# 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 2nd, and Friday July 3th 2020, at Zentrum Mathematik, Technische Universität München.

All talks take place (using zoom)

## Program

Thursday July 2nd, 2020:

• 15:00-15:45 Vittoria Silvestri
• 16:00-16:45 Hermine Biermé
• 17:00-17:45 Arianna Giunti

Friday July 3th, 2020:

• 14:00-14:45 Sara 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.
• Sara 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 cylinders
Abstract: 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).