ZiF Research Group “Stochastic Dynamics” (May–September 2012) Workshop 1 “Stochastic Dynamics in Action” (21–25 May 2012) Programme (speakers in alphabetical order) 1. van Baalen, Minus (Paris, France) “Emerging units of adaptation in spatially extended systems” Abstract: The apparence of discreteness notwithstanding, every living being is in reality a more or less coherent association of members of several species. For instance, although the details are unclear, eukaryotic cells are very likely the result of a fusion of several species of bacte- ria. Another well known example is that of lichens, which are temporal associations of fungi and one or several species of algae. But also a cow is an association: without its intestinal flora it cannot survive. Such symbiotic associations can be found at every level of ecologi- cal organisation, from the level of viruses and cells up to colonies of termites and other social insects, and maybe even larger things. The emergence of such associations in the course of evolution poses theoretical challenges. First of all, it requires cooperation between multiple partners, where unilateral defection (cheating) is often more advantageous. Game theory has nevertheless established that the conditions favouring cooperation are often quite large. More difficult to explain is why members of an association let the common good prevail over their private interests. An association therefore needs a way to resolve potential conflict between partners. Indeed, several mechanisms have evolved to this end: for instance, reinforcing verti- cal transmission, which has the tendency to align evolutionary interests, but also policing and punishment of cheaters, which requires the capacity of active surveillance. If one accepts that what we identify as an ‘individual’ is often in reality a symbiotic association, we are left with the problem of defining what is an adaptation: adaptations are seen as those traits that benefit an individual. Traditional approaches like population genetics and adaptive dynamics typically consider individuals a basic indivisible units and members of a single species. I will briefly discuss how theory has to be extended in order to study the evolution of multispecies symbiotic associations. This extended theory leads to the notion of the unit of adaptation as an emergent structure in evolution. 2. Bakhtin, Yuri (Georgia Tech, USA) “Space-time stationary solutions for the Burgers equation” Abstract: The Burgers equation is one of the basic hydrodynamic models. It describes the evolution of velocity fields of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990’s. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk, I will discuss the quasi-compact case where the random forcing decays to zero at infinity and the completely noncompact case of forcing that is stationary in space-time. The main result is the description of the ergodic components for the dynamics and One Force One Solution principle on each of the components. Joint work with Eric Cator and Kostya Khanin. 3. Daletskii, Alexei (York, UK) “Gibbs states of disordered systems of anharmonic oscillators” Abstract: Let X = Rd and let γ be a random discrete subset of X distributed according to a Poisson measure (or more general point process). We prove the existence of Gibbs measures on Xγ, which are associated with a system of interacting anharmonic oscillators, and discuss their 1 properties. (Based on joint work with Yu. Kondratiev, Yu. Kozitsky and T. Pasurek) 4. Delius, Gustav (York, UK) “Stochastic waves of abundance in a size-structured population model” Abstract: We introduce a stochastic model describing the behaviour and interaction of fish in the ocean, taking into account the size of the individuals and its influence on predation, repro- duction, metabolic loss and mortality. We exploit the formal similarity between this stochastic model of fish and second-quantized models of elementary particle physics. This allows us to use quantum field theory methods to demonstrate the existence of stochastic travelling waves in our model. These describe the propagation of peaks of abundance of fish through the trophic chain from small fish to large fish. We comment on the challenges created by the fact that our model contains non-local interactions. 5. Funaki, Tadahisa (Tokyo, Japan) “Invariant measure for SPDE related to the KPZ equation” Abstract: The particle system approximation to the Cole–Hopf solution of the Kardar–Parisi– Zhang equation due to Bertini and Giacomin shows the invariance of geometric Brownian mo- tion for the stochastic heat equation. I will discuss some direct approach to this problem. This is a joint work with Jeremy Quastel. 6. Gotze,¨ Friedrich (Bielefeld, Germany) “Local universality of repulsive particle systems” Abstract: We show that local correlation universality holds for deformations of the GUE eigen- value density, which don’t allow a spectral interpretation but share the same local repulsion exponents with the GUE eigenvalues. This is joint work with M. Venker. 7. Grigor’yan, Alexander (Bielefeld, Germany) “Markov processes on ultra-metric spaces” Abstract: We consider a discrete ultra-metric space (X, m) with a measure m, and construct 2 in a natural way a symmetric Markov semigroup {Pt}t≥0 in L (X, m) and the corresponding Markov process {Xt}. We prove upper and lower bounds of its transition function and its Green function, give a criterion for the transience, and estimate its moments. Based on a joint work with A. Bendikov and Ch. Pittet. 8. Hatzikirou, Haralampos (Braunschweig, Germany) “On the virtue of simple models: can they help in cancer research?” Abstract: Invasion of malignant brain tumors is typically very aggressive and a highly com- plex phenomenon involving molecular and cellular processes at various spatiotemporal scales, whose precise interplay is still not fully understood. We develop a simple mathematical model that focuses on a crucial aspect of this type of tumor, i.e. the migration/proliferation dichotomy. The focus of the talk is to exemplify that mathematical analysis of such a model may provide us novel biomedical insights. 9. Hurth, Tobias (Georgia Tech, USA) “Invariant densities for dynamical systems with random switching” Abstract: We consider a finite set of smooth vector fields on a smooth manifold. Given an initial point on the manifold, along with an initial vector field, we define a stochastic process that follows the trajectory induced by the point and the vector field for an exponentially dis- tributed time, then switches to a new vector field, again following the corresponding trajectory for an exponentially distributed time. The two-component process whose first component cap- tures the position on the manifold, and whose second component captures the driving vector field, is a Markov process. For invariant measures of its semigroup, we describe Hoermander bracket conditions on the vector fields that guarantee the following: Provided that an invariant measure exists, it is unique and absolutely continuous with respect to Lebesgue measure. We 2 will also present recent results by Benaim, Le Borgne, Malrieu and Zitt that show existence of the invariant measure, as well as convergence of the distribution to the invariant measure in total-variation norm. This is joint work with my advisor, Yuri Bakhtin. 10. Kassmann, Moritz (Bielefeld, Germany) “Regularity theory for parabolic nonlocal operators” Abstract: We extend local regularity results for partial differential operators to nonlocal prob- lems, i.e., we prove Holder-regularity¨ and a Harnack-type inequality for weak solutions to parabolic equations with integro-differential operators of fractional order. Assumptions on the kernel are discussed in detail and several examples are provided. This work is joint with M. Felsinger. 11. Kramer, Peter (RPI, Troy, USA) “Analytical theory for cascade formation in clustered scale-free neuronal network model” Abstract: We will present a second order calculation with explicit reference to clusters (tri- angles) to characterize the susceptibility of a network of stochastic integrate-and-fire neurons to giant cascading events as a function of the underlying network and dynamical parameters. The network is scale-free with a high degree of clustering. The results of the calculation are in excellent agreement with direct numerical simulations, and markedly superior to a calculation assuming a tree-like network. 12. Kuna, Tobias (Reading, UK) “Realizability of point processes” Abstract: To reconstruct from observable quantities, in a systematic way, the underlying effec- tive description of a complex system on relevant scales is a task of enormous practical relevance. The partial question of whether the system can be described by point-like objects on a suitable scale is called the ‘realizability problem’. The realizability problem is identified as an infinite-dimensional version of the classical trun- cated power moment problem. In this talk, different classes of necessary and sufficient condi- tions for solving the finite-dimensional moment problems are discussed. It is illustrated why such results can be easier achieved