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Register a new userImproving constant in end-point Poincare inequality on Hamming cube
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We improve the constant $frac{pi}{2}$ in $L^1$-Poincare inequality on Hamming cube. For Gaussian space the sharp constant in $L^1$ inequality is known, and it is $sqrt{frac{pi}{2}}$. For Hamming cube the sharp constant is not known, and $sqrt{frac{pi}{2}}$ gives an estimate from below for this sharp constant. On the other hand, L. Ben Efraim and F. Lust-Piquard have shown an estimate from above: $C_1le frac{pi}{2}$. There are at least two other independent proofs of the same estimate from above (we write down them in this note). Since those proofs are very different from the proof of Ben Efraim and Lust-Piquard but gave the same constant, that might have indicated that constant is sharp. But here we give a better estimate from above, showing that $C_1$ is strictly smaller than $frac{pi}{2}$. It is still not clear whether $C_1> sqrt{frac{pi}{2}}$. We discuss this circle of questions and the computer experiments.
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We prove that for $alpha in (d-1,d]$, one has the trace inequality begin{align*} int_{mathbb{R}^d} |I_alpha F| ;d u leq C |F|(mathbb{R}^d)| u|_{mathcal{M}^{d-alpha}(mathbb{R}^d)} end{align*} for all solenoidal vector measures $F$, i.e., $Fin M_b(mathbb{R}^d,mathbb{R}^d)$ and $operatorname{div}F=0$. Here $I_alpha$ denotes the Riesz potential of order $alpha$ and $mathcal M^{d-alpha}(mathbb{R}^d)$ the Morrey space of $(d-alpha)$-dimensional measures on $mathbb{R}^d$.
The scalar curvature equation for rotation invariant Kahler metrics on $mathbb{C}^n backslash {0}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $mathbb{C}^n backslash {0}$ in lower dimensions. We also prove that there does not exist negative csck on $mathbb{C}^n backslash {0}$ for $n=2,3$.
A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $Omega$, it is only known that the radius of convergence for the expected signature on sufficiently regular $Omega$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $Omega$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,alpha}$-domain $Omega subset mathbb{R}^d$ with $2leq d leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a domain-averaging hyperbolic development (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}tomathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)equiv1$, then we prove that, under a mild decay condition on $f$, the process $xmapsto u(t,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.
In this paper, a simplified second-order Gaussian Poincare inequality for normal approximation of functionals over infinitely many Rademacher random variables is derived. It is based on a new bound for the Kolmogorov distance between a general Rademacher functional and a Gaussian random variable, which is established by means of the discrete Malliavin-Stein method and is of independent interest. As an application, the number of vertices with prescribed degree and the subgraph counting statistic in the Erdos-Renyi random graph are discussed. The number of vertices of fixed degree is also studied for percolation on the Hamming hypercube. Moreover, the number of isolated faces in the Linial-Meshulam-Wallach random $kappa$-complex and infinite weighted 2-runs are treated.
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