# Colloquium in probability and other talks in winter term 2016/17

**Organisers**: Nina Gantert (TUM), Noam Berger (TUM), Markus Heydenreich (LMU), Franz Merkl (LMU), Silke Rolles (TUM), Konstantinos Panagiotou (LMU),

**Talks**:

**Monday**, 17^{th} October 2016, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Simon Reisser (LMU)

Title: Rumor spreading algorithms on expander graphs

Abstract: In this thesis three popular and well-studied randomized rumor spreading algorithms are studied, the push, pull and push\&pull model. Initially, some vertex owns a rumor and it is passed to one of its neighbours in a way depending on the rumor spreading algorithm. For the push algorithm every vertex that knows the rumor passes it to a random neighbour. For the pull algorithm every uninformed vertex ask the rumor from a random neighbour. The push\&pull algorithm is a combination of both, every vertex either asks or passes the information. This spreading is repeated until every vertex knows the rumor. The main question is how many rounds does it take on a given graphs until all vertices are informed. Here the asymptotic (random) broadcast time for these algorithms on expander graphs, which are almost regular and have small spectral expansion, is studied. The time for the push model was already shown to be $\log_2 n + \log n +o(\log n)$, which is asymptotically the same time it takes on the complete graph. The main result of this thesis is that for the pull and the push\&pull algorithm the same holds, i.e. the asymptotic runtime on expander graphs coincides with the runtime on the complete graph. For the pull model the runtime is $\log_2 n +o(\log n)$ and for the push\&pull model it is $\log_3 n+o(\log n)$.

**Monday**, 24^{th} October 2016, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Stein Andreas Bethuelsen (TUM)

Title: Stochastic domination in space-time for the contact process

Abstract: The contact process is a classical interacting particle system. Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for Z^d and (for infection rate sufficiently large) d-ary homogeneous trees T_d. In this talk we will discuss some space-time versions of their results. In particular, we ask the question whether the contact process may dominate an independent spin-flip process. The answer to this question seems to depend on properties of the graph. We first show that it is not possible if the graph is amenable. On the other hand, we prove some results indicating that it is indeed the case for the contact process on T_d. This talk is based on joint work with Rob van den Berg (CWI and VU Amsterdam).

**Monday**, 14^{th} November 2016

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Friday**, 25^{th} November 2016, 10:00, TUM, room 2.02.11, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Takashi Kumagai (Kyoto University, Kyoto, Japan)

Title: Time changes of stochastic processes associated with resistance forms

Abstract:In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. Fundamental examples of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion, the introduction of which was motivated by the study of the localization and aging properties of physical spin systems, and the two-dimensional Liouville Brownian motion, which is the diffusion naturally associated with planar Liouville quantum gravity. This FIN diffusion is known to be the scaling limit of the one-dimensional Bouchaud trap model, and the two-dimensional Liouville Brownian motion is conjectured to be the scaling limit of simple random walks on random planar maps. We will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals. If time permits, we also discuss heat kernel estimates for the relevant time-changed processes. This is a joint work with D. Croydon (Warwick) and B.M. Hambly (Oxford).

**Monday**, 28^{th} November 2016, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Prof. Dr. Florian Theil (Warwick University)

Title: TBA

**Monday**, 12^{th} December 2016

Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM

**Monday**, 09^{th} January 2017, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Prof. Dr. Fabio Maccheroni (Universita' Bocconi)

Title: Ambiguity and robust statistics

Abstract: We formally clarify the relation between decision theory under ambiguity and the minimax approach of robust statistics.

**Thursday**, 12^{th} January 2017, 14:00,TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Dr. Konrad Kolesko (Universität Darmstadt)

Title: Local behaviour of critical cascade measures

Abstract: In the talk I will present the behavior of a cascade measure near a typical point i.e. the measure of a small ball around a point sampled according to this measure. I will also describe the connections with Liouville measures and random planar maps.

**Monday**, 16^{th} January 2017, 16:30, TUM, room 2.01.10, Parkring 11, Garching-Hochbrück (Technische Universität München)

Franziska Kühn (TU Dresden)

Title: Transition probabilities for Lévy-type processes

Abstract: In this talk we present an existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their, so-called, symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes processes with a given symbol under weak assumptions on the regularity (with respect to the space variable) of the symbol. We derive heat kernel estimates for the transition density as well as its time derivative, and prove the well-posedness of the corresponding martingale problem. Our result gives, in particular, existence results for stable-like, relativistic stable-like and normal tempered stable-like processes. Moreover, in dimension d=1, we obtain existence and uniqueness results for solutions of Lévy-driven SDEs with Hölder continuous coefficients.

**Monday**, 30^{th} January 2017, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Dr. Daniel Valesin (Rijksuniversiteit Groningen)

Title: Extinction time of the contact process on general graphs

Abstract: Complementing earlier work by Mountford, Mourrat, Valesin and Yao, we study metastable behavior of the contact process on general finite and connected graphs. For the contact process with infection rate $\lambda$ on a graph $G$, the extinction time is the random amount of time until the process started from all individuals infected reaches the trap state in which the infection is absent. We prove, without any restriction on $G$, that if $\lambda$ is larger than the critical rate of the one-dimensional process, then the extinction time grows faster than $\exp(|G|/(\log |G|)^a)$ for any constant $a > 1$, where $|G|$ denotes the number of vertices of $G$. Also for general graphs, we show that the extinction time divided by its expectation converges in distribution, as the number of vertices tends to infinity, to the exponential distribution with parameter 1. Joint work with Bruno Schapira.

**Monday**, 13^{th} February 2017, 16:30, LMU, room B 252, Theresienstr. 39, Munich

Dr. Benedikt Jahnel (WIAS Berlin)

Title: The Widom-Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

Abstract: We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-almost sure quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time $t_G>0$ the model is almost sure quasilocal. For the color-symmetric model there is no reentrance. On the constructive side, for all $t>t_G$, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

How to get to Garching-Hochbrück