Berlin Probability Colloquium

Ort
HU Berlin, Institut für Mathematik
Rudower Chaussee 25, Haus 1, Raum 115
12489 BerlinZeit
mittwochs, 17.00 Uhr
Programm
 Christophe Garban (U Lion 1)
 Vortex fluctuations in continuous spin systems and lattice gauge theory
 Abstract: Topological phase transitions were discovered by BerezinskiiKosterlitzThouless (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 integervalued Gaussian Free Field (or Zferromagnet)
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 socalled ''spinwave''. Our approach is nonperturbative 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 BenjamimiSchramm 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 twodimensional 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 timechange of the planar Brownian motion on D. In this talk, we discuss subGaussian 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. Selfrepulsion is introduced via change of measure that penalises particles spending time in an εneighbourhood 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 timeinhomogeneous branching process emerges. The position of the maximum is governed by a FKPP type reactiondiffusion equation with a time dependent reaction term. This is joint work with Anton Bovier.
 17. November 2021
 Boualem Djehiche (KTH Stockholm) ONLINEVortrag
 Propagation of chaos for a class of meanfield 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 meanfield reflected backward stochastic differential equations (BSDEs) with jumps, where the meanfield 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) ONLINEVortrag
 Mean field games with WrightFisher common noise
 Abstract: Motivated by restoration of uniqueness in finite state mean field games, we introduce a common noise which is inspired by WrightFisher models in population genetics. Thus we analyze the master equation of this mean field game, which is a degenerate parabolic secondorder partial differential equation set on the simplex whose characteristics solve the stochastic forwardbackward system associated with the mean field game. We show that this equation, which is a nonlinear 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 Nplayer Nash equilibria converge towards the solution of such a kind of WrightFisher 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 tailormade version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the intensity of a the WrightFisher common noise. Based on joint works with Erhan Byraktar, Asaf Cohen, and François Delarue.
 01. Dezember 2021
 Ilya Chevyrev (Edinburgh) ONLINEVortrag
 Stochastic quantisation of gauge fields
 Abstract: Stochastic portfolio theory was introduced in the nineties by R. Fernholz to model longterm investments in large equity markets. An empirical fact at the center of this theory is the observed stability of the socalled 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 rankbased particle systems, give some historical highlights, and finally discuss some recent developments. No background in mathematical finance will be assumed.
 15. Dezember 2021
 Martin Larsson (Carnegie Mellon)
 Stochastic portfolio theory and rankbased particle systems
 Abstract: Stochastic portfolio theory was introduced in the nineties by R. Fernholz to model longterm investments in large equity markets. An empirical fact at the center of this theory is the observed stability of the socalled 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 rankbased 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 socalled 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 HamiltonJacobiBellmantype 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) ONLINEVortrag
 Weak coupling limit of the Anisotropic KPZ equation
 Abstract: In this talk I will discuss the twodimensional 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 spacetime white noise and $\lambda$ is a socalled coupling constant. Due to the wild oscillations of $\xi$ the above equation is analytically illposed, 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: