Stochastic Dynamics in Action” (21–25 May 2012)

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 ecological 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 motion 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 complex 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 distributed 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 captures 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 problems, 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 effective 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 truncated power moment problem. In this talk, different classes of necessary and sufficient conditions for solving the finite-dimensional moment problems are discussed. It is illustrated why such results can be easier achieved for the full moment problem than for the truncated one, and the best partial solutions known to date are presented. The relation with the classification of pos- itive polynomials is described. A new general approach for truncated moment problems will be presented which overcomes some difficulties. To our knowledge, this approach is also new for finite-dimensional problems, however it may be more adapted for the infinite-dimensional case. Application of these techniques to identification of a random closed set from given two-point coverage probabilities is considered in R. Lachieze-Rey and I. Molchanov (2011). Finally, it is shown how all known restrictions for realizability can be easily derived using our criteria. (In collaboration with Eugene Speer and Joel Lebowitz.) 13. Lachowicz, Mirosław (Warsaw, Poland) “Individual-based modeling of nonlinear systems in mathematical biology” Abstract: A general approach that allows to construct Markov processes describing various processes in mathematical biology (or in other applied sciences) is presented. The Markov processes are of a jump type, and the starting point is the related linear equations. They describe at the micro-scale level the behavior of a large number N of interacting individuals (entities). The large individual limit (“N → ∞”) is studied and the intermediate level (the meso-scale level) is given in terms of nonlinear kinetic-type equations. Finally, the corresponding systems of nonlinear ODEs (or PDEs) at the macroscopic level (in terms of densities of the interacting subpopulations) are obtained. Mathematical relationships between these three possible descrip- tions are presented and explicit error estimates are given. The general framework is applied to propose the microscopic and mesoscopic models that correspond to well-known systems of

3 nonlinear equations in biomathematics. 14. Lythe, Grant (Leeds, UK) “Stochastic dynamics and the adaptive immune system” Abstract: I will discuss mathematical immunology, concentrating on T cells. I will consider models of how a diverse repertoire of T-cell clonotypes is maintained, how signalling through the T-cell receptor activates a T cell, and timescales for the interaction of T cells with dendritic cells in a lymph node. In each case, the model is stochastic and nonequilibrium. The relevant quantities are passage times, collision times and escape times, either in a continuous-time Markov chain model or in a continuous-space Brownian motion model. 15. Lytvynov, Eugene (Swansea, UK) “Stochastic dynamics of binary jumps in continuum” Abstract: Let Γ denote the space of all locally finite subsets (configurations) in Rd. A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over Rd. We will start the talk with a construction of an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing measure. We will next show that a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We will also study another scaling limit, which leads to a spatial birth-and-death process in continuum. In the second part of the talk, we will discuss a non-equilibrium dynamics of binary jumps. We will prove the existence of an evolution of correlation functions on a finite time interval. We will show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density. (Joint with D. Finkelshtein, Yu. Kondratiev and O. Kutoviy) 16. Makhnovskii, Yurii (Moscow, Russia) “Rectification in a periodically tapered tube” Abstract: We study transport of a Brownian particle in a periodically tapering tube, induced by a longitudinal time-periodic force with zero mean. It is shown that under the action of this force the particle drifts in the direction opposite to that in which a permanently applied load force acts. At large amplitude of the driving force, when the effect is maximal, we obtain analytical solutions for the effective drift velocity, the stopping force (the load value which nullifies the effect), and the efficiency of the energy conversion from the input energy into the useful work against the load force. In the range of their applicability, these analytical solutions are in excellent agreement with the results obtained from Brownian dynamics simulations. 17. Molchanov, Stanislav (UNC-Charlotte, USA) “Stochastic dynamics in random nonstationary environment” Abstract: We consider branching random walk on the lattice Zd, d ≥ 1 (with heavy tails in dimensions d = 1, 2). Let β(t, x), µ(t, x) be the rates of particle duplication and death, re- spectively, and assume that the random fields β(·), µ(·) are stationary and short-correlated in time and independent at different x ∈ Zd. Assume also that hβi = hµi (the criticality condition) and that the initial particle field n(0, x) is given by the system of i.i.d. random variables, hn(0, ·)i = ρ > 0. Under appropriate technical assumptions we prove the tightness of the finite- dimensional distributions of n(t, x), i.e., roughly speaking, the existence of the limit n(∞, x) homogeneous in space. Note that for random fields β(·), µ(·) independent of time t, a limiting law does not exist. 18. Ovaskainen, Otso (Helsinki, Finland) “Stochastic dynamics in Ecology and Evolution” Abstract: I discuss stochastic models of animal movement, population dynamics and evolutionary dynamics, focusing on the interplay between environmental heterogeneity and biological processes. I consider both theoretical approaches examining the link between the underlying as-

4 sumptions and the emerging patterns, and statistical approaches aimed at interpreting data. I first discuss how mark-recapture data (e.g. on butterfly movements) relates to diffusion-advection- reaction models and on underlying random walk-based movement models. Here environmental heterogeneity is modeled either through a discrete set of habitat types (with habitat selection at boundaries) or through continuously varying habitat quality. The presence of linear elements (such as movement corridors or barriers) or preferred movement lines can lead to a mixture of two- and one-dimensional diffusions or to anisotropic diffusion. I then discuss how diverse ecological and evolutionary phenomena can be modeled by spatio-temporal point processes. In this framework environmental heterogeneity is modeled e.g. through a smoothed point field, al- lowing one to control parameters such as patch size, patch quality, and patch turnover rate. The spatial and stochastic individual-based models can be analyzed by constructing a perturbation expansion around the mean-field obtained at the limit of global interactions. This approach pro- vides results that match with simulations but it involves unresolved mathematical challenges. As an example I discuss how the evolution of dispersal distance depends on landscape structure, life-history parameters and the approach taken to model evolutionary dynamics (adaptive dynamics vs. mutation-selection-drift balance). 19. Palamara, Gian Marco (Zurich, Switzerland) “Predation effects on mean time to extinction under demographic stochasticity” Methods for predicting the probability and timing of a species’ extinction are based on a combi- nation of theoretical models and empirical data, and focus on single species population dynamics. Of course, species also interact with each other, forming more or less complex networks of interactions. However, models to assess extinction risk often lack explicit incorporation of these interspecific interactions. We derive a birth-and-death process in which the death rate includes a general nonlinear ex- pression for the functional response of predation. We then investigate the effects of the foraging parameters (e.g., attack rate and handling time) on the predicted time to extinction. Mean time to extinction varies by orders of magnitude when we alter the foraging parameters, even when we exclude the effects of these parameters on the equilibrium population size. In particular, we observe an exponential dependence of the mean time to extinction on handling time. These findings clearly show that accounting for the nature of interspecific interactions is likely to be critically important when estimating extinction risk. We intend to extend this work to investigate the interactions between species’ interaction parameters and extinction risk in complex communities. 20. Pasurek, Tatiana (Bielefeld, Germany) “Equilibrium states over the cone of discrete measures” Abstract: We construct Gibbs perturbations of the Gamma process in Rd, which may be used in applications to model systems of densely distributed particles. We propose a definition of Gibbs states over the cone of discrete Radon measures on Rd and analyze conditions for their existence. Our approach works for general Levy´ processes under an assumption that their Levy´ intensity measures have first two moments finite. Also uniform moment estimates are obtained, which are essential for the construction of related diffusions. Based on joint work with D. Hage- dorn, Yu. Kondratiev and M. Rockner.¨ 21. Rockner,¨ Michael (Bielefeld, Germany) “The stochastic quasi-geostrophic equation” Abstract: (Joint work with Rongchan Zhu and Xiangchan Zhu.) Consider the 2D stochastic

5 quasi-geostrophic equation on the torus T2 for general parameter α ∈ (0, 1) dθ(t, ξ) + κ(−∆)αθ(t, ξ) dt + u(t, ξ) · ∇θ(t, ξ) dt = G(t, θ(t, ξ)) dW (t, ξ) 2 ((t, ξ) ∈ [0,T ] × T ), ⊥ − 1 u(t) = −∇ ((−∆) 2 θ(t)), θ(0, ξ) = θ0(ξ),

on the Hilbert space 2 2 R H := θ ∈ L ( )|: 2 θ(ξ) dξ = 0 . T T The talk will give a survey on recent results on this equation. These include existence of weak solutions for additive noise, existence of martingale solutions and Markov selections for multiplicative noise and, under some condition, pathwise uniqueness for all α ∈ (0, 1). Furthermore, 1 in the subcritical case α > 2 , we prove existence and uniqueness of (probabilistically) strong 2 solutions. In addition, we prove ergodicity for α > 3 , provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. We estab- 1 lish the large deviation principle for the stochastic quasi-geostrophic equations for α > 2 for small multiplicative noise. An analogous result is also obtained for the small time asymptotics. (These results are joint with Wei Liu.) The existence of a random attractor for the solutions 1 of the stochastic quasi-geostrophic equation for α > 2 driven by real multiplicative noise and function-valued additive noise is also established. Time permitting, we shall also report on a 1 very recent result about ergodicity for the general subcritical case α > 2 and for degenerate noise. 22. Soshnikov, Alexander (UC Davis, USA) “Outliers in spectrum of finite rank deformations of large random matrices: Delocalized case” Abstract: The talk is based my recent preprint arXiv:1203.5130 math.PR (joint with David Ren- frew) about finite rank deformations of Wigner random matrices. Specifically, we will study the limiting distribution of the outliers in the spectrum and prove its universality provided the eigen- vectors of finite rank perturbation are delocalized. These results are in contrast with the non- universal distribution of the outliers in the localized case that was studied in arXiv:1103.3731, arXiv:1104.1663, arXiv:1103.1170, by Sean O’Rourke, Alessandro Pizzo, David Renfrew, and myself. 23. Stucki, Kaspar (Bern, Switzerland) “On rates for Gibbs point process approximations” Abstract: We use Stein’s method to bound the total variation distance between two Gibbs processes. The key ingredient is a coupling of spatial birth-and-death processes which have as stationary distribution one of the Gibbs processes under consideration, but are started at different configurations. We bound the expected coupling time to obtain the so-called Stein factors. Examples to which our method is applicable are inhibitory pairwise interaction processes, e.g hardcore or Strauss processes, or more generally, locally stable Gibbs processes. This is work in progress, jointly with Dominic Schuhmacher. 24. Vasiliev, Alexander (Bergen, Norway) “Loewner evolution driven by a stochastic boundary point” Abstract: We consider evolution in the complex plane in which the sample paths are represented by the trajectories of points evolving randomly under the generalized Loewner equation. The driving mechanism differs from the SLE equation. 25. Vilela Mendes, Rui (Lisbon, Portugal) “Stochastic solutions of PDEs: A tool for local behavior and parallel computing” Abstract: Stochastic solutions provide new rigorous results for nonlinear PDEs and are a natural tool for parallel computation. There are two different approaches for the construction of

6 stochastic solutions: McKean’s and superprocesses. A review is made of some of the stochastic solutions constructed for kinetic equations as well as the uses and limitations of superprocesses. An extension of superprocesses to signed measures and distributions is proposed. 26. Yang, Wei (Warwick, UK) “Mean-field games and nonlinear Markov processes” Abstract: In a system of N interacting agents (who may have competing or common interests), the complexity of the analysis may become immense as N tends to infinity. However, under certain general assumptions, the asymptotic problem can be described by a well-manageable measure-valued deterministic evolution (a mean-field characteristic) representing a dynamic law of large numbers. Probabilistic analysis of the latter evolution leads to the notion of a nonlinear Markov process. Mean-field games appear in papers by J. M. Lasry and P. L. Lions (France) and by M. Huang, R. P. Malhame´ and P. Cains (Canada), where the underlying processes are Brownian motions. In this talk, we present a general methodology of mean-field games based on the theory of nonlinear Markov processes. The realization of this program includes three steps. First, we prove the well-posedness of general kinetic equations with coupled functional parameters. Second, we prove the well-posedness and sensitivity for the Hamilton–Jacobi–Bellman (HJB) equation. Finally, we justify mean-field games as a proper approximation of large population systems. We develop this methodology for a fully general setting with arbitrary underlying Markov processes of interactions, and prove convergence results with 1/N rate. These estimates are new even for the diffusion case. This is a joint work with Vassili Kolokoltsov and Jiajie Li. 27. Zhizhina, Elena (IITP, Moscow, Russia) “Gibbs field approach and stochastic optimization in image processing (including biological images)” Abstract: We give a brief overview of how Gibbs fields and related stochastic dynamics approaches are applied in image processing and bioinformatics. Problems of detection and fea- ture extraction are considered in the framework of Gibbs point random fields. A new multiple birth-and-death algorithm based on birth-and-death stochastic dynamics in continuum and its discrete time approximation are presented. We discuss new results concerning double annealing regimes in the multiple birth-and-death stochastic algorithms.