# Workshop Women in Probability 2020

The scientific program is organized by Noam Berger, Diana Conache, Nina Gantert, Silke Rolles and Sabine Jansen (LMU). This conference is supported by the "Women for Math Science Program" at Technische Universität München.

There is no conference fee and there are no gender restrictions on the audience, everybody is welcome to attend.

## Location and dates

**Thursday July 2 ^{nd}**, and

**Friday July 3**, at Zentrum Mathematik, Technische Universität München.

^{th }2020All talks take place (using zoom)

## Speakers

- Hermine Biermé (Université de Poitiers)
- Arianna Giunti (Universität Bonn)
- Masha Gordina (University of Connecticut)
- Malin Palö Forsström (KTH Royal Institute of Technology, Stockholm)
- Sara Mazzonetto (Universität Potsdam)
- Patricia Alonso Ruiz (Texas A&M University)
- Vittoria Silvestri (University of Rome La Sapienza)

## Program

**Thursday July 2 ^{nd}**

**, 2020**:

- 14:00-14:45
**Malin Palö Forsström** - 15:00-15:45
**Vittoria Silvestri** - 16:00-16:45
**Hermine Biermé** - 17:00-17:45
**Arianna Giunti**

**Friday July 3 ^{th}**

**, 2020**:

- 14:00-14:45 S
**ara Mazzonetto** - 15:00-15:45
**Masha Gordina** - 16:00-16:45
**Patricia Alonso Ruiz**

## Titles and abstracts

**Hermine Biermé:**Operator-scaling random ball model

Abstract: We study generalized random fields which arise as operator rescaling limits of

spatial configurations of uniformly scattered weighted random balls as the mean radius of the balls tends to $0$ or infinity. Assuming that the radius

distribution has a power law behavior, we prove that the centered and renormalized random balls field admits an $\alpha$-stable limit with strong spatial

dependence, according to the attraction domain of the weights. In particular, our approach provides a unified framework to obtain some operator-scaling $\alpha$-stable random fields,

generalizing the isotropic self-similar case investigated recently in the literature. This is a joint work with Olivier Durieu (IDP, Univ. Tours, France) and Yizao Wang (Univ. Cincinnati, USA).**Arianna Giunti:**Homogenization in randomly perforated domains

Abstract: We consider the homogenization of Poisson and Stokes equations in a bounded domain of $R^d$, $d >2$, perforated by many small random holes. We assume that the holes are generated by properly rescaled balls having random radii and centers (i.e.Boolean process). Our main assumption is that the random radii of the Boolean process have finite (d−2)-moment: This condition is minimal in order to ensure that the average density of capacity generated by the holes is finite, but still allows for the onset of clustering balls with overwhelming probability. By combining analytic and probabilistic, percolation-like methods, we give an homogenization result fora large class of measures as above. These are joint works with R.M. Höfer and J.J.L. Velazquez (University of Bonn).**Masha Gordina:**Ergodicity For Langevin Dynamics With Singular Potentials

Abstract: We discuss Langevin dynamics of N particles on R^d interacting through a singular repulsive potential, such as the

Lennard-Jones potential, and show that the system converges to the

unique invariant Gibbs measure exponentially fast in a weighted Sobolev

norm. The proof relies on an explicit construction of a Lyapunov

function using a modified Gamma calculus. In contrast to previous

results for such systems, our results imply geometric convergence to

equilibrium starting from an essentially optimal family of initial

distributions. This is based on the joint work with F. Baudoin and D. Herzog.**Malin Palö Forsström:**Generalized divide and color representations

Abstract: Summary: I will discuss recent progress made on understanding so called

(generalized) divide and color processes. The results include several

results about existence and uniqueness, as well as specific

consequences for threshold discrete Gaussian free fields and Ising models.- S
**ara Mazzonetto:**Some results on skewed and threshold diffusions

Abstract: The object of this talk is the relationship between two

classes of diffusion processes which are solutions to stochastic differential

equations with discontinuous coefficients and/or involving the local time of the

process itself. The interplay between the two classes plays a key role in

some recent results in Probability and Statistics. We will provide some

examples, for instance in parameter estimation for some diffusions. **Patricia Alonso Ruiz:**Heat kernel analysis on diamond fractals

Abstract: When existent, a heat kernel represents the transition

probability density function of its associated diffusion process. The

behavior of the heat kernel thus encodes many properties of the

corresponding process, including geometric aspects of the underlying space.

In this talk we discuss diffusion processes on a parametric family of

fractals called generalized diamond fractals. They arise as scaling

limits of diamond hierarchical lattices studied in the physics

literature in relation to random polymers, Ising and Potts models.

Earlier investigations of these processes concerned a special class of

parameters for which the associated fractal enjoyed self-similarity. The

latter was key to study the behavior of the heat kernel, in particular

because in this setting some usual assumptions like volume doubling are

not satisfied. For general parameters, also the self-similarity of diamond fractals is

lost. However, their structure as projective limits will make it

possible to derive estimates for the heat kernel, and to establish

functional inequalities that connect analytic and geometric aspects of

diffusion processes.**Vittoria Silvestri:**Fluctuations and mixing of Internal DLA on cylindersAbstract: Internal DLA models the growth of a random cluster by subsequent aggregation of particles. At each step, a new particle starts inside the cluster, and it performs a simple random walk until reaching an unoccupied site, where it settles. When particles move on a cylinder graph GxZ this defines a positive recurrent Markov chain on cluster configurations. In this talk I will address the following questions: How does a typical configuration look like? How long does it take for the process to forget its initial profile? Partly based on joint work with Lionel Levine (Cornell).