Harnack inequality for degenerate balanced random random walks.

Speaker: 

Jean-Dominique Deuschel

Institution: 

Technische Universitat, Berlin

Time: 

Saturday, December 2, 2017 - 2:00pm to 2:50pm

Location: 

NS2 1201

We consider an i.i.d. balanced environment  $\omega(x,e)=\omega(x,-e)$, genuinely d dimensional on the lattice and show that there exist a positive constant $C$ and a random radius $R(\omega)$ with streched exponential tail such that every non negative

$\omega$ harmonic function $u$ on the ball  $B_{2r}$ of radius $2r>R(\omega)$,

we have $\max_{B_r} u <= C \min_{B_r} u$.

Our proof relies on a quantitative quenched invariance principle

for the corresponding random walk in  balanced random environment and

a careful analysis of the directed percolation cluster.

This result extends Martins Barlow's Harnack's inequality for i.i.d.

bond percolation to the directed case.

This is joint work with N.Berger  M. Cohen and X. Guo.

On the Navier-Stokes equation with rough transport noise.

Speaker: 

James-Michael Leahy

Institution: 

USC

Time: 

Saturday, December 2, 2017 - 11:20am to 12:10pm

Location: 

NS2 1201

In this talk, we present some results on the existence of weak-solutions of the Navier-Stokes equation perturbed by transport-type rough path noise with periodic boundary conditions in dimensions two and three. The noise is smooth and divergence free in space, but rough in time. We will also discuss the problem of uniqueness in two dimensions. The proof of these results makes use of the theory of unbounded rough drivers developed by M. Gubinelli et al.

 

As a consequence of our results, we obtain a pathwise interpretation of the stochastic Navier-Stokes equation with Brownian and fractional Brownian transport-type noise. A Wong-Zakai theorem and support theorem follow as an immediate corollary. This is joint work with Martina Hofmanov\'a and Torstein Nilssen.

Deviations of random matrices and applications.

Speaker: 

Roman Vershynin

Institution: 

UCI

Time: 

Saturday, December 2, 2017 - 10:00am to 10:50am

Location: 

NS2 1201

Uniform laws of large numbers provide theoretical foundations for statistical learning theory. This lecture will focus on quantitative uniform laws of large numbers for random matrices. A range of illustrations will be given in high dimensional geometry and data science.

Gaussian comparisons meet convexity: Precise analysis of structured signal recovery

Speaker: 

Christos Thrampoulidis

Institution: 

MIT

Time: 

Tuesday, November 14, 2017 - 11:00am to 11:50am

Host: 

Location: 

RH 306

Gaussian comparison inequalities are classical tools that often lead to simple proofs of powerful results in random matrix theory, convex geometry, etc. Perhaps the most celebrated of these tools is Slepian’s Inequality, which dates back to 1962. The Gaussian Min-max Theorem (GMT) is a non-trivial generalization of Slepian’s result, derived by Gordon in 1988. Here, we prove a tight version of the GMT in the presence of convexity. Based on that, we describe a novel and general framework to precisely evaluate the performance of non-smooth convex optimization methods under certain measurement ensembles (Gaussian, Haar). We discuss applications of the theory to box-relaxation decoders in massive MIMO, 1-bit compressed sensing, and phase-retrieval.

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