# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45

## Monday, September 20

16:30 - 17:30 Noemi Kurt (TU Berlin): Survival and extinction of branching annihilating random walk

Abstract: Branching annihilating random walks (BARW) are stochastic processes where particles perform continuous time random walks, branch at some rate, and annihilate when two of them meet. They occur for example as dual processes in certain population biological models, and are mathematically difficult to treat because of lack of monotonicity. We consider double-branching BARW on Z, which belongs to the so-called "parity preserving universality class", for which physicists conjecture a phase transition between a survival and an extinction regime, depending on the branching rate relative to the migration rate. We present some results about survival and extinction for different branching mechanisms, and give one particular mechanism (which we call "caring"), which allows either survival with positive probability if the branching rate is large enough, or a.s. extinction for small branching rate.

17:30 - 18:30 Stefan Grosskinsky (University of Warwick): Questions in zero-range condensation

Abstract: Zero-range processes are stochastic particle systems where the number of particles per site is unbounded (no exclusion). They can exhibit a condensation transition, where above a critical density a finite fraction of all particles in the system concentrates on a single lattice site. Mathematically, this is related to large deviations of heavy-tailed random variables and a breakdown of the usual law of large numbers. One origin of condensation can be spatial inhomogeneities/disorder in the jump rates for which there is a small spin glass-type open problem of computing the quenched free energy. In spatially homogeneous systems condensation is due to strong enough particle attraction, and there are several interesting dynamical problems related to metastability.

## Monday, September 27

16:30 - 17:30 Zakhar Kabluchko (Universität Ulm): Distribution of levels in high-dimensional random landscapes

Abstract: We prove empirical central limit theorems for the distribution of levels of various random fields defined on high-dimensional discrete structures as the dimension of the structure goes to infinity. The random fields considered include costs of assignments, lengths of Hamiltonian cycles and spanning trees, energies of directed polymers, locations of particles in the branching random walk, as well as energies in the Sherrington-Kirkpatrick and Edwards-Anderson models. The distribution of levels in all models listed above is shown to be essentially the same as in a stationary Gaussian process with regularly varying non-summable covariance function. This type of behavior is different from the Brownian bridge-type limit known for independent or stationary weakly dependent sequences of random variables.

17:30 - 18:30 Louis-Pierre Arguin (New York University): Ground states of disordered systems

Abstract: Ground states are configurations on a given state space that minimize a given functional on that space. I will consider the Edwards-Anderson model where the state space is {-1,+1}^{Z^d} and the functional (the energy) is random. An open question is to determine the number of ground states in dimension d. I will show how basic properties of ground states greatly restrict their number and present ideas that recently led to prove that the ground state is unique (in a probabilistic sense) in d=2 on the half-plane. Pictures will be drawn.

## Monday, October 4

16:30 - 17:30 Nicola Kistler (Universität Bonn): Some solvable models of Spin Glasses

Abstract: I will present some mean field Spin Glasses which are amenable to a rigorous study, and shed some light on the crucial issues of the Parisi Theory, such as the ultrametricity and the chaotic behavior.

17:30 - 18:30 Lihu Xu (EURANDOM): Ergodicity of 3D stochastic Navier-Stokes equation

Abstract: In this talk, I will show the ergodicity of 3D stochastic Navier-Stokes equation. Because of the time restriction, I will first show the sketch procedure of the approach. Then I will concentrate on how to prove the strong Feller property by Malliavin calculus.

## Monday, October 11

16:30 - 17:30 Eulalia Nualart (Université Paris 13): Hitting probabilities and capacity for SPDEs

Abstract: It is well-known that the probability that a d-dimensional Brownian motion hits a given set is related to the Newtonian capacity of the set. In this talk we will show how one can relate the hitting probabilities of random fields that are solutions to systems of non-linear SPDEs to the Newtonian capacity. We will then focus in the particular case of the non-linear stochastic heat equation.

17:30 - 18:30 Arno Siri-Jégousse: The length of the Beta-coalescent

Abstract: The Beta-coalescent is the process appearing as the limit of genealogies of a certain population model. Estimation of mutation rates of species is an interesting problem in population genetics, which is highly connected to the study of the length of the coalescent tree. We will give a few asymptotic results on this type of ancestral tree, when the initial population grows to infinity.

## Monday, October 18

16:30 - 17:30 Christoph Aistleitner (TU Graz): Weak dependence in probabilistic number theory

Abstract: We present several problems and results from probabilistic number theory. In particular we show how techniques from number theory, Fourier analysis and probability theory are combined to obtain precise asymptotic results in probabilistic discrepancy theory and metric Diophantine approximation.

## Monday, October 25

16:30 - 17:30 David Cohen (Universitaet Basel): Stochastic Trigonometric Methods

Abstract: We will discuss the numerical discretization of stochastic oscillators with a high frequency. The proposed numerical schemes permit the use of large step sizes and offer various additional properties. These new numerical integrators can be viewed as a stochastic-generalization of the trigonometric integrators for highly oscillatory deterministic problems.

17:30 - 18:30 Carlo Marinelli (Università di Bolzano): Tools for SPDEs (part I)

Abstract: After a brief discussion on the relationship among different notions of solution for stochastic PDEs, we are going to show several techniques to obtain maximal inequalities for stochastic convolutions (also with respect to processes with discontinuous paths). This is the first of 3 lectures focusing on some "universal" tools for the study of stochastic evolution equations.

## Tuesday, November 16

18:15 - 19:15 Anton Klimovsky: Parisi-type formulae as traces of hierarchical self-organization in range-free spin glasses with Gaussian Hamiltonians

Abstract: The celebrated Parisi formula gives a variational expression for the free energy of the Sherrington-Kirkpatrick (SK) model, i.e., the logarithmic asymptotics of the sum of exponentials of a strongly correlated high-dimensional Gaussian field, as the dimension tends to infinity. The expression has a remarkable structure involving a solution of a terminal-value problem for a Burgers-like PDE with a control parameter. The original derivation by Parisi relied upon several unrigorous heuristics: the so-called replica trick and certain ingenious "hierarchical replica symmetry breaking Ansatz". These heuristics led to many important applications well beyond the SK model. In particular, to the problems that can be recasted as the search of extrema on high-dimensional rugged landscapes with range-free correlations (e.g., coming from NP hard combinatorial optimization problems). We will review some rigorous techniques (mainly due to Guerra and Talagrand) that have recently led to a proof of the Parisi formula in the context of the SK model. The techniques are based on relating the extreme value structures that occur in the thermodynamic limit of the (hierarchical and completely solvable) generalized random energy model (GREM) with those of the SK model. For that purpose, we employ the illuminating comparison scheme of Aizenman, Sims & Starr and a reformulation of Guerra’s scheme using the same "language".

## Monday, November 22

16:30 - 17:30 Onur Gün (Université de Provence-Aix Marseille 1): Extremal Aging for Sherrington-Kirkpatrick model and p-spin models of spin glasses

Abstract: We consider Random Hopping Time (RHT) dynamics of mean field spin glass models that can be seen as a (random) time change of the simple random walk on the state space. For Sherrington - Kirkpatrick spin glass and p-spin models, we prove that under a proper normalization the clock process (time change process) converges to an extremal process and the system exhibits aging like behavior.

17:30 - 18:30 Andrej Depperschmidt (Universität Freiburg): Asymptotics of a Brownian ratchet

Abstract: Motivated by the protein translocation model introduced by Peskin, Odell and Oster (1993) we consider a Brownian ratchet. This process is defined as a reflecting Brownian motion Bt with moving reflection boundary given by a non-decreasing jump process Rt. At rate proportional to Bt-Rt a new reflection boundary is chosen uniformly in the interval [Rt,Bt]. In the talk we outline the proof of the law of large numbers and the central limit theorem for the Brownian ratchet. As time permits, we also discuss a biologically relevant generalization in which the reflected Brownian motion is replaced by reflected Brownian motion with negative drift.

The talk is based on joint work with Peter Pfaffelhuber and Sophia Götz.

## Monday, November 29

16:30 - 17:30 Véronique Gayrard (Université de Provence): Convergence of clock processes in random environments and ageing in the p-spin SK model

Abstract: I will first present a general criterion for the convergence of clock processes in random dynamics in random environments that is applicable in cases when correlations are not negligible. I will then show how this criterion can be applied to the random hopping time dynamics of the p-spin SK model to prove that (on a wide range of time scales) the clock process converges to a stable subordinator almost surely with respect to the environment. This result was recently obtained in joint work with Anton Bovier.

17:30 - 18:30 Anton Klymovskiy (Technische Universiteit Eindhoven): Hierarchically interacting spatial Lambda-Flemming-Viot processes

Abstract: We introduce a class of models for spatially extended populations with occasional extreme reproduction (or survival bottleneck) events on a hierarchically structured geographical space. An important tool in the analysis of this model is duality. This model is amenable for multiple space-time scale analysis and displays universal behavior on macroscopic space-time scales. Joint work in progress with A. Greven, F. den Hollander and S. Kliem.

## Monday, December 6

16:30 - 17:30 Anita Winter (Universität Essen): Brownian motion on real trees

Abstract: The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact real tree equipped with a Radon measure. We specify criterion under which the Brownian motion is recurrent or transient.

(this is joint work with Siva Athreya and Michael Eckhoff)

## Wednesday, December 8

10:00 - 10.45 Carlo Marinelli (Università di Bolzano): Tools for SPDEs (part II)

11:00 - 11.45 Carlo Marinelli (Università di Bolzano): Tools for SPDEs (part III)

Abstract: We continue the discussion on how to obtain various estimates for stochastic convolutions (i.e. solutions of linear stochastic evolution equations with additive noise), and how these estimates can be used to obtain well-posedness for some classes of stochastic PDEs with nonlinear accretive coefficients (without any Lipschitz nor local Lipschitz assumption).

## Monday, December 13

16:30 - 18.30 Lluís Quer i Sardanyons (Universitat Autònoma de Barcelona): Stochastic integrals and SPDEs I & II

Abstract: At the first session, we will present some extensions of Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and we will we explore the links between these theories. Somewhat surprisingly, the end results of both theories turn out to be essentially equivalent. Then, at the second session, we will show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat equation driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.

## Tuesday, December 14

18:15 - 19:15 N. Kurt: The competing species model and its duals

Abstract: The competing species model introduced by Blath, Etheridge and Meredith in 2007 describes the evolution of two species on a lattice, competing for resources. It is modeled by certain interacting diffusions. For some special choices of the various parameters involved, dualities to other models, such as random walk in Markovian random environment, or branching annihilating random walks can be found. In this talk, I will present the model and derive the main dualities. For a special case, comparison with oriented percolation yields coexistence in the competing species model, which via duality carries over to a result about survival of branching annihilating random walk on Z.

19.15 - 20:15 A. Depperschmidt: RWRE performed by ancestral lines of the discrete contact process

Abstract: The discrete contact process on Zd can be seen as a toy example of so called locally regulated population models. In contrast to stepping stone models in which the population size at any site and at any time is kept constant, in locally regulated models the population size at any site may change over time. Looking backwards in time the ancestral lines of individuals perform a random walk in the case of stepping stone model and a random walk in a random environment (RWRE) in the case of locally regulated models. To our knowledge, such RWRE do not fall in a class of RWRE covered in the literature.

In the talk we describe the discrete contact process and show that the RWRE performed by the ancestral line of an individual satisfies a law of large numbers and a central limit theorem when averaging over both the randomness of the walks steps and the environment.

This is work in progress with M. Birkner, J. Cerny and N. Gantert.

## Monday, December 20

16:30 - 17:30 Zeev Sobol (Swansea University): Well-posedness of Markov solution to a Martingale problem (canceled)

Abstract: The subject of the talk is construction of a Feller transition probability semi-group in a weighted space of continuous functions, whose resolvent is consistent with the one of the generator of the martingale problem. The prime idea is that a strongly continuous Markov semi-group on a weighted space of (weakly) continuous functions, with the weight of compact level sets (that is, infinite at infinity), is a transition probability semi-group of a standard process. The main problem is that the resolvent of the generator of the martingale problem defined in a weighted space of continuous function may have a non-trivial kernel (pseudo-resolvent). Nevertheless, its restriction to a narrower weighted space appears a strongly continuous Feller resolvent, associated the transition probability semi-group of the unique Markov solution to the martingale problem.

17:30 - 18:30 Christian Litterer (University of Oxford): Rough paths in geometric settings

Abstract: We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-Gamma manifolds.