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Discrete Comput. On the topology of moduli spaces of non-negatively curved R iemannian metrics. The domination monoid in o-minimal theories. Multilevel P icard iterations for solving smooth semilinear parabolic heat equations. Partial Differ. Weak convergence rates for spatial spectral G alerkin approximations of semilinear stochastic wave equations with multiplicative noise.

Reconstruction of twisted S teinberg algebras. International Mathematics Research Notices , November A counterexample to the unit conjecture for group rings. Annals of Mathematics , 3 —, November Weakly binary expansions of dense meet-trees. MLQ Math. Nonlinearity , 34 12 —, November Cannon— T hurston maps for CAT 0 groups with isolated flats. On the range of the relative higher index and the higher rho-invariant for positive scalar curvature.

On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values. Stochastic Process. Parabolic induction via the parabolic pro-p I wahori- H ecke algebra. Representation theory , —, October Existence of metrics maximizing the first eigenvalue on non-orientable surfaces.

Theory , 11 3 —, September Spatial S obolev regularity for stochastic B urgers equations with additive trace class noise. Nonlinear Anal. Non-negatively curved GKM orbifolds. Mathematische Zeitschrift , September On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete Contin. Deep splitting method for parabolic PDE s. SIAM J. Perturbative and geometric analysis of the quartic K ontsevich model. Methods Appl.

Transactions of the American Mathematical Society , September Transfer maps in generalized group homology via submanifolds. A new look at random projections of the cube and general product measures. Bernoulli , 27 3 —, August Journal of Open Source Software , 6 64 , August Tall cardinals in extender models.

Notre Dame J. Osaka J. Math , August Simplicity of the automorphism groups of order and tournament expansions of homogeneous structures. Algebra , —62, August A first-order framework for inquisitive modal logic. The typical cell of a V oronoi tessellation on the sphere. A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of K olmogorov partial differential equations with constant diffusion and nonlinear drift coefficients.

Journal of the Institute of Mathematics of Jussieu , pages 1—80, July Ostaszewski, K. Glassmeier, C. Goetz, P. Heinisch, P. Henri, S. Park, H. Ranocha , I. Richter, M. Rubin, and B. Annales Geophysicae , 39 4 —, July Conical tessellations associated with W eyl chambers.

International Mathematics Research Notices , 14 —, July Phase-field approximation of functionals defined on piecewise-rigid maps. Nonlinear Sci. Solving the K olmogorov PDE by means of deep learning. Phase transition for the volume of high-dimensional random polytopes. Random Structures Algorithms , 58 4 —, July Tate motives on W itt vector affine flag varieties. Selecta Math. Non-convergence of stochastic gradient descent in the training of deep neural networks.

Wolters, and Christian Engwer. DUNEuro - A software toolbox for forward modeling in bioelectromagnetism. Signaling gradients in surface dynamics as basis for planarian regeneration. Journal of Mathematical Biology , June Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes.

Communications on Applied Mathematics and Computation , June Solving high-dimensional optimal stopping problems using deep learning. European J. On unitary representations of algebraic groups over local fields. Representation Theory , —, June The motivic S atake equivalence. Large deviations, a phase transition, and logarithmic S obolev inequalities in the block spin P otts model. Renormalization in combinatorially non-local field theories: the H opf algebra of 2-graphs.

Mathematical Physics, Analysis and Geometry , 24 2 , May A well-posedness result for a system of cross-diffusion equations. Journal of Evolution Equations , 21 2 —, May Winters, Hendrik Ranocha , and Gregor J. A purely hyperbolic discontinuous G alerkin approach for self-gravitating gas dynamics. Journal of Computational Physics , , May The S charfetter— G ummel scheme for aggregation—diffusion equations.

Communications on Pure and Applied Mathematics , 74 12 —, May On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity. Nonlinearity , 34 5 —, May Entropy, S hannon orbit equivalence, and sparse connectivity. Mathematische Annalen , 3 —, May A non-conforming dual approach for adaptive T rust- R egion reduced basis approximation of PDE -constrained parameter optimization. Vortex dynamics for 2 D E uler flows with unbounded vorticity. Annals of Mathematics , 3 —, May Iterability for transfinite stacks.

Journal of Mathematical Logic , 21 02 , April A conservative fully discrete numerical method for the regularized shallow water wave equations. LeFloch and Hendrik Ranocha. Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts.

Journal of Scientific Computing , 87 2 , April Friedrich , M. Perugini , and F. L ower semicontinuity for functionals defined on piecewise rigid functions and on GSBD. Journal of Functional Analysis , April Fehr, C. Himpe , S. Rave , and J. Sustainable research software hand-over. Journal of Open Research Software , April Inventiones mathematicae , 1 —, April A new class of a stable summation by parts time integration schemes with strong initial conditions.

Journal of Scientific Computing , 87 1 , March N o phase transition in tensorial group field theory. Physics Letters B , , March Alexandrov spaces with integral current structure. Kabluchko and J. T he maximum entropy principle and volumetric properties of O rlicz balls. Journal of Mathematical Analysis and Applications , March Angles of random simplices and face numbers of random polytopes.

Advances in Mathematics , March On the homotopy type of L -spectra of the integers. Some remarks on replicated simulated annealing. Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit. Archive for Rational Mechanics and Analysis , 2 —, March Self-similar behavior of the exchange-driven growth model with product kernel. Communications in Partial Differential Equations , 46 3 —, March The gromov-lawson-chernysh surgery theorem.

Structure of unital 3-fields. Mathematische Semesterberichte , 68 1 —53, March Cox, and Martin Huesmann. Model-independent pricing with insider information: a S korokhod embedding approach. Journal of Functional Analysis , 4 , 34, February Wolters, John C. Mosher, and Richard M.

International Society for Optics and Photonics, February Rokhlin dimension: duality, tracial properties, and crossed products. Ergodic Theory and Dynamical Systems , 41 2 —, February N on-openness of v-adic G alois representation for A -motives.

International Journal of Number Theory , 17 1 —53, February A mathematical model for bleb regulation in zebrafish primordial germ cells. Stochastic Processes and their Applications , —, February Lorenz, Hinrich Mahler, and Benedikt Wirth.

Entropic regularization of continuous optimal transport problems. Journal of Mathematical Analysis and Applications , 1 , 22, February Ben-Zvi and R. Right-angled artin group boundaries. Proceedings of the American Mathematical Society , 2 —, February Faces in random great hypersphere tessellations.

Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation. In Fred J. Springer International Publishing, January Strong error analysis for stochastic gradient descent optimization algorithms. IMA J. Product formulas for periods of CM abelian varieties and the function field analog. Journal of Number Theory , —, January Numerical approximation of von K arman viscoelastic plates. International Mathematics Research Notices , January The one-dimensional log-gas free energy has a unique minimizer.

Communications on Pure and Applied Mathematics , 74 3 —, January Strategies for the V ectorized B lock C onjugate G radients method. M odeling glioma invasion with anisotropy-and hypoxia-triggered motility enhancement: F rom subcellular dynamics to macroscopic PDE s with multiple taxis. A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions.

Localized model reduction for parameterized problems. In Model order reduction. V olume 2: S napshot-based methods and algorithms , pages — January Computers and Mathematics with Applications , —, January Cartier modules and cyclotomic spectra. Journal of the American Mathematical Society , 34 1 :1—78, January Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values.

Journal of Mathematical Analysis and Applications , 1 , 33, January Asymptotic normality for random simplices and convex bodies in high dimensions. Proceedings of the American Mathematical Society , 1 —, January Width, largeness and index theory. Exponential moment bounds and strong convergence rates for tamed-truncated numerical approximations of stochastic convolutions. Algorithms , 85 4 —, December Higher-order estimates for collapsing C alabi- Y au metrics.

Cambridge Journal of Mathematics , —, December Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations. Overcoming the curse of dimensionality in the numerical approximation of A llen- C ahn partial differential equations via truncated full-history recursive multilevel P icard approximations.

Journal of Numerical Mathematics , 28 4 —, December Equivariant KK -theory and the continuous R okhlin property. International Mathematics Research Notices , December Plethysm and cohomology representations of external and symmetric products.

Advances in Mathematics , , December Expected number of real zeroes of random T aylor series. The maximal injective crossed product. Ergodic Theory and Dynamical Systems , 40 11 —, November Numerical simulations for full history recursive multilevel P icard approximations for systems of high-dimensional partial differential equations.

Communications in Computational Physics , 28 5 —, November Journal of Mathematical Analysis and Applications , 1 , November Potential theory on minimal hypersurfaces I : singularities as M artin boundaries. Potential Anal. Intersection of unit balls in classical matrix ensembles. Israel J. Multi-group binary choice with social interaction and a random communication structure- A random graph approach.

A , , October A premouse inheriting strong cardinals from V. Annals of Pure and Applied Logic , 9 , October Gusakova , Z. Kabluchko , and D. Distribution of complex algebraic numbers on the unit circle. Journal of Mathematical Sciences , 1 —66, October Identification of the polaron measure in strong coupling and the P ekar variational formula. Ordered structures and large conjugacy classes.

Algebra , —96, September Exact asymptotic volume and volume ratio of S chatten unit balls. Theory , , 13, September First-order continuous- and discontinuous- G alerkin moment models for a linear kinetic equation: M odel derivation and realizability theory. Journal of Computational Physics , , September Martingale B enamou- B renier: a probabilistic perspective. Quantitative analysis of finite-difference approximations of free-discontinuity problems.

Interfaces Free Bound. How long is the convex minorant of a one-dimensional random walk? Dimension, comparison, and almost finiteness. JEMS , 22 11 —, August Mathematika , 66 4 —, August A , 53 35 , 37, August Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. Hugh Woodin. Mice with finitely many W oodin cardinals from optimal determinacy hypotheses. Absorption probabilities for G aussian polytopes and regular spherical simplices.

Numerical upscaling of perturbed diffusion problems. Statistical shape analysis of tap roots: a methodological case study on laser scanned sugar beets. BMC Bioinformatics , 21 1 , July Algebra Number Theory , 14 5 —, July Analysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of B lack- S choles partial differential equations.

Pricing and heding american-style options with deep learning. Journal of Risk and Financial Management , July Exa-dune — flexible PDE solvers, numerical methods and applications. Cham, July Category of C -motives over finite fields. Journal of Number Theory , July Convergence rates for the S tochastic G radient D escent method for non-convex objective functions. Journal of Machine Learning Research , 21 :1—48, June On weakly complete group algebras of compact groups.

Lie Theory , 30 2 —, June Spectral H irzebruch- M ilnor classes of singular hypersurfaces. An extension result for generalised special functions of bounded deformation. Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations. BIT , 60 4 —, June Characteristic numbers of manifold bundles over surfaces with highly connected fibers.

Journal of the London Mathematical Society , June Inner-model reflection principles. Studia Logica , 3 —, June Hartl and A. Pink's theory of H odge structures and the H odge conjecture over function fields , pages 35— Hartl and W.

Injectivity, crossed products, and amenable group actions. May S anov-type large deviations in S chatten classes. Weak convergence rates for E uler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients.

Foundations of Computational Mathematics , May Dynamic cell imaging in PET with optimal transport regularization. Exact recovery in block spin I sing models at the critical line. Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally L ipschitz continuous nonlinearities. A note on the vanishing viscosity limit in the Y udovich class. Canadian Mathematical Bulletin , pages 1—11, April Crystallization in a one-dimensional periodic landscape.

Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity. On the L ittlewood —P aley spectrum for passive scalar transport equations. Journal of Nonlinear Science , 30 2 —, April Conjugating automorphisms of graph products: K azhdan's property T and SQ -universality. Bulletin of the Australian Mathematical Society , 2 —, April A note on K uttler— S igillito's inequalities.

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. Yet another note on the arithmetic-geometric mean inequality. Studia Math. On the topology of the space of R icci-positive metrics. Proceedings of the American Mathematical Society , —, April Lower error bounds for the stochastic gradient descent optimization algorithm: S harp convergence rates for slowly and fast decaying learning rates.

Journal of Complexity , , April Convex lifting-type methods for curvature regularization. April Almost finiteness and the small boundary property. Braun, Karl H. Hofmann, and L. Automatic continuity of abstract homomorphisms between locally compact and polish groups. Groups , 25 1 :1—32, March Twisted spin cobordism and positive scalar curvature. On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients.

Asymptotic properties of linear field equations in anti—de S itter space. Communications in Mathematical Physics , March Definability in the group of infinitesimals of a compact L ie group. Confluentes Mathematici , 11 2 :3—23, March Real zeroes of random analytic functions associated with geometries of constant curvature. Fluctuation results for general block spin I sing models.

NIP henselian valued fields. Logic , 59 —, February Equi-energy sampling does not converge rapidly on the mean-field P otts model with three colors close to the critical temperature. Finite crystallization and W ulff shape emergence for ionic compounds in the square lattice.

Nonlinearity , 33 3 —, February Two-well rigidity and multidimensional sharp-interface limits for solid—solid phase transitions. Partial Differential Equations , 59 2 :Paper No. Identification of the P olaron measure I : F ixed coupling regime and the central limit theorem for large times.

Communications on Pure and Applied Mathematics , 73 2 —, February Higher geometry for non-geometric T -duals. Communications in Mathematical Physics , 1 —, February Quantum field theory on noncommutative spaces. Configuration categories and homotopy automorphisms. Periods of D rinfeld modules and local shtukas with complex multiplication.

Jussieu , 19 1 —, January Zelevinsky operations for multisegments and a partial order on partitions. Pacific J. Journal of Algebraic Geometry , 29 1 :1—52, January Griffith energies as small strain limit of nonlinear models for nonsimple brittle materials. Mathematics in Engineering , 2 1 —, January Journal of High Energy Physics , , January Discrete R iemannian calculus on shell space. A note on locally elliptic actions on cube complexes. International Mathematics Research Notices , 1 —, January P-adic limits of renormalized logarithmic E uler characteristics.

Groups, Geometry, and Dynamics , January The L eutwyler— S milga relation on the lattice. Modern Physics Letters A , , January Mathematical analysis of transmission properties of electromagnetic meta-materials. Phase field approximations of branched transportation problems. Partial Differential Equations , 59 1 :Paper No. On number of ends of graph products of groups. Communications in Algebra , pages 1—10, January Classification of rationally elliptic toric orbifolds.

Basel , 6 —, January On moduli spaces of positive scalar curvature metrics on highly connected manifolds. Homogenisation of high-contrast brittle materials. Localization of the G aussian multiplicative chaos in the W iener space and the stochastic heat equation in strong disorder. Isogenies of abelian A nderson A -modules and A -motives. A solvable tensor field theory.

Letters in Mathematical Physics , pages , December Witt groups of abelian categories and perverse sheaves. Annals of K-Theory , 4 4 —, December The automorphism group of the universal C oxeter group. Expositiones Mathematicae , December Cones generated by random points on half-spheres and convex hulls of P oisson point processes.

Theory Related Fields , —, December System-theoretic model order reduction with pyMOR. PAMM , November Hitting times, commute times, and cover times for random walks on random hypergraphs. A B enamou— B renier formulation of martingale optimal transport.

Bernoulli , 25 4A —, November Abstract homomorphisms from locally compact groups to discrete groups. Journal of Algebra , —, November Journal of Mathematical Analysis and Applications , 2 —, November Uniformizing the moduli stacks of global G -shtukas. A locally conservative reduced flux reconstruction for elliptic problems. Journal of Mathematical Logic , 0 0 , November Adaptive energy-saving approximation for stationary processes.

Izvestiya: Mathematics , 83 5 —, October Global minimizers for anisotropic attractive-repulsive interactions. European Journal of Applied Mathematics , pages 1—17, October October On hyperbolicity and virtual freeness of automorphism groups. Geometriae Dedicata , October Cell Reports , 29 4 — Abstract bivariant C untz semigroups II.

Forum Mathematicum , September Crystallization in the hexagonal lattice for ionic dimers. A F irst J ourney through L ogic. Student Mathematical Library. American Mathematical Society, 1. Advances in model order reduction for large scale or multi-scale problems. Oberwolfach Reports , —40, September Phase field models for two-dimensional branched transportation problems.

Symmetric operads in abstract symmetric spectra—erratum. Jussieu , 18 5 , September Tilting chains of negative curves on rational surfaces. Nagoya Math. The geometry of multi-marginal S korokhod E mbedding. Probability Theory and Related Fields , August The R icci flow on solvmanifolds of real type. Advances in Mathematics , —, August A local model for the trianguline variety and applications.

IHES , 1 —, August Twisted differential cohomology. Semisimplicial spaces. Multiscale methods for perturbed diffusion problems. Oberwolfach Reports , August Hofmann, Linus Kramer , and Francesco G. The S ylow structure of scalar automorphism groups.

Topology and its Applications , —43, August Nonperturbative evaluation of the partition function for the real scalar quartic QFT on the M oyal plane at weak coupling. Journal of Mathematical Physics , 60 8 , August Journal of Statistical Physics , 1 —94, August Building-like geometries of finite M orley rank. Three-field mixed finite element formulations for gradient elasticity at finite strains. Smooth stability and sphere theorems for manifolds and E instein manifolds with positive scalar curvature.

Weak convergence rates of spectral G alerkin approximations for SPDE s with nonlinear diffusion coefficients. Weinan, and Arnulf Jentzen. Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations.

Some extensions of linear approximation and prediction problems for stationary processes. Limit theorems for the least common multiple of a random set of integers. Transactions of the American Mathematical Society , 7 —, July The R icci flow under almost non-negative curvature conditions.

Inventiones Mathematicae , —, July Optimal curvature estimates for homogeneous R icci flows. International Mathematics Research Notices. IMRN , 14 —, July On the large N limit of S chwinger- D yson equations of a rank-3 tensor field theory. Journal of Mathematical Physics , 60 7 , July The intersection motive on the moduli stack of shtukas. Forum of Mathematics Sigma , July Dynamic models of W assersteintype unbalanced transport.

Communications in Mathematical Physics , July On coherence of graph products of groups and C oxeter groups. Discrete Mathematics , 7 —, July Symmetric operads in abstract symmetric spectra. Jussieu , 18 4 —, July Characterization of optimal carbon nanotubes under stretching and validation of the C auchy- B orn rule. Padova , —, June Non- G aussian disorder average in the S achdev- Y e- K itaev model.

Physical Review D , 99 12 , June Superexpanders from group actions on compact manifolds. Geometriae Dedicata , 1 —, June On the limiting spectral density of random matrices filled with stochastic processes. Random Operators and Stochastic Equations , 27 2 —, June Planarity of C ayley graphs of graph products of groups.

Discrete Mathematics , 6 —, June On multilevel P icard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations. The dynamics of inextensible fibers in turbulent airflows is of interest, e.

In this talk, a model based on higher-index stochastic differential-algebraic equations is presented. It involves an implicitly given Lagrange multiplier process, the explicit representation of which leads to an underlying stochastic ordinary differential equation with non-globally monotone coefficients. Strong convergence is established for a half-explicit drift-truncated Euler scheme which fulfills the algebraic constraint exactly.

Thomas Kruse, University of Giessen Multilevel Picard approximations for high-dimensional semilinear parabolic partial differential equations. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of one of the proposed methods grows polynomially both in the dimension and in the reciprocal of the required accuracy.

We illustrate the efficiency of the approximation methods by means of numerical simulations. This model is a simplicial complex model that generalises Apollonian networks and the random recursive trees, by, in particular, adding random weights to the nodes. For this general model, we prove limiting theorems for the degree distribution, and confirm the conjecture of Bianconi and Rahmede on the scale-free propoerties of this random graph.

Infinite divisible laws play a crucial role in many areas of probability theory. This class of laws corresponds exactly to all the possible weak limits of triangular arrays of random variables, due to the generalized central limit theorem, and hence infinite divisible laws are natural generalizations of the Gaussian laws. We will see that these representations depend heavily on the Banach space under consideration. Many examples of high dimensional probability measures come from stochastic process theory, and we will illustrate the results within this framework.

Activated random walks is a system of particles which perform random walks and can spontaneously fall asleep, staying put. When an active particle falls on a sleeping one, the sleeping particle becomes active and continues moving. The system displays a phase transition in terms of the density of particles. If the density is small, all particles will eventually sleep forever, while, if the density is high, the system can sustain a positive proportion of active particles.

In this talk we describe the critical behavior of the model in the totally asymmetric case. Joint work with Leo Rolla. Phylogenetic gene trees are contained within the branches of the species trees. In order to model genealogy backwards in time, of both, gene trees and species trees, simple exchangeable coalescent snec process are defined and characterized in talk.

In particular, we study the coming down form infinity property for the so called nested Kingman. Finally, we present a model to include population structure in gene lineages. We give conditions for the convergence of the profile aka the sequence of generation sizes as the size of the tree goes to infinity. This gives a more general formulation and a probabilistic proof of a conjecture due to Aldous for conditioned Galton-Watson trees.

Our formulation contains results in this direction obtained previously by Drmota-Gittenberger and Kersting. The technique, based on path transformations for exchangeable increment processes, also gives us a partial compactness criterion for the inhomogeneous continuum random tree. Joint work with Osvaldo Angtuncio. Lackner and Panholzer introduced the parking process on trees as a generalization of the classical parking process on the line.

The cars go down the tree and try to park on empty vertices as soon as possible. The proof relies on tools in percolation theory such as differential in equalities obtained through increasing couplings combined with the use of many-to-one lemmas and spinal decompositions of random trees. This is joint work with Nicolas Curien Paris Saclay.

The relation between these two important families of processes has been investigated in some cases. Using Gillispie's sampling method, we find that an analogous relation holds for every lambda coalescent. Furthermore, functionals of independent CSBPs with different laws lead to frequency processes of coalescents with selection, mutation, efficiency and more. We discuss the problem of the Monte Carlo simulation of sample paths. The Embeddable Markov Chain Schemes simulate the exact positions of a diffusion at some hitting times which are not known but only approximated.

Such schemes aim at overcoming the situations in which the Euler-Maruyama scheme cannot be used, because of the presence of discontinuous coefficients, interfaces, sticky points, … In this talk, we present two such schemes. The first one uses the explicit expressions of the resolvent kernel instead of the density and deal with discontinuous coefficients. It leads to fast convergence. The second one generalises the Donsker approximation in presence of degenerate coefficients and appears to be fully flexible.

Both schemes heavily rely on the underlying infinitesimal generator. In this talk, I consider d -dimensional random vectors Y 1 , … , Y n that satisfy a mild general position assumption a. The hyperplanes. I will present a formulas for the number of cones which holds almost surely. For a random cone chosen uniformly at random from this random tessellation, I will address expectations for a general series of geometric functionals. These include the face numbers, as well as the conical intrinsic volumes and the conical quermassintegrals.

All these expectations turn out to be distribution-free. I will present analogous formulas the number of cones in this tessellation and the expectations of the same geometric functionals for the random cones obtained from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected by a certain linear subspace in general position.

Osvaldo Angtuncio, UNAM On multitype random forests with a given degree sequence, the total population of branching forests and enumerations of multitype forests. In this talk, we introduce the model of uniform multitype forests with a given degree sequence MFGDS. The construction is done using the results of Chaumont and Liu , and a novel path transformation on multidimensional discrete exchangeable increment processes, which is a generalization of the Vervaat transform.

We also obtain the joint law of the number of individuals by types in a MGW forest, generalizing the Otter-Dwass formula. This allows us to get enumerations of multitype forests with a combinatorial structure plane, labeled and binary forest , having a prescribed number of roots and individuals by types. Finally, under certain hypotheses, we give an easy algorithm to simulate CMGW forests, generalizing the unitype case given by Devroye in First, we discuss general methods for solving singular SPDEs endowed with boundary conditions.

Then, starting from the Neumann problem for the KPZ equation, we discuss how and why boundary renormalisation effects arise in this context. We are interested in the infinite limit of the alpha-Ford model, which is a family of random cladograms, interpolating between the coalescent tree or Yule tree and the branching tree or uniform tree.

For this, we use the notion of algebraic measure trees, which are trees without edge length and equipped with a sampling measure. In the space of algebraic measure trees, the limit of the alpha-Ford model is well defined. We then describe some statistics on the limit trees, allowing for tests of hypotheses on real world phylogenies.

Furthermore, the alpha-Ford algebraic measure trees appear as the invariant distributions of Markov processes describing the evolution of phylogenetic trees. Joint work with Mark Veraar Singular integral operators play a prominent role in harmonic analysis. In the context of robust optimal stopping one aim is to prove minimax identities, which play an important role in financial mathematics, especially in the characterization of arbitrage-free prices for American options.

Normally, the proof relies on the assumption that the underlying set of probability measures priors satisfies the property of time-consistency which can be regarded as an extension of the tower property for conditional expectations. Unfortunately, time-consistency is very restrictive. In this talk we present a different kind of conditions that ensure the desired minimax result. The key is to impose a compactness assumption on the set of priors.

The presented conditions reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales. Furthermore, we give a short outlook how to extend the minimax results to the model free situation where no reference probability measure is given in advance.

In this work, we investigate the asymptotic distribution of likelihood ratio tests in models with several groups, when the number of groups converges with the dimension and sample size to infinity. We derive central limit theorems for the logarithm of various test statistics and compare our results with the approximations obtained from a central limit theorem where the number of groups is fixed.

In this talk, we will consider two testing problems, namely testing for a block diagonal covariance matrix and for equality of normal distributions. We can tackle this problem by considering a space-time transformation of the Bessel process. We provide martingale estimation functions based on eigenfunctions of the diffusion generator for this transformed Bessel process.

Furthermore, we compare the martingale estimation functions through a simulation study and discuss the emerging complications. The drift and the inverse of the limit covariance matrix are expressed in terms of the zeros of classical Jacobi polynomials. We also rewrite the CLT in trigonometric form and determine the eigenvalues and eigenvectors of the limit covariance matrices. Croydon extended this result to symmetric Feller processes associated with a resistance metric.

Both approaches are tailored to discrete or basically linear state spaces. They fail in higher dimensions, where the resistance metric is not well-defined. In this talk we lay out a path towards a general invariance principle. We introduce a class of occupation time functionals and a notion of convergence of the state spaces based on these functionals.

In the standard optimal stopping, model uncertainty is usually handled by considering as an objective the expected return. In this talk, we pursue a more versatile approach towards uncertainty and consider optimal stopping problems with conditional convex risk measures including average value-at-risk and other risk measures.

Based on a generalization of the additive dual representation of [Rogers ] to the case of optimal stopping under uncertainty, we develop a novel Monte Carlo algorithm for the approximation of the corresponding value function. The algorithm involves optimization of a genuinely penalized dual objective functional over a class of adapted martingales. This formulation allows to construct upper bounds for the optimal value with a reduced complexity.

Further we discuss the convergence analysis of the proposed algorithm. In this talk we consider the stochastic thin-film equation. The stochastic thin-film equation is a fourth-order, degenerate stochastic PDE with nonlinear, conservative noise. This makes the existence of solutions a challenging problem. Due to the fourth order nature of the equation, comparison arguments do not apply and the analysis has to solely rely on integral estimates.

In this talk we will prove the existence of weak solutions in the case of quadratic mobility. The construction of a solution will be based on an operator splitting technique, which at the same time gives rise to an easy to implement numerical method. In this talk we introduce a new distance for stochastic processes taking values on compact metric measure trees based on hitting times. We show how this new metric can be bounded in terms of the Gromov-Prokhorov distance and argue how this yields rates of convergence in the f.

We complete the metric by adding a functional that captures tightness. We discuss how this complete metric can be used to derive rates for weak convergence in path-space. As an application, we use this distance to derive a rate for weak convergence in path-space for SRW to BM on compact intervals.

We present Sinai's random walk in random environment, and focus on its functional limit theorem. We begin by recalling SInai's model and its long time behaviour. We illustrate that the proof of the known functional limit theorem can be shortened, when verifying the convergence of the resistance metric measure spaces rather than that of the random walks associated with these. Viktor Schulmann , TU Dortmund Life span estimation for randomly moving particles based on their places of death.

Consider the following problem from physics: A radiation source is placed at the center of a screen. At certain time intervals the source releases particles. For that case an estimator was given by Belomestny and Schoenmakers using the Mellin and Laplace transforms. A closed solution of the symmetric model will be provided.

After this, we talk about the dynamical Gibbs-non-Gibbs transitions for the time-evolved model under independent stochastic symmetric spin-flip dynamics. The main goal is to show for appropriate percolation regimes that there exists a unique giant cluster that is of a size comparable of that of the entire tree where size is defined as either the number of vertices or the number of balls. Instead, they are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws.

The approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which allows us to apply a classical limit theorem for the convergence of triangular arrays to infinitely divisible distributions.

Jan Nagel , University Dortmund Random walk on barely supercritical branching random walk. The motivating question behind this project is how a random walk behaves on a barely supercritical percolation cluster, that is, an infinite percolation cluster when the percolation probability is close to the critical value. As a more tractable model, we approximate the percolation cluster by the embedding of a Galton-Watson tree into the lattice.

When the random walk runs on the tree, the embedded process is a random walk on a branching random walk. Now we can consider a barely supercritical branching process conditioned on survival, with survival probability approaching zero.

In this setting the tree structure allows a fine analysis of the random walk and we can prove a scaling limit for the embedded process under a nonstandard scaling. The talk is based on a joint work with Remco van der Hofstad and Tim Hulshof. At any time t, the value of g is unknown and it is only with the realisation of the whole process when we can know when the last zero of the process occurred. However, this is often too late, we usually are interested in know how close is the process to g at time t and take some actions based on this information.

We state some basic properties of the last zero process and prove the existence of the solution of the optimal stopping problem. Then we show the solution of the optimal stopping problem and therefore the optimal prediction problem is given as the first time that the process crosses above a non-increasing and non-negative curve dependent on the time of the last excursion away from the negative half line.

Carina Betken , University Bochum Stein's method and preferential attachment random graphs. We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sub- linear function of the indegree of the older vertex at that time.

Using Stein's method for Poisson and Normal approximation we also show limit theorems for the outdegree distribution as well as for the number of isolated vertices. We study the behaviour of level sets of 0-mean Gaussian free field on regular expanding graphs, and at least partially prove that they exhibit a similar phase transition as Bernoulli percolation and the vacant set of random walk on such graphs.

The topic of this talk is mean-field spin glasses, in particular the Sherrington-Kirkpatrick SK model. I will revisit the Thouless-Andersson-Palmer approach to the these models from the physics literature, and report on our efforts to use it as a basis for an alternative mathematically rigorous treatment. Max Grieshammer , University Erlangen Measure representation of evolving genealogies.

We present a method of describing evolving genealogies, i. We relate measure-valued processes to certain genealogical quantities and use this connection to prove a tightness result for evolving genealogies. Finally, we sketch how this result applies to Moran models. In , Fyodorov, Khoruzhenko and Sommers introduced the weakly non-Hermitian regime, which provides a natural bridge between Hermitian and normal random matrix theories. In this talk, I will discuss ensembles of a similar appearance and their universality, in particular from the viewpoint of Ward equations.

Vitalii Konarovskyi , Leipzig A particle model for Wasserstein type diffusion. Wioletta Ruszel , Delft Netherlands A zoo of scaling limits of odometers in divisible sandpile models. Noemi Kurt , Berlin Modelling the Lenski experiment. In recent years deep artificial neural networks DNNs have very successfully been used in numerical simulations for a numerous of computational problems including, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations PDEs.

Such numerical simulations indicate that DNNs seem to admit the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the dimension of the function which the DNN aims to approximate in such computational problems. In this talk we show that DNNs do overcome the curse of dimensionality in the numerical approximation of Kolmogorov PDEs with constant diffusion and nonlinear drift coefficients.

We prove that the number of parameters used to describe the employed DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the PDE dimension. Martin Slowik , Berlin Green kernel asymptotics for two-dimensional random walks among random conductances. Michael Hinz , Bielefeld Hydrodynamic limits of weakly asymmetric exclusion processes on fractals. Sebastian Riedel , Berlin A random dynamical system for stochastic delay differential equations.

Thomas Mikosch , Copenhagen Denmark and Bochum Regular variation and heavy-tail large deviations for time series. Johannes Heiny , Bochum RTG Assessing the dependence of high-dimensional time series via autocovariances and autocorrelations. In modelling, boundaries appear naturally and in 1D we answer the question of what type of boundary condition for the non-local operator corresponds to what type of boundary behaviour of the process by using numerical approximation schemes.

Many models of complex systems can be seen as a system of many interacting micro variables evolving in time. We focus on the situation, where the network of interactions between the variables is complex and possibly itself evolves in time. We discuss a modeling framework for interacting particle systems on evolving networks based on such familiar ingredients as exchangeability and Markovianity.

In some simple cases, we discuss the genealogies of such population models. The block-counting process traces back the number of potential ancestors of a sample of the population at present. Selection and mutation translate into additional branching and pruning.

Under some conditions the block-counting process is positive-recurrent and its stationary distribution is described via a linear system of equations. We solve the latter for the Kingman model and the star-shaped model. Based on joint work with M. The framework introduced in this talk is motivated by applications in biology. It is well suited for modeling the random evolution of the genealogy of a population in a hierarchical system with two levels, for example, host-parasite systems or populations which are divided into colonies.

We discuss spatially structured Wright-Fisher type diffusions modelling the frequency of an altruistic defense trait. These arise as the diffusion limit of spatial Lotka-Volterra type models with a host population and a parasite population, where one type of host individuals the altruistic type is more effective in defending against the parasite but has a weak reproductive disadvantage. For the many-demes limit mean-field approximation hereof, we prove a propagation of chaos result in the case where only a few diffusions start outside of an accessible trap.

In this "sparse regime", the system converges in distribution to a forest of trees of excursions from the trap. The Curie-Weiss model of self-organized criticality was introduced by Cerf and Gorny , as a modification of the Curie-Weiss model of ferromagnetism that drives itself into a criticalstate. We consider a dynamic variant of this model, i. In this talk we are interested in limit objects of graph-theoretic trees as the number of vertices goes to infinity.

Depending on which notion of convergence we choose different objects are obtained. One notion of convergence with several applications in different areas is based on encoding trees as metric measure spaces and then using the Gromov-weak topology. Apparently this notion is problematic in the construction of scaling limits of tree-valued Markov chains whenever the metric and the measure have a different scaling regime. We therefore introduce the notion of algebraic measure trees which capture only the tree structure but not the metric distances.

Convergence of algebraic measure trees will then rely on weak convergence of the random shape of a subtree spanned a sample of nite size. We will be particularly interested in binary algebraic measure trees which can be encoded by triangulations of the circle. We will show that in the subspace of binary algebraic measure trees sample shape convergence is equivalent to Gromov-weak convergence when we equip the algebraic measure tree with an intrinsic metric coming from the branch point distribution.

The main motivation for introducing algebraic measure trees is the study of a Markov chain arising in phylogeny whose mixing behavior was studied in detail by Aldous and Schweinsberg We give a rigorous construction of the diffusion limit as a solution of a martingale problem and show weak of the Markov chain to this diffusion as the number of leaves goes to infinity. The only requirements is to be able to sample a particular random variable whose density is given by the resolvent kernel of the stochastic process that one wants to simulate.

This mainly means that an analytical form of the resolvent kernel is required. On this particular case of Feller's processes, we show through numerical experiments that the GEARED method has a quite fast convergence and that it conserves important properties in physics such a good repartition of mass. It is well known from the Feynman-Kac formula that a classical solution of the Kolmogorov backward equation can be written as the expectation of the solution of the corresponding SDE.

In M. Hairer, M. Hutzenthaler, and A. Moreover, they proved in the finite dimensional case that under suitable assumption the Kolmogorov backward equation has a unique viscosity solution which can be represented as the expectation of the solution of the corresponding SDE.

Therefore I will use a more general notation of viscosity solution introduced by H. Ishii and show that under suitable assumptions the expectation of the solution of an SPDE is the unique viscosity solution of the corresponding Kolmogorov backward equation.

About some skewed Brownian diffusions: explicit representation of their transition densities and exact simulation. In this talk we first discuss an explicit representation of the transition density of Brownian dynamics undergoing their motion through semipermeable and semireflecting barriers, called skewed Brownian motions.

We use this result to present an exact simulation of these diffusions, and comment some still open problems. Eventually we consider the exact simulation of Brownian diffusions whose drift admits finitely many jumps. The aim of the talk is to briefly introduce some ideas in my PhD thesis. Random motions in random media is an interesting topic that has been studied intensively since several decades.

Although these models are relatively simple mathematical objects, they have a wide variety of interesting properties from the theoretical point of view. Reversibility provides the model a variety of interesting connections with other fields in mathematics, for instance, percolation theory and especially stochastic homogenization. Many questions coming from this model have been answered by techniques from partial differential equations and harmonic analysis.

As seen in the name of the talk, I would like to consider this model under ''degenerate conditions''. Here, ''degenerate'' has essentially two meanings. First, the conductances are not assumed to be bounded from above and below and stochastically independent. Since there are percolation clusters, where the existence of the infinite cluster does not rely on stochastic independence, it is reasonable to accept the lack of stochastic independence.

In the first part of the talk I introduce quenched invariance principles joint work with Jean-Dominique Deuschel and Martin Slowik. We assume that the positive conductances have some certain moment bounds, however, not bounded from above and below, and give rise to a unique infinite cluster and prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infinite cluster.

An essential ingredient of our proof is a new anchored relative isoperimetric inequality. In the second part I would like to talk about Liouville principles. As in the first part, I also assume some moment bounds and prove a first order Liouville property for this model. Using the corrector method introduced by Papanicolaou and Varadhan, the first and the second part are closely related to each other at the technical level. I also introduce a discrete analogue of the Dirichlet-to-Neumann estimate, which compares the tangential and normal derivatives of a harmonic function on the boundary of a domain.

Although it is a purely deterministic classical result, it is used in the second part and perhaps useful for numerical analysis. With this tool we can generalize the well-known diffusion limit of a Wright-Fisher model with randomly fluctuating selection. Whereas the classical result assumes the selection coefficients to be independent for different generations, we allow the environment to persist with a positive probability.

The diffusion limit turns out to depend on this probability. This talk is based on joint work with Martin Hutzenthaler and Peter Pfaffelhuber. Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation. We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment.

This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint.

The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities. The expected value of partial perfect information EVPPI expresses the value gaining by acquiring further information on certain unknowns in a decision-making process. Further, we introduce an unbiased estimator based on a randomized version multi-level Monte-Carlo algorithm. We will show some numerical result in evaluating the benefit of further information when deciding between two different treatment options.

In many models of Applied Probability and Statistical Physics quantities of interest satisfy distributional limit theorems i. Sometimes, there is even no convergence in law, but periodical fluctuations centered at some limit law occur. Common to all these models is that the limiting distributions satisfy so-called smoothing equations. I will start my talk by presenting several examples where this phenomenon occurs.

These results are then applied to the examples mentioned in the introduction. This talk is based on joint work with Matthias Meiners, University Innsbruck. The asymptotic behavior of the ground state energy of the Anderson model on large regular trees. It is related to random walks in a random environment. We give a detailed description of the ground state energy on large finite symmetric subtrees. While the majority of results for this problem deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications.

In recent years a number of positive results for this problem has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions. In our talk we present negative results for this problem.

First we show that there exist SDEs with bounded smooth coefficients such that their solutions can not be approximated by means of any kind of adaptive method with a polynomial rate of convergence. While the diffusion coefficients of these pathological SDEs are globally Lipschitz continuous, the first order partial derivatives of the drift coefficients are, essentially, of exponential growth.

In the second part of the talk we show that sub-polynomial rates of convergence may happen even when the first order partial derivatives of the coefficients have at most polynomial growth, which is one of the typical assumptions in the literature on numerical approximation of SDEs with globally monotone coefficients. In this talk, I will present certain implementations of conformal field theory CFT in a doubly connected domain. We consider a classical deterministic model for the evolution of a haploid population with two allelic types which is subject to mutation, selection, and a special form frequency-dependent selection.

The deterministic model arises also as the large population limit of the Moran model, in which neither parameters nor time are rescaled. Despite the deterministic nature of this limiting process, the ancestry of single individuals in the population is still stochastic. In the case with mutation and selection, we describe it via a killed ancestral selection graph and connect it to the deterministic process via duality; this leads to a stochastic representation of the deterministic solution.

In particular, the stationary state obtains a nice probabilistic interpretation. We generalise the construction to the case with frequency-dependent selection. Thereby we obtain a class of multilevel Picard approximations. The talk is based on joint work with Weinan E. It can be shown using the Malliavin-Stein approach that the rate of convergence is dominated by the differences of the third and fourth cumulants.

In , I. Nourdin and G. This SPDE arises for instance as the high density limit of particle systems which undergo branching random walks and allow for extra death due to overcrowding. Above this critical value, the probability of global survival is strictly positive. What can be said for solutions with finite initial mass if we condition on their survival?

In this talk I start with an overview of the main probabilistic ideas and arguments of the above mentioned concepts. Here, the choice of the wave front marker plays an important role. Finally, I outline why an understanding of the behaviour of travelling wave solutions can help to answer questions on convergence of solutions with arbitrary initial conditions. We complete the analysis of the phase diagram of the complex branching Brownian motion energy model by studying Phases I, III and boundaries between all three phases I-III of this model.

All results are shown for any given correlation between the real and imaginary parts of the random energy. In this talk, I will present a time-discretization method of the stochastic incompressible Navier--Stokes problem using a penalty-projection method. Basically, the talk will consist of three parts. A brief introduction of the mathematical problem. Then an overview of the main computational issue that shares the stochastic and the deterministic form of Navier--Stokes.

Different algorithm will be introduced: a main algorithm and in order to treat the nonlinear character of the equation two auxiliary algorithms. At the end of the day we will arrive at the convergence with rate in probability and a strong convergence of the main algorithm. Generalizing the classical discrete arcsine law for the position of the maximum due to E. This is done in two different settings: for random walks with symmetrically exchangeable increments and for random bridges with exchangeable increments.

This probability, in turn, is related to the number of Weyl chambers of a product-type reflection group that are intersected by a linear subspace in general position. Joint work with Vladislav Vysotsky and Dmitry Zaporozhets. We prove a variational formula for the corresponding annealed pressure. We furthermore study classes of models with second order phase transitions which include models on an interval and rotation-invariant models on spheres, and classify their critical exponents.

For large classes of models they are the same as for the Ising model, but we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval. Simple nested coalescent has been introduced to model backwards in time the genealogy, of both, species trees and genes trees. In this setting, the Kingman case corresponds to binary coalescences in the species trees or the genes trees, but not simultaneously.

On the other hand, two level Fleming Viot with two level selection arises in Dawson as the limit in distribution of multilevel multitype population undergoing mutation, selection, genetic drift and spatial migration. In this talk, I will establish a duality relation between Kingman nested coalescents and the two level Fleming Viot associated with a two-level multitype population with genetic drift. Using this relation we can read off the genealogy backwards in time of a Kingman nested.

Imagine that we destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. We then record the genealogy induced by this destruction process in a random rooted binary tree, the so-called cut-tree. The goal of this talk will be to show a general criterion for the convergence of the rescaled cut-tree in the Gromov-Prohorov topology to a real tree, when the underlying tree has a small height.

In particular, we consider uniform random recursive trees, binary search trees, scale-free random trees. The approach relies in the introduction of a continuous version of the cutting-down procedure which we allow us to represent the destruction process up to a certain finite time as Bernoulli bond percolation. The contact process is a classical model for the spread of infections in a population. In this talk, we focus on the contact process in the supercritical regime for which infections may spread forever with positive probability.

Our goal is to understand how this process behaves compared with a process having no spatial correlations. In particular, does the contact process stochastically dominate a non-trivial independent in space spin-flip process? We present some space-time versions of their results for the contact process on general graphs. From our methods, we furthermore conclude strong uniform mixing properties for certain space-time projections of the contact process.

We review recent results on non-asymptotic analysis for the distributions of quadratic and almost quadratic forms in random elements with values in a Hilbert space. The study of almost quadratic forms is motivated by approximation problems in multidimensional mathematical statistics.

A number of results are optimal - they can not be improved without additional assumptions. A unified approach will be proposed for constructing non-asymptotic approximations on the basis of the general result on approximation accuracy for symmetric functions of several variables. Moreover, I intend to discuss the mixing properties, which draw particular interest from the theoretical point of view. We explain two models for biased random walks in random environment biased random walk on percolation clusters, biased random walk among random conductances which describe transport in an inhomogenous medium.

We give an overview of typical questions and present several results about Einstein relation and monotonicity of the speed. Probabilistic, Bayesian methods provide important approaches to data modelling in the field of machine learning. Inference on unobserved variables or parameters typically requires the performance of high-dimensional integrals or sums. If the dimensionality of the problem is very large, one often has to apply approximate inference methods which approximate intractable multivariate probability distributions by tractable ones.

I will give an introduction to the basic idea behind this approach and then discuss a combination of ideas from random matrix theory and dynamical functional methods of statistical physics which could be used to improve the efficiency of such methods and to study the convergence of inference algorithms for large systems. While convergence of finite-dimensional Markov processes e. In the first part of the talk we explain how the classical methods which are based on the construction of a Lyapunov function can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric.

This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow In the second part of the talk we provide some specific applications to SPDEs and stochastic delay equations and discuss the arising challenges. Joint work with Alexey Kulik and Michael Scheutzow. Butkovsky Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, Butkovsky, M.

Invariant measures for stochastic functional differential equations. The Donnelly-Kurtz lookdown model contains an evolving genealogy. The genealogical tree of the population at each time can be described by an isomorphy class of a metric measure space to obtain a tree-valued Fleming-Viot process.

The states of this process can be viewed as ergodic components of the genealogical distance matrices. In this context, we also discuss a general representation for exchangeable ultrametrics in terms of sampling from marked metric measure spaces. Simple symmetric random walk on a finite, connected graph is positive recurrent and the mean time to visit all vertices is finite. Recently, variable speed motions on metric measure trees have been constructed. In this talk we will raise the question what happens if we replace the assumption on the finiteness of the graph by completeness of the tree.

The Kesten-Stigum theorem describes the growth of supercritical Galton-Watson processes. Its classical proof relies on the analysis of probability generating functions. We look at a conceptual proof due to Lyons, Pemantle and Peres that uses size-biased Galton-Watson trees and basic measure theory. We also discuss its generalization to the case of stationary and ergodic environments.

The methods used are multilinear eigenfunction estimates and a spectral decomposition of the Laplacian, which allows the construction of suitable function spaces. In these spaces we can prove space-time estimates of Strichartz type.

As the main result, we will see that the Cauchy problem is locally well-posed for initial data in an appropriate Sobolev space. In the talk we discuss problem of simplification of musical signals which arise in cochlear implant music transmission. The main peculiarity of musical signals is highly varying temporal and spectral structure. Also the hearing loss and the hearing aid limitations should be used as an a-priori knowledge.

The simplification of the music signals may be achieved by an adaptive dimension reduction of spectral data and by the extraction of important musical features. To this end, we develop a novel unsupervised segmentation procedure for music signals which relies on an explained variance criterion in the eigenspace of the constant-Q spectral domain. We study super-replication of contingent claims in markets with fixed transaction costs.

The first result in this paper reveals that in reasonable continuous time financial market the super—replication price is prohibitively costly and leads to trivial buy—and—hold strategies. Our second result is derives non trivial scaling limits of super—replication prices in the binomial models with small fixed costs. Joint work with P. In order to construct tree-valued stochastic processes, one needs a topological space of trees as state space.

While this is not an issue for finite or countable graph-theoretic trees, we want to consider global limits as the number of vertices tends to infinity. We call the limiting objects also "tree" but have to make precise what we mean by this. We argue that sometimes it is more natural to consider a different type of structure, namely the tree-structure instead of the distance. Second, because sometimes one might want to preserve structural properties such as being binary in the limit.

We present a framework for a space of such continuum trees possessing no metric- but only a tree-structure. We call them algebraic trees , because we formalise the tree-structure by a tertiary operation on the tree, namely the branch point map. We construct a natural, topology on spaces of sufficiently nice algebraic trees. In the binary case, the resulting space is compact and intimately related to the set of triangulations of the circle as introduced by Aldous, equipped with the Hausdorff metric.

The main result states the validity of the proposed procedure for finite samples with an explicit error bound on the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high dimension. Numeric results confirm a nice performance of the method in realistic examples.

These are the joint results with V. The goal of this talk is to characterize the weak convergence of Brownian motions in terms of a geometric convergence of the underlying spaces. As main results, we show several equivalences between the weak convergence of Brownian motions and the pointed measured Gromov convergence of the underlying spaces satisfying the Riemannian Curvature-Dimension RCD condition.

Today's markets do not provide simple instruments that can be used for hedging volumetric risk, i. In the current literature hedging of volumetric risk is settled via utility functions. Here we consider this task using coherent risk measures and obtain explicit hedging strategies in some cases. If we extract oil in one country and sell it in another one, then we additionally have currency risk. Motivated by the dependence between oil prices and currency exchange rates we settle a problem of hedging both currency and volumetric risks via multidimensional coherent risk measures, with put options on oil prices as hedging instruments, and present its solution in some situations.

Malliavin Calculus is an infinite dimensional calculus defined on the Wiener space. It was developed in by Paul Malliavin. In this talk we consider some properties of the Mallivin derivative and its adjoint operator, known as the Skorokhod integral.

We then apply this theory to study distributional properties of the Stochastic Heat Equation. More specifically, we will see how this theory helps us find conditions under which the solution to this equation has a smooth density.

Random walks provide some of the most rich examples of stochastic processes. Starting with a random walk with real values zero mean and finite variance , we are interested in finding out how it behaves if we restrict the state space. In particular, what happens when we are only interested in the positive real line.

This version of the process is known as random walk conditioned to stay positive. Given the analogies between random walks with finte variance and Brownian motion, we also conditioned the latter to stay positive. The resulting process can be seen as 3-d Bessel process. Once these conditioned processes have been introduced we analyse the asymptotic behaviour of their occupation times; to be more precise, we seek for deterministic functions that describe the limiting behaviour of these processes.

We consider super-replication with small transaction costs in complete multi-asset multinomial markets. These results can be interpreted as a multidimensional extension of Kusuoka This is a joint work with P. Bank and A. Percolation is one of the simplest ways to define models in statistical physics and mathematics which displays a non-trivial critical behaviour. What happens at criticality, i. A diffusion subject to a deterministic movement and and external supply of particles can be modelled by a Brownian motion with drift and killing.

In probability this can be studied by Girsanov and Feynman-Kac transforms, in analysis by a heat equation with first order and potential terms. If the drift depends on time, we can no longer operate with one-parameter semigroups, and if a related Markov process exists, it will not be homogeneous.

Combining evolution semigroups, time-dependent Dirichlet forms and space-time processes we deduce probabilistic representations for classical solutions of abstract evolution problems. Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography.

To address this need, I consider the Wright-Fisher diffusion under selection and piecewise constant population sizes. In the case of genic selection, I will sketch the derivation of the transition density. For general diploid selection, I will apply a moment-based approach to devise an efficient and fast algorithm for the computation of the allele frequency spectrum. I will discuss several applications that are of interest for the analysis of whole-genome sequences.

In respect of a further application, I will introduce and discuss a system of coupled differential equations describing the coevolution of host and parasite alleles. A novel approach towards variance reduction for discretised diffusion processes is presented. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance of the terminal functionals. These theoretical results are illustrated by several numerical examples.

We consider a sequential testing problem of two hypotheses concerning the drift value of a fractional Brownian motion. We show that it can be reduced to the optimal stopping problem for a standard Brownian motion with non-linear cost of observation. Using standard technique one may characterize optimal stopping boundaries as an unique solution of a system of non-linear integral equations, which can be used for numerical evaluation via backward induction technique.

However, for this, one needs to have some end-point T, after which the values of optimal boundaries are known. The main point of the talk is how to obtain some upper and lower estimates for the boundaries which provide the choice of such a T. In contrast to known methods based on the study of corresponding free-boundary problem, our approach is more probabilistic.

Examples of such processes include the fractional Brownian motion and some of its relatives. We establish upper and lower bounds for the expected maximum of such a process and investigate the rate of convergence to that quantity of its discrete approximation. Further properties of these two maxima are established in the special case of the fractional Brownian motion.

#### Positivity of S egre- M ac P herson classes.

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Round-the-clock forex advisor | For that case an estimator was given by Belomestny and Schoenmakers using the Mellin and Laplace transforms. Annales Geophysicae39 4 —, July Numerical methods for autonomous SPDEs are thoroughly investigated in the literature, while to the best of our knowledge the non-autonomous cases are not yet well understood. Ostaszewski, K. Large deviations, a phase transition, and logarithmic S obolev inequalities in the block spin P otts model. |

Binary options anton gromov | An essential ingredient of our proof is a new anchored relative isoperimetric inequality. In this talk we consider some properties of the Mallivin derivative and its adjoint operator, known as the Skorokhod integral. Journal of Statistical Physics1 —94, August We can tackle this problem by considering a space-time transformation of the Bessel process. Ulyanov Moscow State University Non-asymptotic analysis of non-linear forms in random elements We review recent results on non-asymptotic analysis for the distributions of quadratic and almost quadratic forms in random elements with values in a Hilbert space. In the associated application areas, it is then necessary to finely resolve the singularities. |

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