolev inequality. The characterization of the equality cases is new for p ≥ n p\geq n even in the unweighted setting and E = R n E=\mathbb R^n . As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases.

Matrix-valued modified logarithmic Sobolev inequality for sub-Laplacian on [formula omitted]

We prove that the canonical sub-Laplacian on SU(2) admits a modified log-Sobolev inequality on matrix-valued functions, independent of the matrix sizes. This establishes the first example of a matrix-valued modified log-Sobolev inequality for a sub-Laplacian. We also show that on Lie groups the heat kernel measure pt at time t satisfies matrix-valued modified log-Sobolev inequality with constants in order O(t−1).

The Log-Sobolev Inequality for a Submanifold in Manifolds With Asymptotic Non-Negative Intermediate Ricci Curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong et al. (Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194 (2024)) which already generalizes Yi and Zheng (To appear in Chin. Ann. Math., (2023)) and Brendle (Comm. Pure Appl. Math. 75(3), 449–454 (2022)).

Ornstein[formula omitted]Uhlenbeck type processes on Wasserstein spaces

Let P2 be the space of probability measures on Rd having finite second moment, and consider the Riemannian structure on P2 induced by the intrinsic derivative on the L2-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein−Uhlenbeck (OU) type Dirichlet form on P2 whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is L2-compact. Perturbations of the OU Dirichlet form are also studied.

Sharp log-Sobolev inequalities in [formula omitted] spaces with applications

Given p,N>1, we prove the sharp Lp-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0,N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0,N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf–Lax semigroup in CD(0,N) spaces. The proof of this result uses Hamilton–Jacobi inequality and Sobolev regularity properties of the Hopf–Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0,N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Riemannian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

Non-linear Log-Sobolev Inequalities for the Potts Semigroup and Applications to Reconstruction Problems

Non-linear Log-Sobolev Inequalities for the Potts Semigroup and Applications to Reconstruction Problems

A monotone numerical flux for quasilinear convection diffusion equation

We propose a new numerical 2-point flux for a quasilinear convection–diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results.

Upgrading MLSI to LSI for reversible Markov chains

For reversible Markov chains on finite state spaces, we show that the modified log-Sobolev inequality (MLSI) can be upgraded to a log-Sobolev inequality (LSI) at the surprisingly low cost of degrading the associated constant by log⁡(1/p), where p is the minimum non-zero transition probability. We illustrate this by providing the first log-Sobolev estimate for Zero-Range processes on arbitrary graphs. As another application, we determine the modified log-Sobolev constant of the Lamplighter chain on all bounded-degree graphs, and use it to provide negative answers to two open questions by Montenegro and Tetali (2006) [27] and Hermon and Peres (2018) [17]. Our proof builds upon the ‘regularization trick’ recently introduced by the last two authors.

Sharp stability of log-Sobolev and Moser-Onofri inequalities on the sphere

In this paper, we are concerned with the stability problem for endpoint conformally invariant cases of the Sobolev inequality on the sphere Sn. Namely, we will establish the stability for Beckner's log-Sobolev inequality and Beckner's Moser-Onofri inequality on the sphere. We also prove that the sharp constant of global stability for the log-Sobolev inequality on the sphere Sn must be strictly smaller than the sharp constant of local stability for the same inequality. Furthermore, we also derive the non-existence of the global stability for Moser-Onofri inequality on the sphere Sn.

Some inequalities on Riemannian manifolds linking Entropy, Fisher information, Stein discrepancy and Wasserstein distance

For a complete connected Riemannian manifold M let V∈C2(M) be such that μ(dx)=e−V(x)vol(dx) is a probability measure on M. Taking μ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.

Modified log-Sobolev inequalities for strong-Rayleigh measures

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $\pi$ satisfies a form of negative dependence known as the stochastic covering property. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where $\pi$ is $k-$homogeneous, our results imply the celebrated concentration inequality for Lipschitz functions due to Pemantle & Peres (2014). As another application, we deduce that the natural Monte-Carlo Markov Chain used to sample from $\pi$ has mixing time at most $kn\log\log\frac{1}{\pi(x)}$ when initialized in state $x$. To the best of our knowledge, this is the first work relating negative dependence and modified log-Sobolev inequalities.

Bochner formulas, functional inequalities and generalized Ricci flow

As a consequence of the Bochner formula for the Bismut connection acting on gradients, we show sharp universal Poincaré and log-Sobolev inequalities along solutions to generalized Ricci flow. Using the two-form potential we define a twisted connection on spacetime which determines an adapted Brownian motion on the frame bundle, yielding an adapted Malliavin gradient on path space. We show a Bochner formula for this operator, leading to characterizations of generalized Ricci flow in terms of universal Poincaré and log-Sobolev type inequalities for the associated Malliavin gradient and Ornstein-Uhlenbeck operator.

Sharp uniform-in-time propagation of chaos

We prove the optimal rate of quantitative propagation of chaos, uniformly in time, for interacting diffusions. Our main examples are interactions governed by convex potentials and models on the torus with small interactions. We show that the distance between the k-particle marginal of the n-particle system and its limiting product measure is \(O((k/n)^2)\), uniformly in time, with distance measured either by relative entropy, squared quadratic Wasserstein metric, or squared total variation. Our proof is based on an analysis of relative entropy through the BBGKY hierarchy, adapting prior work of the first author to the time-uniform case by means of log-Sobolev inequalities.

Quantum Gaussian maximizers and log-Sobolev inequalities

Quantum Gaussian maximizers and log-Sobolev inequalities

Hairer-Quastel universality in non-stationarity via energy solution theory

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers equation by providing a solution theory that builds on the energy solution theory [15, 19, 20, 23]. The perspective we adopt is to study the stochastic Burgers equation by writing its solution as a probabilistic solution [22] plus a term that can be studied with deterministic PDE considerations. One motivation is universality of KPZ and stochastic Burgers equations for a certain class of stochastic PDE growth models, first studied in [28]. For this, we prove universality for SPDEs with general nonlinearities, thereby extending [28, 29], and for many non-stationary initial data, thereby extending [21]. Our perspective lets us also prove explicit rates of convergence to white noise invariant measure of stochastic Burgers for non-stationary initial data, in particular extending the spectral gap result of [23] beyond stationary initial data, though for non-stationary data our convergence will be measured in Wasserstein distance and relative entropy, not via the spectral gap as in [23]. Actually, we extend the spectral gap in [23] to a log-Sobolev inequality. Our methods can also analyze fractional stochastic Burgers equations [23]; we discuss this briefly. Lastly, we note our perspective on the KPZ and stochastic Burgers equations gives a first intrinsic notion of solutions for general continuous initial data, in contrast to Hölder regular data needed for regularity structures [26], paracontrolled distributions [14], and Hölder-regular Brownian bridge data for energy solutions [19, 20].

Sharp Poincaré and log-Sobolev inequalities for the switch chain on regular bipartite graphs

Consider the switch chain on the set of d-regular bipartite graphs on n vertices with $$3\le d\le n^{c}$$ , for a small universal constant $$c>0$$ . We prove that the chain satisfies a Poincaré inequality with a constant of order O(nd); moreover, when d is fixed, we establish a log-Sobolev inequality for the chain with a constant of order $$O_d(n\log n)$$ . We show that both results are optimal. The Poincaré inequality implies that in the regime $$3\le d\le n^c$$ the mixing time of the switch chain is at most $$O\big ((nd)^2 \log (nd)\big )$$ , improving on the previously known bound $$O\big ((nd)^{13} \log (nd)\big )$$ due to Kannan et al. (Rand Struct Algorithm 14(4):293–308, 1999) and $$O\big (n^7d^{18} \log (nd)\big )$$ obtained by Dyer et al. (Sampling hypergraphs with given degrees (preprint). arXiv:2006.12021 ). The log-Sobolev inequality that we establish for constant d implies a bound $$O(n\log ^2 n)$$ on the mixing time of the chain which, up to the $$\log n$$ factor, captures a conjectured optimal bound. Our proof strategy relies on building, for any fixed function on the set of d-regular bipartite simple graphs, an appropriate extension to a function on the set of multigraphs given by the configuration model. We then establish a comparison procedure with the well studied random transposition model in order to obtain the corresponding functional inequalities. While our method falls into a rich class of comparison techniques for Markov chains on different state spaces, the crucial feature of the method—dealing with chains with a large distortion between their stationary measures—is a novel addition to the theory.