# Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Andrea Fiebig

Berlin Probability Colloquium

## Berlin Probability Colloquium

#### Ort

HU Berlin, Institut für Mathematik
Rudower Chaussee 25, Haus 1, Raum 115
12489 Berlin

#### Zeit

mittwochs, 17.00  Uhr

#### Programm

27. Oktober 2021
Christophe Garban (U Lion 1)
Vortex fluctuations in continuous spin systems and lattice gauge theory
Abstract: Topological phase transitions were discovered by Berezinskii-Kosterlitz-Thouless (BKT) in the 70’s. They describe intriguing phase transitions for classical statistical physics models such as
the $2d XY$ model (spins on ${Z\!\!\!\!Z}^2$ with values in the unit circle)
the 2d Coulomb gas
the integer-valued Gaussian Free Field (or Z-ferromagnet)
Abelian lattice gauge theory on ${Z\!\!\!\!Z}^4$
In this talk, I will explain a new technique to obtain quantitative lower bounds on the fluctuations induced by the topological defects (vortices) on such systems at low temperature. We will see in particular that the fluctuations generated by the vortices are at least of the same order of magnitude as the ones produced by the so-called ''spin-wave''. Our approach is non-perturbative but it gives matching lower bounds with the fluctuations predicted from $RG$ analysis. I will start the talk by giving an overview of the above models. The talk is based on joint works with Avelio Sepúlveda.
03. November 2021
15 Uhr - Geoffrey Grimmett (Cambridge)
Site percolation on hyperbolic planar graphs
Abstract: Percolation continues to generate important new problems. This talk is directed towards solutions of a number of old problems associated with site percolation on (hyperbolic) planar graphs, and it begins from scratch. The first result is a relationship between the critical point of a graph and the uniqueness critical point of its matching graph. Our methods enable proofs of two more of the notable Benjamimi-Schramm conjectures from 1996 concerning percolation in the hyperbolic plane. A key technique is a method for expressing a site model as a bond model, in the context of a matching pair of graphs. (Joint work with Zhongyang Li.)
16 Uhr - Sebastian Andres (Manchester)
Heat kernel bounds for Liouville Brownian motion
Abstract: In two-dimensional Liouville quantum gravity the main object of study is a random geometry on a domain D, which can be formally described by a Riemannian metric tensor of the form $e^{γh(x)}dx^2$ where h is a Gaussian free field on D and γ ∈ (0, 2) is a parameter. The natural diffusion process in this random geometry, called Liouville Brownian motion (LBM), can be constructed as a time-change of the planar Brownian motion on D. In this talk, we discuss sub-Gaussian heat kernel estimates for the LBM when $γ = \sqrt{8/3}$, which are sharp up to a polylogarithmic factor in the exponential. This value of γ is special because $γ = \sqrt{8/3}$-Liouville quantum gravity is equivalent to the Brownian map. This talk is based on a joint work in progress with Naotaka Kajino (Kyoto) and Jason Miller (Cambridge).
17 Uhr - Lisa Hartung (Mainz)
Branching Brownian motion with self repulsion
Abstract: We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an ε-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term. This is joint work with Anton Bovier.
17. November 2021
Boualem Djehiche (KTH Stockholm) ONLINE-Vortrag
Propagation of chaos for a class of mean-field reflected BSDEs with jumps
Abstract: I will review a recent result on the propagation of chaos property for weakly interacting nonlinear Snell envelopes which converge to a class of mean-field reflected backward stochastic differential equations (BSDEs) with jumps, where the mean-field interaction in terms of the distribution of the Y -component of the solution enters in both the driver and the lower obstacle.
24. November 2021
Cecchin Alekos (Ecole Polytechnique Paris) ONLINE-Vortrag
Mean field games with Wright-Fisher common noise
Abstract: Motivated by restoration of uniqueness in finite state mean field games, we introduce a common noise which is inspired by Wright-Fisher models in population genetics. Thus we analyze the master equation of this mean field game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo (AMS, 2013) has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. This is enough to conclude that the mean field game with such type of common noise is uniquely solvable. Then we introduce the finite player version of the game and show that N-player Nash equilibria converge towards the solution of such a kind of Wright-Fisher mean field game. The analysis is more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which hence may differ from 1/N and which, most of all, evolves with the common noise. Finally, we give an idea on how the randomly forced and uniquely solvable mean field game is used to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the intensity of a the Wright-Fisher common noise. Based on joint works with Erhan Byraktar, Asaf Cohen, and François Delarue.
01. Dezember 2021
Ilya Chevyrev (Edinburgh) ONLINE-Vortrag
Stochastic quantisation of gauge fields
Abstract: Stochastic portfolio theory was introduced in the nineties by R. Fernholz to model long-term investments in large equity markets. An empirical fact at the center of this theory is the observed stability of the so-called capital distribution curve. Over the past twenty years, the search for models capable of capturing this stability has generated a rich literature on particle systems interacting through their ranks. In such a system, each particle follows a diffusion process whose coefficients depend on the particle’s rank within the population. In this talk I will introduce the capital distribution curve, describe the connection with rank-based particle systems, give some historical highlights, and finally discuss some recent developments. No background in mathematical finance will be assumed.
15. Dezember 2021
Stochastic portfolio theory and rank-based particle systems
Abstract: Stochastic portfolio theory was introduced in the nineties by R. Fernholz to model long-term investments in large equity markets. An empirical fact at the center of this theory is the observed stability of the so-called capital distribution curve. Over the past twenty years, the search for models capable of capturing this stability has generated a rich literature on particle systems interacting through their ranks. In such a system, each particle follows a diffusion process whose coefficients depend on the particle’s rank within the population. In this talk I will introduce the capital distribution curve, describe the connection with rank-based particle systems, give some historical highlights, and finally discuss some recent developments. No background in mathematical finance will be assumed
12. Januar 2022
Max Nendel (Bielefeld)
Markovian transition semigroups under model uncertainty
Abstract: When considering stochastic processes for the modeling of real world phenomena, a major issue is so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing information about the future in a single stochastic framework. In a dynamic setting, this leads to the task of constructing consistent families of nonlinear transition semigroups. In this talk, we present two ways to incorporate model uncertainty into Markovian dynamics. One approach considers parameter uncertainty in the generator of a Markov process while the other considers perturbations of a reference model within a Wasserstein proximity. In both cases, we are able to compute the infinitesimal generator of the semigroup, and show that, in typical situations, these two a priori different approaches de facto lead to the same nonlinear transition semigroup. Moreover, we discuss the connection between upper envelopes of Markovian transition semigroups, arbitrage bounds for prices of European contingent claims, and abstract Hamilton-Jacobi-Bellman-type differential equations. The talk is based on joint works with Robert Denk, Sven Fuhrmann, Michael Kupper, and Michael Röckner.
26. Januar 2022
Dirk Erhard (Universidade Federal de Bahia) ONLINE-Vortrag
Weak coupling limit of the Anisotropic KPZ equation
Abstract: In this talk I will discuss the two-dimensional anisotropic KPZ equation formally given by $\partial_th = \frac{1}{2} \Delta h + \lambda [(\partial_xh)^2 - (\partial_y h)^2] + \xi.$ Here, $\xi$ denotes a space-time white noise and $\lambda$ is a so-called coupling constant. Due to the wild oscillations of $\xi$ the above equation is analytically ill-posed, and is critical in the sense that it just falls out of the framework of the theory of regularity structures. Considering a regularised version of the above equation I will explain that as $\lambda$ goes to zero in a suitable way as one removes the regularisation parameter, then the sequence of equations converges to the stochastic heat equation $\partial_th = \frac{\nu}{2} \Delta h + \sqrt{\nu\xi},$ where $\nu > 1$ is an explicit constant. The above result in particular shows that the nonlinearity does not simply vanish with $\lambda$ but produces a Laplacian and a noise term. This is joint work with Giuseppe Cannizzaro and Fabio Toninelli.
09. Februar 2022
Peter Pfaffelhuber (Freiburg)
tba
Abstract: