## Future Seminar Talks

## 11/10 Carl Mueller (University of Rochester)

**Title**: The Radius of a Moving Polymer

**Abstract:**This is joint work with Eyal Neuman.

Polymer models give rise to some of the most challenging problems in probability and statistical physics. For example, the typical end-to-end distance of a self-avoiding simple random walk is known only in one dimension and in dimensions greater than or equal to 5. The parameter \(n\) in the walk does not represent physical time, but rather the distance from one end of the polymer. There has been very little work on moving polymers, in spite of the obvious physical motivation.

We consider Funaki's random string, which was also known to polymer scientists as the continuum limit of the Rouse model. Consider the stochastic heat equation with vector-valued solutions \(u(t,x)\in\mathbf{R}^d\) for \(x\in[0,J]\). Then \(t\) represents physical time and \(x\) is the length along the polymer, while \(u(t,x)\) is the position of the polymer. The SPDE is:

\[

\partial_tu=\partial_x^2u+\dot{W}(t,x)

\]

with Neumann boundary conditions, where \(\dot{W}\) is a \(d\)-dimensional vector of independent white noises. Next we impose an exponential weighting which penalizes self-intersection. In dimension \(d=1\), we study the radius \(R=R(t)\) of the polymer at a typical time \(t\), where \(R(t)=\sup_xu(t,x)-\inf_xu(t,x)\).

## 12/8 Yu Gu (Carnegie Mellon University)

**Title**: Gaussian fluctuations from nonlinear SPDEs

**Abstract:**I will talk about some joint work with Alex Dunlap, Jiawei Li, Lenya Ryzhik and Ofer Zeitouni on the large scale fluctuations of nonlinear SPDEs including the KPZ equation and nonlinear SHE in two and higher dimensions. We derive the Edwards-Wilkinson limit in certain weak disorder regime.

## Past Seminar Talks

## 09/15 David Nualart (University of Kansas)

**Title:**Ergodicity and Gaussian fluctuations for the stochastic wave equation [Slides]

**We consider the stochastic wave equation in dimensions 1, 2 and 3 driven by a noise which is white in time and it has a Riesz covariance kernel in the space variable. We will discuss the spatial ergodicity of the solution and present quantitative central limit theorems for spatial averages in dimensions 1 an 2. These results are based on moment estimates of the derivative of the solution in the sense of Malliavin calculus.**

****Abstract:

## 09/29 Carsten Chong (Columbia University)

**Title**: Recent advances on power variations for the stochastic heat equation [Slides]

**Abstract:**We discuss some unexpected limit theorems related to the power variations of the solution to the stochastic heat equation. In the first part of the talk, where power variations are taken in time at a fixed spatial point, interesting phenomena appear in the associated central limit theorems (CLT). Assuming a Gaussian noise that is white in time and has the Riesz kernel of order \(\alpha \in (0,2\wedge d)\) as spatial correlation function, we obtain a CLT for all values of \(\alpha\) in the case of additive noise but three different regimes under multiplicative noise: if \(\alpha\in(0,1)\) or \(p=2\), an analogous CLT holds true; if \(\alpha =1\) and \(p\neq 2\), we still have a CLT but the asymptotic mean is no longer zero; if \(\alpha \in (1,2)\) and \(p\neq 2\), we have to subtract certain renormalization terms that are intimately related to those arising in the analysis of singular stochastic PDEs.

In the second part of the talk, we consider power variations of the solution viewed as a process with values in Sobolev spaces \(H_r\). Here a phase transition already occurs on the level of the law of large numbers (LLN): while the power variations in \(H_r\) satisfy a genuine LLN for all \(r<-d/2\), a degenerate LLN holds true for \(r \geq -d/2\).

This talk is partially based on joint work with Robert Dalang (EPFL).

## 10/13 Masha Gordina (University of Connecticutt)

**Title**: Uniform doubling on SU(2) and beyond [Slides]

**Abstract:**Suppose G is a compact Lie group equipped with a left-invariant Riemannian metric. Such metrics usually form a finite-dimensional cone. The Riemannian volume measure corresponding to such a metric is the Haar measure of the group (up to a multiplicative constant). Because of compactness, each left-invariant metric g has the doubling property, that is, there exists a doubling constant D=D(G, g) such that the volume of the ball of radius 2r is at most D times the volume of the ball of radius r. We are concerned with the following question: does there exist a constant D(G) such that, for all left-invariant metrics g on G, the constant D(G, g) is bounded above by D(G)? This is what we call uniform doubling. The conjecture is that any compact Lie group is uniformly doubling. The only cases for which the conjecture is known are Riemannian tori and the group SU(2). The talk will describe a number of analytic/probabilistic consequences of uniform doubling such as Harnack inequalities, heat kernel bounds, spectral gap and Weyl's eigenvalue counting function estimates etc. These hold in absence of curvature bounds. The work in progress for U(2) might be mentioned as well. This is a joint work with Nathaniel Eldredge (University of Northern Colorado) and Laurent Saloff-Coste (Cornell University). Reference: Left-invariant geometries on SU(2) are uniformly doubling, GAFA 2018.

## 10/27 Xiaochuan Yang (University of Luxembourg)

**Title**: Two-scale stabilization on the Poisson space

**Abstract:**We prove a collection of analytic inequalities, allowing one to assess the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one – that we evaluate and compare at two different scales – and are specifically tailored for assessing the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization – see Penrose and Yukich (2001) and Penrose (2005). Our main bounds generalize and simplify the estimates recently exploited by Chatterjee and Sen (2017), for proving a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbor graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. This is a joint work with Raphaël Lachièze-Rey and Giovanni Peccati.