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Register a new userEntropic multipliers method for langevin diffusion and weighted log sobolev inequalities
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In his work about hypocercivity, Villani [18] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L 2 are given in a quite general setting, convergence in entropy requires some boundedness condition on the Hessian of the Hamiltonian. We will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality. As a byproduct, we also give tractable conditions for this functional inequality, which is a particular instance of a weighted logarithmic Sobolev inequality, to hold.
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We prove that if ${(P_x)}_{xin mathscr X}$ is a family of probability measures which satisfy the log-Sobolev inequality and whose pairwise chi-squared divergences are uniformly bounded, and $mu$ is any mixing distribution on $mathscr X$, then the mixture $int P_x , mathrm{d} mu(x)$ satisfies a log-Sobolev inequality. In various settings of interest, the resulting log-Sobolev constant is dimension-free. In particular, our result implies a conjecture of Zimmermann and Bardet et al. that Gaussian convolutions of measures with bounded support enjoy dimension-free log-Sobolev inequalities.
We introduce a notion called entropic independence for distributions $mu$ defined on pure simplicial complexes, i.e., subsets of size $k$ of a ground set of elements. Informally, we call a background measure $mu$ entropically independent if for any (possibly randomly chosen) set $S$, the relative entropy of an element of $S$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $S$, a constant multiple of its ``share of entropy. Entropic independence is the natural analog of spectral independence, another recently established notion, if one replaces variance by entropy. In our main result, we show that $mu$ is entropically independent exactly when a transformed version of the generating polynomial of $mu$ can be upper bounded by its linear tangent, a property implied by concavity of the said transformation. We further show that this concavity is equivalent to spectral independence under arbitrary external fields, an assumption that also goes by the name of fractional log-concavity. Our result can be seen as a new tool to establish entropy contraction from the much simpler variance contraction inequalities. A key differentiating feature of our result is that we make no assumptions on marginals of $mu$ or the degrees of the underlying graphical model when $mu$ is based on one. We leverage our results to derive tight modified log-Sobolev inequalities for multi-step down-up walks on fractionally log-concave distributions. As our main application, we establish the tight mixing time of $O(nlog n)$ for Glauber dynamics on Ising models with interaction matrix of operator norm smaller than $1$, improving upon the prior quadratic dependence on $n$.
We prove a Lieb-Thirring type inequality for potentials such that the associated Schr{o}dinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.
We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakrys $Gamma-$ calculus. As a byproduct, the systematic method for constructing entropies which we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.
Let $X$ be a ball Banach function space on ${mathbb R}^n$. In this article, under the mild assumption that the Hardy--Littlewood maximal operator is bounded on the associated space $X$ of $X$, the authors prove that, for any $fin C_{mathrm{c}}^2({mathbb R}^n)$, $$sup_{lambdain(0,infty)}lambdaleft |left|left{yin{mathbb R}^n: |f(cdot)-f(y)| >lambda|cdot-y|^{frac{n}{q}+1}right}right|^{frac{1}{q}} right|_Xsim | abla f|_X$$ with the positive equivalence constants independent of $f$, where $qin(0,infty)$ is an index depending on the space $X$, and $|E|$ denotes the Lebesgue measure of a measurable set $Esubset {mathbb R}^n$. Particularly, when $X:=L^p({mathbb R}^n)$ with $pin [1,infty)$, the above estimate holds true for any given $qin [1, p]$, which when $q=p$ is exactly the recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which even when $q< p$ is new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and Gagliardo--Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincare inequality, the extrapolation, the exact operator norm on $X$ of the Hardy--Littlewood maximal operator, and the geometry of $mathbb{R}^n.$
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