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Register a new userUniform Poincar{e} and logarithmic Sobolev inequalities for mean field particles systems
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Arnaud Guillin
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In this paper we establish some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean field particles system, uniform in the number of particles, when the confinement potential have many local minimums. Our uniform log-Sobolev inequality, based on Zegarlinskis theorem for Gibbs measures, allows us to obtain the exponential convergence in entropy of the McKean-Vlasov equation with an explicit rate constant, generalizing the result of [10] by means of the displacement convexity approach, or [19, 20] by Bakry-Emery technique or the recent [9] by dissipation of the Wasserstein distance.
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If Poincar{e} inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods: Bobkovs argument and super-Poincar{e} inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee-Vempala in the logconcave bounded case are refined for radial measures.
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.
Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheegers inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheegers inequality, which is a $L_{p}$-Poincar{e} inequality for $p=2$, to any real $pgeq 1$.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. The geometric invariant appearing in the sharp lower bound is shown to be nonnegative. We also characterize the expanders when such invariant is zero. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. In particular, we prove that the {it asymptotic volume ratio} of any gradient shrinking soliton with nonnegative Ricci curvature must be zero.
The nonlinear Schrodinger equation NLSE(p, beta), -iu_t=-u_{xx}+beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space Lp^2(mT). For pleq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure mu^{beta}_N on balls Omega_N= {phi in Lp^2(mT) : | phi |^2_{Lp^2} leq N} in phase space such that the Cauchy problem for NLSE(p,beta) is well posed on the support of mu^{beta}_N, and that mu^{beta}_N is invariant under the flow. This paper shows that mu^{beta}_N satisfies a logarithmic Sobolev inequality for the focussing case beta <0 and 2leq pleq 4 on Omega_N for all N>0; also mu^{beta} satisfies a restricted LSI for 4leq pleq 6 on compact subsets of Omega_N determined by Holder norms. Hence for p=4, the spectral data of the periodic Dirac operator in Lp^2(mT; mC^2) with random potential phi subject to mu^{beta}_N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hills equation when the potential is random and subject to the Gibbs measure of KdV.
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