# Poster Session and Reception

Tuesday, April 28, 2015 - 4:30pm - 6:00pm

Lind 400

**On Holder Continuity of the Solution of Stochastic Wave Equations in Dimension Three**

Jingyu Huang (University of Kansas)

We study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path H\older continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean H\older continuity of the covariance function or in terms of its spectral measure. Some examples of the covariance functions are proved to satisfy our conditions. In particular, we obtain the H\older continuity results for the solution of the stochastic wave equations driven by (space inhomogeneous) fractional Brownian noises. For this particular noise, the optimality of the obtained H\older exponents is also discussed.**Log-Sobolev Inequalities on the Horizontal Path Space of a Totally Geodesic Foliation**

Qi Feng (Purdue University)

We develop a Malliavin calculus on the horizontal path space of a totally geodesic Riemannian foliation. As a first application, under suitable assumptions, we prove a log-Sobolev inequality for a natural one-parameter family of infinite-dimensional Ornstein-Uhlenbeck type operators. As a second application, we obtain concentration and tail estimates for the horizontal Brownian motion of the foliation.**Surface Area of a Convex Body in $mathbb R^n$ with Respect to Log Concave Spherically Invariant Measures**

Galyna Livshyts (Kent State University)**An Improvement of the McDiarmid Inequality**

Emmanuel Rio (Université Versailles/Saint Quentin-en-Yvelines)**An Optimal Form of the Li-Yau Inequality Under a Ricci Curvature Bounded from Below**

Ivan Gentil (Institut Camille Jordan, Université Lyon 1)

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

Collaboration with D. Bakry (France, Toulouse 3) and F. Bolley (France, Paris 6)**Quantitative Relationship Between Noise Stability and Influences for General Models**

Raphael Bouyrie (Université de Toulouse III (Paul Sabatier))

Recently, N. Keller and G. Kindler proved a quantitative version of the famous Benjamini--Kalai--Schramm Theorem on noise sensitivity of Boolean functions. The result was extended latter on to the continuous Gaussian space. We manage to do a direct approach of these results, both in discrete and continuous settings. It extends to further discrete models of interest such as Cayley or Schreier graphs, or in continuous setting subclass of log-concave measures.**Bounding Marginal Densities via Affine Isoperimetry**

Peter Pivovarov (University of Missouri)

We show that any probability measure with a bounded density has many well-bounded marginals. This probabilistic fact is based on affine isoperimetric inequalities for certain averages of the density on the Grassmannian and affine Grassmannian. Such inequalities can be viewed as functional analogues of affine isoperimetric inequalities for convex sets, due to Busemann-Straus, Grinberg and Schneider. These results have applications to small-ball probabilities. Based on joint work with Susanna Dann and Grigoris Paouris.**An $L^p$-Brunn-Minkowski Inequality for Convex Measures in the Unconditional Case**

Arnaud Marsiglietti (University of Minnesota, Twin Cities)

We consider the $L^p$-Minkowski combination of convex sets in $\R^n$ introduced by Firey and we prove an $L^p$-Brunn-Minkowski inequality, $p \in [0,1]$, for a general class of measures called convex measures that includes log-concave measures, under symmetry assumptions.**Discrete Curvature and Abelian Groups**

Peter Ralli (Georgia Institute of Technology)

Our aim is to replicate results from Riemannian geometry in discrete structures. We define a discrete analogue of Ricci curvature in graphs, building on the Γ2-calculus of Bakry and Émery. Many different notions of discrete Ricci curvature have previously been suggested. Ours has the virtue that it is straightforward to compute the curvature for many graphs of general interest. We also derive an analogue of Buser’s inequality, relating isoperimetric and functional constants in the case of non-negative curvature.**CLT for Sample Covariance Matrices in the Tensor Product Case**

Ganna Lytova (University of Alberta)

For any $k$, $m$, $n$, we consider $n^k\times n^k$ real symmetric random matrices of the form

$$

{M}_{n}=M_{n,m,k}=\sum_{\alpha

=1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }^{(1)}\otimes...\otimes

\mathbf{y}_{\alpha }^{(k)}(\mathbf{y}_{\alpha }^{(1)}\otimes...\otimes

\mathbf{y}_{\alpha }^{(k)})^T,

$$

where $\tau _{\alpha }$ are real numbers and $\{\mathbf{y}_\alpha^{(p)}\}_{\alpha, p=1}^{m,k}$ are i.i.d. copies of a normalized isotropic random vector in $\mathbb{R}^n$. We suppose that $k$ is fixed and $m\rightarrow\infty$, $m/n^k\rightarrow c\in [0,\infty)$ as $n\rightarrow\infty$. This tensor analog of the sample covariance matrices appeared in quantum information theory and was firstly introduced to the random matrix theory by Hastings, Ambainis, and Harrow. For the case corresponding to uniformly distributed on the unit sphere vectors, they proved the Marchenko-Pastur law for the limit $\mathcal{N}$ of the expectation of the normalized counting

measure of eigenvalues and convergence of extreme eigenvalues to the endpoints

of the support of $\mathcal{N}$. We find

a class of random vectors satisfying some moment conditions such that for any smooth enough test-function $\varphi$ the linear

statistics $Tr \varphi(M_n)$ of eigenvalues of corresponding matrices $M_n$ being centered and properly normalized converge in distribution to a Gaussian random variable.**On the Brunn-Minkowski Inequality**

Piotr Nayar (University of Minnesota, Twin Cities)

We prove the Brunn-Minkowski inequality in its classical form for unconditional log-concave measures and unconditional convex sets in R^n. Moreover, we can remove the unconditionality assumption in the case n=2.**Mixing Rates of Random Walks with Little Backtracking**

Peng Xu (University of Delaware)