Global existence and large time behaviour of nonlinear reaction-diffusion systems

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: can be found here

Bridging rough SDEs and rough semimartingales: A martingale fixed point approach

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract:  A crucial ingredient in the study of rough SDEs is the notion of a rough stochastic integral. Alongside the stochastic sewing approach to rough SDEs (see [FrizHocquetLê2021] and [AllanPieper2026]), a different notion of rough stochastic integration was introduced in the framework of rough semimartingales in [FrizZorin-Kranich2023]. While the stochastic sewing approach relies on strong integrability assumptions on the driving martingale, the rough semimartingale framework requires only mild integrability assumptions. These reduced requirements allow for an Itô calculus for rough semimartingales under weaker assumptions than those required for the corresponding calculus developed through the stochastic sewing approach, especially without the need for boundedness or growth conditions on the test functions. In this talk, I will discuss the connection between the solution theory for càdlàg rough SDEs via stochastic sewing and the theory of rough semimartingales. I will first discuss the pathwise regularity properties of rough Itô processes by establishing suitable versions of the Kolmogorov continuity criterion in p-variation spaces. Second, I will present a new approach to rough SDEs formulated directly in the space of rough semimartingales through a martingale fixed point argument. As an application, I will discuss a novel rough stochastic Duhamel formula with jumps, which allows one to quickly verify the assumptions of this approach in the setting of semilinear rough SDEs. This is a joint work in progress with Jannis R. Dause and Peter K. Friz.

Time-inconsistent stochastic games under mean-variance preferences

Research Seminar Stochastic Analysis & Finance at TU Berlin, IMoS, Fasanenstraße 89, 10623 Berlin, Raum IMoS 2003

Abstract: We study continuous-time stochastic games with mean-variance preferences. We first investigate a non-Markovian finite-player game in which each player’s objective functional depends non-linearly on the expected value of the state process. Since the classical Bellman optimality principle no longer applies, we adopt a two-layer game-theoretic framework and seek sub-game–perfect Nash equilibria at both the intra-personal and inter-personal levels. We characterise these equilibria and the corresponding value processes through a system of coupled backward stochastic differential equations (BSDEs). We then analyse the mean-field counterpart, described by a system of McKean-Vlasov BSDEs, and prove the convergence of the finite-player equilibria and value processes to their mean-field counterparts. Building on this BSDE characterisation, we then study a continuous-time principal-agent problem in which a principal hires a sophisticated agent with mean-variance preferences and promises a lump-sum payment at a finite terminal time. To address the moral hazard problem faced by the principal, we show that the analysis can be restricted, without loss of generality, to a class of revealing contracts for which the agent’s sub-game–perfect equilibrium can be characterised explicitly. This reduction allows us to use the BSDEs associated with the agent’s continuation utility as a controlled state variable for the principal and to reformulate the moral hazard problem as a standard stochastic control problem with stochastic target constraints.

From super-Brownian motions to time-fractional super-Brownian motions

Research Seminar Stochastic Analysis & Finance at TU Berlin, IMoS, Fasanenstraße 89, 10623 Berlin, Raum IMoS 2003

Abstract: Consider a sequence of nearly critical interacting particle systems with chain-reaction and decay. Under some fast decay assumption, we show that the particle systems can be well approximated by a super-Brownian motion after a suitable time-spatial scaling. On the other hand, under some regular variation condition on the decay, we proved the weak convergence of the rescaled interacting particle systems to a novel non-Markovnian super-process, named as time-fractional super-Brownian motion, that can be fully characterized by the Fourier-Laplace functional given in the form of unique solution to a time-fractional F-KPP equation. In particular, the time-fractional super-Brownian motion on the real line is proved to be absolutely continuous with respect to Lebesgue measure. Moreover, the density process is the unique weak solution to a time-fractional parabolic semilinear SPDE driven by a Gaussian white noise.

Rough BSDEs

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748 online, 11:00 AM

Abstract:  We establish an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven by a geometric rough path of finite p-variation, with p < 3. In contrast to earlier approaches based on transformation methods or approximation of smooth drivers, the equation is here, for the first time, solved intrinsically. In particular, this gives meaning to the solution pair (Y, Z) as solution to a hybrid rough stochastic integral equation. The proof relies on an original combination of rough stochastic techniques (via stochastic sewing) and p-variation estimates for martingale transforms (in the spirit of Lépingle estimates), supplemented by Malliavin calculus and John–Nirenberg inequalities for VMOp-var processes. Under suitable regularity assumption, we prove a strict comparison principle and a priori estimates, followed by existence, uniqueness, and local stability with respect to the terminal condition, the drift coefficient, and the driving rough path. Brownian randomization of the rough path leads connects naturally to BDSDEs. (Joint work with P. Friz, J. Song and H. Zhang)

Dynamics of a 3D rimming flow

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: We study the dynamic behaviour of a thin viscous fluid film coating the inner wall of a rotating cylinder– a so-called rimming flow. The resulting PDE for the height h > 0 of the fluid film is quasilinear, degenerate parabolic and of fourth order. Various competing effects drive the dynamics of the interface– viscosity, surface tension and gravity. For a positive surface tension parameter γ > 0 of order one we investigate existence and stability of positive steady states, depending on the influence of gravity (δ ≥ 0).

The talk is based on joint works with Juri Joussen (Stuttgart), Janne Laudien (Stuttgart) and Juan Velázquez (Bonn).

Asymptotic democracy for finite nonlinear Pólya urns

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: In this presentation, we will study Pólya urn models -- specifically, a generalization involving an arbitrary finite number of urns and where the addition of a ball to each urn is proportional to a (non-linear) function f of the number of balls already present in that urn. After introducing the continuous-time modeling of this system and presenting two fundamental results regarding monopoly and leadership -- more precisely, monopoly (resp. leadership) occurs if and only if 1/f(i)  (resp. 1/f(i)2) is summable --, we will outline a new "democracy" theorem that emerges with weighting functions characterized by relatively slow growth. 

A Bernstein-von Mises Theorem for indirect measurement models with nonlinear unbounded forward operators

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: Motivated by recently proposed Gaussian process solvers for semilinear elliptic PDEs [1], we consider Bayesian nonparametric inference for indirect measurement models with nonlinear unbounded forward operators. We establish optimal posterior contraction rates and a nonparametric Bernstein-von Mises theorem in strong Sobolev norms $(H^{2-d/2})$ for $d \leq 3$. Our results show that unbounded forward maps may lead to stronger Bernstein-von Mises results in $L^p$-type norms. This is interesting in light of the conventional statistical wisdom that the nonparametric Bernstein-von Mises phenomenon is only feasible in ‘weak’ metrics.

[1] Yifan Chen, Bamdad Hosseini, Houman Owhadi, Andrew M. Stuart. Solving and learning nonlinear PDEs with Gaussian processes, Journal of Computational Physics, 447: 110668 (2021)

Pattern formation and symmetry breaking for a family of functionals and their intrinsic nonlocal curvature

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: In this talk, we will introduce a rigorous approach to the study of the symmetry breaking and pattern formation phenomenon for isotropic functionals with local/nonlocal interactions in competition. More precisely, we consider a general class of nonlocal variational problems in general dimension d, in which an isotropic surface term favouring pure phases competes with an isotropic nonlocal term with power law kernel favouring alternation between different phases. Close to the critical regime in which the two terms are of the same order, we give a rigorous proof of the conjectured structure of global minimizers, in the shape of domains with flat boundary (e.g. stripes or lamellae). In order to show such structure we will use nonlocal curvatures which appear naturally in these types of problems. This work is in collaboration with S. Daneri.

Metastability for the 2-pattern Hopfield model and general product chains

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: The Hopfield model is a mean-field spin system designed to store a set of patterns as ground states of its Hamiltonian, providing a simple mathematical model of associative memory. At low temperature, the associated Glauber--Metropolis dynamics is metastable: started near a stored pattern, the system remains in its vicinity for an exponentially long time before an abrupt transition towards another pattern. In this talk I discuss the metastable behavior of the model with two stored patterns. After a brief introduction to the potential-theoretic approach to metastability -- capacities, equilibrium potentials, and the variational principles that connect them to metastable transition times -- I show how the decomposition of the spin sites into the sets where the two patterns agree and disagree reduces the dynamics, via an exact lumping, to a product of two Curie--Weiss-type chains. We will show how this tensorization structure allows metastability to be propagated from the component chains to the full model capacities of the product chain are controlled by those of its marginals, and one obtains sharp Eyring--Kramers-type estimates for the transition times between the stored patterns.

Based on ongoing work with Elena Pulvirenti and Martin Slowik.

TBA

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

A path roughness approach to observability in deterministic control systems

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: We consider deterministic control systems in which only partial observations of the state are available. A classical result of Sussmann (1978) states that, for analytic systems, the class of maximally observing (“universal”) controls is generic among smooth inputs. We revisit this result using path-roughness criteria originating in stochastic analysis and rough path theory. This viewpoint yields direct probabilistic consequences: for instance, Brownian and fractional Brownian sample paths provide universal controls, almost surely. We also show how the same roughness mechanism can be used in output-feedback stabilization, and illustrate this in the case of partially observed bilinear systems. Joint work with Lucas Brivadis (L2S, CentraleSupélec).

Microlocal analysis of Klein-Gordon fields and timelike boundaries

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: In this talk (based on work in progress with Alexander Strohmaier) we discuss singularities for solutions of the Klein-Gordon equations with the Dirichlet boundary condition imposed at a timelike boundary on a “globally hyperbolic” spacetime. We will highlight the role of the Mellin transform and of the boundary condition in the analysis of said singularities. If time permits, we shall relate this to quantum field theories.

Computational lower bounds in clustering and ranking

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: In this talk, we will speak about computational lower bounds. After an extended general introduction that presents these tools, and we will focus more particularly on the problem of clustering.

Renormalization: Locality, Multiplicativity, and Birkhoff factorizations

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: I will discuss the interactions between three conditions that can be imposed on the renormalization of Feynman amplitudes.

The first is the Epstein--Glaser factorization condition. In Lorentzian signature, this condition encodes casuality, in Riemannian signature it encodes a strong form of locality. The second, known as "multiplicativity", means that counterterms can be absorbed into the Lagrangian of the theory. Finally, the Connes--Kreimer approach to renormalization is organized around the idea that the renormalized Feynman rules are given by a Birkhoff factorization on the group of characters on the Hopf algebra of Feynman graphs.

These three conditions represent different ways of thinking about renormalization that are sometimes in tension. Nevertheless, I will attempt to show how they can be reconciled.

Quantum error correction and entangled states in mixed dimension Hilbert spaces

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 4:00 PM

Abstract: In  this talk I will begin with a general comparison between classical error correction and quantum error correction. Then I will introduce a mathematical framework to be able to work with quantum error-correcting codes and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. This leads to a general version of the Singleton bound, the first of which appeared in [2510.17231] and the second in [2604.25790]. I will present a quantum MacWilliams identity for mixed-dimensional spaces, the main results of [2604.25790] and define AME states for such spaces and detail the results obtained in [2510.17231] and [2604.25790].

AME states in constant dimension spaces of dimension n on 4 systems are related to mutually orthogonal latin spaces of order n, a classical combinatorial object first studied by Euler. As conjectured by Euler and proved by Tarry in 1900, there do not exist mutually orthogonal latin squares of order 6. However, by proving the existence of an AME state on four systems in constant dimension spaces of dimension 6, there do exist mutually orthogonal quantum latin squares of order 6 if we allow for entanglement between the squares [2104.05122]. In the recent article [2603.02334], we prove that two mutually orthogonal quantum latin squares of order 6 must be entangled.

TBA

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: TBA

Some variations on Brownian motions on star graphs

Berlin Probability Colloquium at TU Berlin, room MA-041, 5:15 PM

Abstract: I will present the construction of some stochastic processes in a star graph, that is a non-euclidean structure where some features of the classical modelling fail. In particular, I will define Brownian motions on graphs and how to construct them. Based on this construction, I will discuss sticky behaviour in the vertex of the graph, first in the Markovian framework, then in a more general setting. We shall first discuss the diffusion property of the sticky Brownian motion; then, we introduce a parabolic problem on the star graph with non-local dynamic conditions in the vertex. In the last part, I will discuss some preliminary results about the Ornstein-Uhlenbeck process and its transition semigroup.

This is based on joint works with Fausto Colantoni (Ulm), Mirko D’Ovidio (Sapienza University, Rome, Italy) and Gianni Pagnini (Bilbao).

Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Computation lower bounds in latent models: example of Clustering Gaussian Mixture Models

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: In many high-dimensional problems, like sparse-PCA, planted clique, or clustering, the best known algorithms with polynomial time complexity fail to reach the statistical performance provably achievable by algorithms free of computational constraints. This observation has given rise to the conjecture of the existence, for some problems, of gaps - so called statistical-computational gaps - between the best possible statistical performance achievable without computational constraints, and the best performance achievable with poly-time algorithms. A powerful approach to predict the best performance achievable in poly-time is to investigate the best performance achievable by polynomials with low-degree. In this talk, I will introduce this low degree framework for estimation problems, which was first defined in 2022 in a paper by Alex Wein and Tselil Schramm. In a second time, I will talk about a specific estimation problem; Clustering Gaussian Mixture Models, for which there is a statistical-computational gap when the dimension and the number of clusters are large.

Superdiffusivity for time-dependent critical drift: a stochastic homogenization approach

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

p-adic multiple zeta values and double shuffle relations for integer indices

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 4:00 PM

Abstract: In the study of multiple zeta values, admissible indices are usually positive indices satisfying a convergence condition. In this talk, I will discuss an extension of this framework to integer indices, allowing zero and negative entries under a suitable admissibility condition. The main point is to show that the two product structures underlying the double shuffle relations remain compatible with this extended notion of admissibility. This gives double shuffle relations for admissible integer indices. I will also explain how this construction leads to a definition of p-adic multiple zeta values at admissible integer indices and why the resulting values satisfy the same double shuffle relations.

Ito Stochastic Differentials

Berlin Probability Colloquium at TU Berlin, room MA-041, 5:15 PM

Abstract: Stochastic calculus is an integral calculus, but classically lacks a corresponding differential calculus. This talk will describe how to define a local notion of an Ito stochastic differential role which gives a meaning, at an instant in time, to symbols such as dWt which appear in the standard shorthand for SDEs. Thus SDEs will become differential equations and a link between the integral calculus and differential calculus is given by an appropriate fundamental theorem of calculus. We will give examples of applications in geometry and finance that show how a differential calculus allows various concepts to be more clearly articulated.

Statistical Analysis of Spatial Graphs

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: A spatial graph is a specific type of graph with spatial attributes associated with the nodes and the edges. It is a smart modelling choice for capturing the skeleton of a shape, a blood vessel network, a porous tissue, and many other data objects with intrinsically complex geometry. In this talk, we describe how spatial graphs can be analysed using a specific metric (the Fused Gromov–Wasserstein metric). We extend a testing procedure between distributions of spatial graphs, a depth measure to describe the distribution of spatial graphs, and a dimensionality reduction procedure based on preserving key topological features. We present this variety of methods on a dataset of cardiac fibrosis tissue and on a dataset of fungus mycelium networks.

The variational approach for 2D Abelian Higgs measure

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Gradient Flow dynamics for rough control systems and Deep ResNets

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 4:00 PM

Abstract: We consider the (sub-Riemannian type) control problem of finding a path going from an initial point x to a target point x, by only moving in certain admissible directions. We assume that the corresponding vector fields satisfy the bracket-generating (Hörmander) condition, so that the classical Chow-Rashevskii theorem guarantees the existence of such a path. One natural way to try to solve this problem is via a gradient flow on control space. However, since the corresponding dynamics may have saddle points, any convergence result must rely on suitable (e.g. random) initialisation. We consider the case when this initialisation is irregular, which is conveniently formulated via Lyons' rough path theory. We show that one advantage of this initialisation is that the saddle points are moved to infinity, while minima remain at a finite distance from the starting point. In the step 2-nilpotent case, we further manage to prove that the gradient flow converges to a solution, if the initial condition is the path of a Brownian motion (or rougher). The proof is based on combining ideas from Malliavin calculus with Łojasiewicz inequalities. A possible motivation for our study comes from the training of Deep Residual Neural Networks, where suitable stochastic scalings of randomly initialised networks give rise to rough limiting dynamics.

Towards a Marcinkiewicz Theorem for Expected Signatures with Applications to PDE

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: In 1939 Marcinkiewicz published a celebrated result which establishes that a random variable with polynomial cumulants is necessarily distributed as a (possibly degenerate) Gaussian. We reformulate this result in the language of signatures and rough paths, and explore the properties of random variables with values in the group like elements of the tensor series algebra. We obtain a partial characterization of such random variables with polynomial cumulants (in the tensor series algebra) as Brownian motions diffused along the free Lie algebra. This endeavor is then connected to positivity-preserving properties of some parabolic PDEs with unbounded coefficients.

A selfcontained mathematical analysis of the Schrödinger-Poisson system

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Recent developments in likelihood inference for stochastic differential equations

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: The complexity of likelihood inference for stochastic differential equations based on discrete time samples often necessitates the use of approximations or computational techniques. Approximate likelihood methods for high frequency data have often been used in financial econometrics, but these methods usually do not perform well for strongly nonlinear models. 

New developments of approximate likelihood methods based on splitting schemes are presented. These methods perform well also for strongly nonlinear models and at moderate sampling frequencies. Splitting schemes were originally introduced to solve ODEs and SDEs numerically, but in Pilipovic, Samson and Ditlevsen (2024) it was proposed to use them for statistical inference. In the talk a more general approach is presented that is applicable to a broad class of diffusion models. The theory is developed in the framework of approximate martingale estimating functions, which provide approximations to the score function and estimators that are efficient for high frequency data. For Strang splitting an approximate martingale estimating function of order 3 is obtained. 

Sometimes useful models with an explicit likelihood function can be found. This enables exact likelihood inference, which works at all sampling frequencies. As an example of this, a class of stochastic differential equation models on the torus is presented, which can be used to analyse time series of angular data. These diffusion processes are ergodic and time-reversible and can be constructed for any pre-specified stationary distribution on the torus. If time permits, applications to biological data will be briefly presented.   

The lecture is based on joint work with Susanne Ditlevsen, Adeline Samson and Eduardo García-Portugués.

Reference: 

Pilipovic, P., Samson, A. And Ditlevsen, S. (2024): Efficient estimation for ergodic diffusion processes sampled at high frequency. Ann. Statist., 52, 842 - 867. 

Role of the maximum principle and geometry in Dean-Kawasaki equations

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

A Hopf algebra on nonplanar binary forests

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 4:00 PM

Abstract: We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of admissible cuts. We show that the terms in this coproduct have an additional combinatorial interpretation in terms of subsets of leaves, which facilitates the construction of Hopf algebra morphisms involving this Hopf algebra, and creates a connection with a Hopf algebra of Bruned used in the renormalization of stochastic processes. Finally, we show that this Hopf algebra is dual to the universal enveloping algebra of a Lie algebra arising from a pre-Lie operator on binary trees based on edge-insertion.

Bayesian dynamic portfolio optimization with informative constraints

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 5:15 PM

Abstract: There is a recent debate on whether sustainable investing necessarily impact portfolio performance negatively. We model the financial impact of portfolio constraints by attributing the performance of dynamic portfolios to contributions from individual constraints. We consider a mean-variance portfolio problem with unknown asset returns. Investors impose a dynamic constraint based on a firm characteristic that contains information about returns, such as the environmental, social, and governance (ESG) score. We characterize the optimal investment strategy through two stochastic Riccati equations. Using this framework, we demonstrate that, depending on the correlation between returns and firm characteristics, incorporating the constraint can, in certain cases, enhance portfolio performance compared to a passive benchmark that disregards the information embedded in these constraints. Our results shed light on the role of implicit information contained in constraints in determining the performance of a constrained portfolio.

Magic strikes

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 4:15 PM

Abstract: Rolloos and Arslan showed that the volatility swap strike can be approximated by the implied volatility evaluated at a single (magic) strike- the zero-vanna strike. We extend this idea systematically to a broad class of attainable claims, including variance swaps, gamma swaps, the leverage contract, and stochasticity. Using the forest expansion of Alòs, Gatheral and Radoičić, we derive a Bergomi-Guyon-type smile expansion and show that each contract’s fair value can be approximated, to arbitrary order, by the total implied variance evaluated at a small number of magic strikes determined by simple fixed-point equations. At leading order, these magic strikes recover known results; at higher orders, they provide corrections that are model-independent in the sense that they depend on the model only through the implied volatility smile. Numerical tests under the Heston and rough Bergomi models demonstrate these approximations are highly accurate even for strongly skewed smiles. The method does not require strike inversion and can be applied directly to interpolated market smiles.

Multivariate Self-Exciting Processes with Dependencies and application to cyber loss processes

Berlin Probability Colloquium at TU Berlin, room MA-041, 5:15 PM

Abstract: We introduce the class of multidimensional self-exciting processes with dependencies (MSPD), which is a unifying writing for a large class of processes: counting, loss, intensity, and also shifted processes. The framework takes into account dynamic dependencies between the frequency and the severity components of the risk, and therefore induces theoretical challenges in the computations of risk valuations. We present a general method for calculating different quantities related to these MSPDs, which combines the Poisson imbedding, the pseudo-chaotic expansion and Malliavin calculus. The methodology is illustrated on cyber loss process and the computation of cyber stress scenarios, based on the accumulation of claims or vulnerabilities.

Based on joint works with Yousra Cherkaoui, Thomas Peyrat and Anthony Réveillac.

Discrete Anderson Hamiltonians with correlated Gaussian potentials

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Overcoming the spatial order barrier for nonlinear SPDEs with additive space-time white noise

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: When approximating solutions of SPDEs, a basic challenge is that the rate of convergence is limited due to the low time and space regularity of the solution. Considering semilinear SPDEs with additive space-time white noise in space dimension d=1, we introduce a novel numerical scheme which improves the spatial convergence rate from the classical rate 1/2 to rate 3/2. The temporal convergence rate is proven to be 3/4 for our scheme, which enhances the classical rate 1/4 using ideas of previous works (Jentzen, Kloeden ‘08, Jentzen ’11, Djurdjevac, Kremp, Gerencser '24) . The talk is based on a work in progress together with Lukas Anzeletti and Mate Gerencser.

The Stefan problem with a phase transition between visco-elastic fluids and finitely-strained solids

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains in the Eulerian frame. The Jeffreys (also called anti-Zener) rheology in the deviatoric part with an additional viscosity is used. The main philosophy for the mechanical solid-liquid transition is that the viscous (or viscoplastic) response depends on temperature and may completely degenerate towards a viscoelastic fluid during thawing, which then allows for free flow of the fluid and its freezing in a new configuration and possibly again subsequent melting towards the fluid unlimitedly repeating such cycles. The classical Stefan problem related with the latent heat of the 1st-order (i.e. here thawing-freezing) phase transition is augmented by involving kinetic overheating and undercooling. The (sketched) analysis by a time discretization with a suitable truncation is applied to a higher-gradient modification of the original problem, i.e. involving the concept of multipolar nonsimple material. Some enhancements of the basic model as phase-field fracture in the solid phase or a diffusant dependency (like a salinity variation within the sea water/ice transition or a nickel content variation within the Earth inner/outer core transition) will be outlined, too.

Rigidity of Riesz gases

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: In this talk, we will survey the rigidity of Riesz gases, which are equilibrium states (or Gibbs measures) in R^d of continuum particle systems subjected to the Riesz potential. Here, rigidity refers to hyperuniformity, number rigidity, or symmetry breaking (such as crystallization or cyclic factor property). Several results will be presented, along with a number of conjectures. 

Asymptotic equivalence for nonparametric additive regression

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: We prove asymptotic equivalence of nonparametric additive regression and an appropriate Gaussian white noise experiment in which a multidimensional shifted Wiener process is observed, whose dimension equals the number of additive components. The shift depends on the additive components of the regression function and solely the one- and two-dimensional marginal distributions of the covariates via an explicitly specified bounded but non compact linear operator. The number of additive components is allowed to increase moderately with respect to the sample size. In the special case of pairwise independent components of the covariates, the white noise model  decomposes into independent univariate processes. Moreover, we study  approximation in some semiparametric setting where the operator splits into a  multiplication operator and an asymptotically negligible Hilbert-Schmidt  operator.

This talk is based on a joint work with Moritz Jirak (Universität  Wien) and Angelika Rohde (Albert-Ludwigs-Universität Freiburg).

The algebra for quantum fault-tolerance

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 3:00 PM

Abstract: Quantum computers promise considerable speed-ups over classical systems, but realising these speed-ups requires protecting fragile quantum states through quantum error correction (QEC). While QEC has advanced rapidly, most research focuses on preserving quantum memories rather than enabling entire fault-tolerant computations. Developing intuitive, reliable tools for designing and reasoning about fault-tolerant quantum systems is therefore critical. In this introductory talk, I will describe some of the key difficulties in fault-tolerant quantum computing. I will then outline some recent algebraic formalisms for attacking these difficulties: the homological approach to Calderbank-Shor-Steane codes and (time-permitting) the symplectic approach to stabiliser codes.

Collective Pensions

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 5:15 PM

Abstract: This talk will explain when it is (and when it is not) possible for a group of investors to gain mutual benefit from a collective pension design. We will see that investors can obtain mutual benefit by completing the market with additional insurance products and will estimate the potential benefit that collective designs can provide over traditional pension products.

Multivariate Rough Volatility

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 4:15 PM

Abstract: We review some empirical facts of financial markets that have motivated the rough volatility paradigm for modelling financial volatility, both from the point of view of financial time series and options pricing. 

Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model proposed in Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model, propose parameter estimators, derive their asymptotic theory and  perform a simulation study that confirms the asymptotic theory in finite sample. We carry out an extensive empirical investigation on emprical realized volatility time series, showing that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories.

Regularisation by noise for stochastic wave equations

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: We consider nonlinear Stochastic Wave Equations (SWE), posed on d-dimensional bounded domains and perturbed by additive noise that is white in time, colored in space, with spatial covariance given by a Riesz kernel. Leveraging regularisation-by-noise techniques in infinite dimensions, we show pathwise well-posedness of SWE with Hölder nonlinearities b for d ≤ 4 and (probabilistically) weak existence of solutions for distributional b in all dimensions. Our approach relies on a shifted version of Lê’s stochastic sewing lemma in Hilbert spaces. Time permitting, we shall discuss an application to Smoluchowski-Kramers approximation with irregular coefficients. 

Ongoing joint work with Oleg Butkovsky (WIAS and HU Berlin) and Michael Salins (Boston University).

An alpha-potential game framework for dynamic N-player games

Berlin Probability Colloquium at TU Berlin, room MA-041, 5:15 PM

Abstract: Game theory has a long history, yet identifying Nash equilibria in dynamic non-cooperative games remains a fundamental challenge with significant computational and conceptual complexity. Over the past decade, mean field game theory has emerged as a pivotal framework, offering important theoretical insights and computational advances for the analysis of large-scale stochastic games. However, mean field games require homogeneity and weak interactions among players and focus only on the limiting behavior when the number of players goes to infinity.

In this talk, we present a new paradigm for dynamic N-player games, called alpha-potential games, where the change of a player’s objective function resulting from a unilateral deviation of her strategy is equal to the change of an alpha potential function up to an error alpha. Within this framework, the problem of computing approximate Nash equilibria reduces to a stochastic control problem for the alpha-potential function, significantly simplifying both analysis and computation. The parameter alpha also reveals important structural properties of the game, such as the population size, the intensity of player interactions, and the degree of heterogeneity across players. We will discuss through simple examples some recent theoretical and algorithmic developments, along with a few open problems for this new game framework.

Cascade equation in the Stefan problem and equilibria of mean-field games

Berlin Probability Colloquium at TU Berlin, room MA-041, 4:15 PM

Abstract: After motivating the Stefan problem from the random growth model perspective, I will discuss its discontinuities in time. These turn out to be characterized by the cascade equation, a second-order hyperbolic PDE.

Questions of existence and regularity for the latter can be answered by expressing its solution as the value function of a player in an equilibrium of a suitable mean field game.

Based on joint work with Yucheng Guo and Sergey Nadtochiy.

BV curves of measures

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: Representation results for absolutely continuous curves with values in the Wasserstein space of Borel probability measures in R^d with finite p-moment, p > 1, provide a crucial tool to study evolutionary PDEs in a measure-theoretic setting. They are strictly related to the superposition principle for measure-valued solutions to the continuity equation.

This talk revolves around the extension of these results to the case p = 1, and to curves that are only of bounded variation in time. Based on a joint collaboration with Stefano Almi (Napoli) and Giuseppe Savaré (Milano).

Kurtz' Markov Mapping Theorem 

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: When working with Markov processes from the point of view of martingale problems, the question of well-posedness (existence+uniqueness) is central. One central idea in this setting is to endow a martingale problem with more structure that helps proving uniqueness. Done correctly, the extended process "maps" down to the original one and transfers many properties. In this talk, we will delve into the main ideas behind Kurtz' Markov Mapping and Extension Theorems and take some time to go through possible applications, starting from the equivalence between the concepts of weak solutions and martingale solutions to an SDE. 

Unbounded operator integrals and Quantum Field Theory

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: Operator integrals appear in abundance in many areas of functional analysis. The typical way to deal with these is through a theory called Multiple Operator Integration (MOI). Noncommutative Geometry is no exception, but the involved operator arguments are here typically unbounded, and they do not neatly fit the usual theory of MOI. To solve this problem, in joint work with Edward McDonald and Teun van Nuland, we provided a construction of MOIs which can handle (unbounded) abstract pseudodifferential operators. I will sketch how this construction can play a role in a Quantum Field Theory model based on noncommutative geometry (through the spectral action). I will then discuss recent joint result with Teun van Nuland and Jesse Reimann regarding the power counting in this model.

A new neural network architecture for learning convex functions

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: For small covariate dimension, shape constrained inference is a well-established topic within nonparametric statistics. For large covariate dimensions, one naturally wants to introduce machine learning based methods. Input convex neural networks (ICNNs) were designed as a network architecture to learn convex functions. In this talk, we introduce Hyper Input Convex Neural Networks (HyCNNs). HyCNNs combine the principles of Maxout networks with ICNNs to create a neural network that is always convex in the input, theoretically capable of leveraging depth, and performs reliable when trained at scale compared to ICNNs. Concretely, we prove that HyCNNs require exponentially fewer parameters than ICNNs to approximate quadratic functions up to a given precision. Throughout a series of synthetic experiments, we demonstrate that HyCNNs outperform existing ICNNs and MLPs in terms of predictive performance for convex regression and interpolation tasks. WWe further apply HyCNNs to learn high-dimensional optimal transport maps for synthetic examples and for single-cell RNA sequencing data, where they oftentimes outperform ICNN-based neural optimal transport methods and other baselines across a wide range of settings.

For more details, see arxiv.org/pdf/2604.26942.

This is joint work with Shayan Hundrieser and Insung Kong

Mean field type singular SPDEs

Oberseminar Analysis-Probability at MPI, E1 05 (Leibniz-Saal), 3:15 PM

Abstract: I will explain how to set up a variant of the first order paracontrolled calculus to prove local in time well-posedness for some mean field type (gPAM)-like equation, and prove propagation of chaos for an associated system of interacting fields.

On pitchfork bifurcations in Phi-4-2

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Quantum fault-tolerance by construction

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 3:00 PM

Abstract: Quantum data is extremely sensitive to noise. This has led to an increasingly widespread believe that quantum error correction and fault-tolerant quantum computation will be a crucial ingredient in the vast majority of useful quantum computations. I will talk about a recent approach to designing and working with fault-tolerant quantum circuits called "fault tolerance by construction". In this approach, you can specify a target quantum computation and transform it iteratively using special rules called fault equivalences, which preserve a computations behaviour under noise. We can define fault equivalences for circuits, but also for diagrams in the ZX calculus, a particularly useful tool for quantum circuit optimisation. After giving a flavour for what it's like working with the ZX calculus and explaining the basics of quantum fault tolerance, I will give some examples of calculations involving fault equivalences and survey some of the state of the art results and open problems in the area. This talk assumes no prior knowledge of quantum computing or quantum circuits.

An Introduction to the Seiberg–Witten Equations

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: The Seiberg–Witten equations provide a powerful tool for studying the topology of smooth four-manifolds via gauge-theoretic methods. In this talk, we introduce the equations and the geometric framework in which they are defined, focusing on connections on line bundles, spin^c structures, and the associated Dirac operator. We then discuss the moduli space of solutions modulo gauge transformations and outline how Seiberg–Witten invariants arise from counting these solutions. Time permitting, we briefly mention some applications and connections to other approaches in four-manifold topology.

For more information and log in details please contact Christian Molle. 

Energy dissipation and shocks in ideal fluid systems

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: In an ideal density dependent fluid system, is the total energy dissipated on shock type discontinuities? To this end, we study the local energy balance for weak solutions to the isentropic compressible Euler system and derive fine properties of the defect measure, that describes the (possible) loss of the total energy. This is done by a careful analysis of the small scale properties of the solutions at the shock discontinuity. By means of the same technique, we also consider the inhomogeneous incompressible case, and, comparing the result, we confirm the general principle that the accumulation of the total energy on codimension one singular structures is a feature that distinguishes compressible and incompressible models.

Two phase transitions for catalytic branching Markov chains

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: Consider a continuous-time branching Markov chain (Zt,t≥0) on a locally finite graph G rooted at o. Each particle moves according to an irreducible Markov process ξ and branches at a rate that depends on their location: the branching rate is

λrt ≥0 at the root and λ≥0 elsewhere. The offspring distribution is supercritical with mean m>1, has no extinction and finite second moment. We characterize the recurrence/transience phase transition for this catalytic branching Markov chain. Furthermore, under suitable assumptions we prove a second phase transition concerning the asymptotic behaviour of the relative empirical density,

(Zt(G))−1Zt, where Zt is the empirical measure of the particles and Zt(G) is the total population size. If (m−1)(λ0−λ)>γesc, where

γesc is the escape probability that ξ never returns to the root, then (Zt(G))−1Zt converges almost surely to a deterministic probability measure. If (m−1)(λ0−λ)∈(0,γesc], then (Zt(G))−1Zt converges almost surely to zero. When the graph is the integer lattice G=Zd and ξ is the simple random walk, our results confirm several conjectures of Mailler and Schapira [Ann. Appl. Probab. 2026], which studied this model via a different approach.

Based on a joint work in progress with Xinxin Chen (Bejing Normal Unversity), Nina Gantert (Technical University of Munich) and Haojie Hou (Beijing Institute of Technology). 

Multiple change point detection in functional data with applications to biomechanical fatigue data

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: Injuries to the lower extremity joints are often debilitating, particularly for professional athletes. Understanding the onset of stressful conditions on these joints is therefore important in order to ensure prevention of injuries as well as individualised training for enhanced athletic performance. We study the biomechanical joint angles from the hip, knee and ankle for runners who are experiencing fatigue. The data is cyclic in nature and densely collected by body worn sensors, which makes it ideal to work with in the functional data analysis (FDA) framework. 

We develop a new method for multiple change point detection for functional data, which improves the state of the art  with respect to at least two  novel aspects. First, the curves are compared with respect to their maximum  absolute deviation, which leads to a better interpretation of local changes in the functional data compared to classical $L^2$-approaches. Secondly, as slight aberrations are to be often expected in a human movement data, our method will not detect arbitrarily small changes but hunts for relevant changes, where maximum absolute deviation between the curves exceeds a specified threshold, say $\Delta >0$. We recover multiple changes in a long functional time series of biomechanical knee angle data, which are larger than the desired threshold $\Delta$, allowing us to identify changes purely due to fatigue. In this work, we analyse data from both controlled indoor as well as from an uncontrolled outdoor (marathon) setting.

Semimartingales forced onto a manifold by a large drift

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Mean-field control of non exchangeable systems

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 5:15 PM

Abstract: We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us  to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows  of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a L²-set of Wasserstein space-valued functions.  

This talk is based on a joint work with A. De Crescenzo, M. Fuhrman and H. Pham.

Quantification of limit theorem for Hawkes processes

Research Seminar Stochastic Analysis & Finance at TU Berlin, room MA-043, 4:15 PM

Abstract: Hawkes processes are a popular model for self-exciting phenomena, from earthquakes to finance. In this talk, I will first present them in a simple way, using a Poisson imbedding construction. I will then review what is known about their long-time behavior, through limit theorems for both linear and non-linear cases. The focus will be on three regimes that appear when the process has a long memory and the branching ratio gets close to or above one: the Nearly Unstable, the Weakly Critical, and the Supercritical Nearly Unstable Hawkes processes. These regimes have been studied qualitatively, but quantitative convergence results have been missing. I will explain how we obtain explicit convergence rates, relying on a coupling with a Brownian sheet, Fourier analysis, and a careful approximation of the absolute value function.

Rough Stochastic Kalman-Bucy Filter

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: In this talk, we adopt a rough path perspective on the classical Kalman-Bucy filtering problem with conditionally linear dynamics and correlated noise. By treating the observation process as a deterministic rough path, the signal dynamics are formulated as Rough Stochastic Differential Equations (RSDEs) with linear coefficients, in the sense of Friz, Hocquet, and Lê. We show that under a Gaussian initial condition, the distribution of the rough signal process remains Gaussian, and we characterize its mean and covariance as solutions to deterministic Rough Differential Equations (RDEs). In particular, the covariance satisfies a linear-quadratic Riccati RDE, for which we establish global well-posedness and non-explosion results that are of independent interest and, to the best of our knowledge, the first for this class of equations. Finally, we show that, when randomised, the RDEs for the mean and the covariance recover the classical Kalman-Bucy filtering equations. Joint work with P.K. Friz, K. Lê, and H. Zhang (arXiv:2509.11825).

Sharp pathwise nonuniqueness for additive SDEs

Berlin Probability Colloquium at TU Berlin, room MA-041, 5:15 PM

Abstract: We construct a family of random velocity fields demonstrating the sharpness of the classical Zvonkin–Veretennikov–Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift u and show that for any α < 0, there exists a velocity field u ∈ Lt∞ Cxα  that admits a unique weak solution but does not satisfy pathwise uniqueness (and hence has no strong solutions). This contrasts with the case α ≥ 0, for which the existence of a unique strong solution is guaranteed. The velocity field construction is random, and the proof essentially uses cen tral limit theorem scaling through the Berry–Esseen theorem. We also give natural extensions to non-Brownian driving noises, including nonuniqueness for arbitrary driving noises with certain H¨older regularities and an analo gous sharpness of the strong well-posedness by noise regime for fractional Brownian motions. 

Estimation of a smooth functional for inverse problems

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: Since the seminal paper by Ibragimov and Khasmisnkii (1980) it is well known that the plug-in approach is sub-optimal in estimating a nonlinear functional like the squared norm of the signal. A kind of bias correction is necessary to achieve root-n optimality. Series of recent papers by Koltchinskii with coauthors revisited this problem from modern prospective with a high or even dimensional parameter space and a complex structure of the functional to be estimated. This talk discusses the problem of estimation of a smooth functional in the inverse problem setup given by an objective function L(θ).

A different view is offered. Namely, the functional φ(θ) is included in the parameter list as the target value, while the parameter θ is treated as a nuisance parameter. The structural relation x = φ(θ) is replaced by the structural penalty (λ|x − φ(θ))2. The resulting estimator is obtained as a third-order correction (xˆ, θˆ) of the full-dimensional penalized MLE (x ̃, θ ̃) = arg inf(x,θ) L(θ) + λ|x − φ(θ)|2/2. The approach enables us to obtain an accurate expansion with an explicit leading term and self-consistent remainder and provides sharp risk bounds for this estimator. In particular, we present sufficient conditions for root-n consistency of the procedure.

Stochastic control for singular diffusions: viscosity solutions & BSDEs

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: We introduce a calculus of space–time controlled functions for rough stochastic systems, providing a unified composition rule for evaluating random fields along rough semimartingales. This framework yields a rough stochastic Itô–Wentzell formula under natural and verifiable regularity assumptions. Motivated by recent work of Hude et al. (2024) and, independently, Del Moral and Singh (2022), which established forward–backward perturbation identities for diffusion processes, we show how Itô–Alekseev–Gröbner and backward Itô–Wentzell type representations arise naturally within a rough stochastic framework. This is joint work with Peter K. Friz, Arnulf Jentzen and Jian Song.

Time-asymptotic self-similarity of the damped Euler equations in parabolic scaling variables

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: In this talk, we study the long-time behavior of solutions to the compressible Euler equations with frictional damping on the whole space, assuming nonzero direction-dependent values for the density at spatial infinity. By introducing parabolic scaling variables, we reformulate the system and derive a relative entropy inequality. This framework allows us to show that the density converges to a self-similar solution of the porous medium equation, while the limiting momentum is governed by Darcy’s law. We also obtain convergence rates that explicitly depend on the flatness of the limiting profile. The main part of the analysis focuses on weak solutions in the one-dimensional case, and we further extend the results to energy-variational solutions in the multidimensional setting.

This research is joint work with Thomas Eiter (WIAS).

Non-Standard Fluctuations of the edge density in the edge-triangle Model

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: Exponential Random Graph Models extend the classical (dense) Erdős–Rényi random graph by incorporating higher-order structural features through a Hamiltonian formalism inspired by statistical mechanics. Among these, the edge-triangle model is a basic but non trivial example, where the competition between edge and triangle terms leads to a rich phase structure. In this talk we will give an overview of known limit theorems and concentration results for subgraph densities, with particular emphasis on an open problem: a non-standard CLT for the edge density. The conjecture will be supported by a mean-field analysis, which allows for explicit computations and suggests a different fluctuation scaling, while also providing some insight into related open questions.

This talk is based on joint works with A. Bianchi, F. Collet, and G. Passuello.

The signature group for paths of higher variation as an inverse limit

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: E. Le Donne and R. Züst (2021) showed that the space of signatures of rectifiable paths is a geodesic metric tree. We extend their result originally formulated in terms of curve length to the setting of p-variation for p > 1. This extension relies on results of Boedihardjo, Geng, Lyons, and Yang (2016) concerning signatures of weakly geometric rough paths, which provide the appropriate analytical control in the p-variation framework. The key components of the original construction are reinterpreted using the language of metric groups, allowing us to define and work with the signature group adapted to p-variation.

Within this framework, we revisit the lifting procedure introduced by Le Donne and Züst - namely, the construction of a canonical lift of a path into a suitable group structure - and show that the main result, concerning the existence and properties of such lifts, continues to hold when length is replaced by p-variation.

Extremes of Structural Causal Models

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, 10:00 AM

Abstract: The behaviour of extreme observations is well-understood for time series or spatial data, but little is known if the data generating process is a structural causal model (SCM). We study the behavior of extremes in this model class, both for the observational distribution and under extremal interventions. We show that under suitable regularity conditions on the structure functions, the extremal behavior is described by a multivariate Pareto distribution, which can be represented as a new SCM on an extremal graph. Importantly, the latter is a sub-graph of the graph in the original SCM, which means that causal links can disappear in the tails. We further introduce a directed version of extremal graphical models and show that an extremal SCM satisfies the corresponding Markov properties. Based on a new test of extremal conditional independence, we propose two algorithms for learning the extremal causal structure from data. The first is an extremal version of the PC-algorithm, and the second is a pruning algorithm that removes edges from the original graph to consistently recover the extremal graph. The methods are illustrated on river data with known causal ground truth.

Discrete Anderson Hamiltonians with correlated Gaussian potentials

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

How to measure quantum fields? Implementing a causal measurement scheme

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 1:00 PM

Abstract: While measurement processes in standard quantum mechanics are well understood, the extension of these ideas to quantum field theory (QFT) remains a key challenge. In particular, ensuring that measurements respect fundamental principles such as relativistic causality is crucial. A persistent issue concerning measurements in QFT is, though, that the usual axioms for QFT alone are insufficient to prevent superluminal signaling. In this talk, I will discuss a recent proposal by Fewster and Verch for a local, covariant and causal measurement framework in algebraic QFT. In particular, I will discuss completeness of the framework and motivate its underlying assumptions focussing on the concrete setting of a free scalar field and Gaussian measurements. We conclude that the Fewster-Verch approach is suitable to model typical measurements in QFT.

A Rough Functional Breuer–Major Theorem

Research Seminar Rough Analysis and Stochastic Dynamics at TU Berlin, room MA-748, 11:00 AM

Abstract: We extend the functional Breuer–Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983). This is joint work with Henri Elad Altman (Paris XIII) and Nicolas Perkowski (FU Berlin).

Regularity of the score and convergence rates of generative diffusion models

Forschungsseminar Mathematische Statistik at WIAS Berlin, Erhard-Schmidt lecture room, & ZOOM, 10:00 AM

Abstract: We show that diffusion-based generative models adapt to the smoothness of the target distribution: the score function inherits the target’s regularity. Leveraging this adaptivity, we obtain a concise proof that diffusion models achieve minimax-optimal rates for density estimation.

The critical 2-d Stochastic Heat Flow and some of its properties

Oberseminar Analysis-Probability at MPI, E2 10 (Leon-Lichtenstein), 3:15 PM

Abstract: The critical 2-d Stochastic Heat Flow arises as a non-trivial solution of the Stochastic Heat Equation (SHE) and its discretisation via directed polymers at the critical dimension 2 and at a phase transition point. It is a log-correlated field, which is neither Gaussian nor a Gaussian Multiplicative Chaos. We will review the phase transition of the 2-d SHE, describe the main points of the construction of the Critical 2-d SHF and outline some of its features (e.g. singularity, regularity, moments, support etc.) Most of the talk will be based on joint works with Francesco Caravenna and Rongfeng Sun but contributions of other researchers will be presented.

Variational methods for problems with inertia and their limits

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: For static and quasi-static problems, (iterated) minimization has long been one of the most important tools to prove existence of solutions. The main advantage of these variational approaches is that they able to deal with complicated nonlinearities and nonconvexities in a rather natural fashion, directly relying on the description of a problem in terms of its physical energy. In contrast, for dynamic problems, i.e. those involving inertia, such variational approaches so far have been much less used in practical existence proofs. 

The aim of this talk is to present our recent and not so recent attempts at bridging this gap, using a ”time-delayed” approach which uses energetical descriptions and minimization as both a modelling approach, as well as a way of showing existence of solutions. This will be illustrated in with a number of problems from recent publications, involving solids, fluids and their interaction. Furthermore we will see how the same ideas can be used to study limit systems of parameter-dependent families of such problems in a similarly general fashion.

This is based on joint works with, among others, B. Benešová, D. Breit, A. Češík, G. Gravina, M. Kružík and S. Schwarzacher.

Elliptic operators with distributional coefficients

Research Seminar Stochastics at FU Berlin, SR 115, Arminallee 3, 10 AM

Relational Learning with Covariance Information

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 3:00 PM

Abstract: Covariance information is commonly used in machine learning to reveal data interdependencies such as network topology inference (e.g., graphical lasso) and dimensionality reduction (e.g., Principal Component Analysis (PCA)). However, such information is often only the first step in a machine learning pipeline that is performed separately from the task. Because of finite-data estimation errors, we end up working with a sample covariance matrix that leads to uncertainties in its spectrum. For example, PCA is notoriously unstable to covariance estimation errors, i.e., small data perturbations might lead to large changes in principal directions. To address this, coVariance Neural Networks (VNNs) were introduced. These networks perform graph convolutions on the sample covariance matrix, an operation that, similarly to PCA, modulates the data principal components, but with enhanced representation power and greater stability against covariance estimation errors. However, in sparse, high-dimensional settings with limited data, covariance estimation is particularly difficult, which hinders VNNs’ performance despite their stability. Sparse VNNs overcome this by using theoretically grounded covariance sparsification, which improves stability, reduces the impact of spurious correlations on performance and improves computation and memory efficiency. The success of VNNs motivates their extension to different settings. SpatioTemporal VNNs, for instance, process multivariate time series by applying graph convolutions on the online estimated covariance and temporal convolutions over time, achieving stability to estimation errors in both covariance and model parameters due to streaming data. Finally, VNNs’ stability promotes fairness in datasets with poorly represented groups. Building on this, Fair VNNs leverage equitable covariance estimates and fairness penalties in the loss function to ensure a more balanced treatment of these groups.

Primal dual methods for Wasserstein gradient flows

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: Many nonlinear evolution equations arising in porous media flow, materials science, and collective behavior can be understood as gradient flows in the space of probability measures. In this talk, we present a new numerical framework that exploits this variational structure by combining ideas from optimal transport with modern operator splitting methods. Our approach is based on the Jordan-Kinderlehrer-Otto (JKO) scheme and the Benamou-Brenier formulation of the Wasserstein distance, which together recast the solution of certain nonlinear, nonlocal PDEs as a sequence of convex optimization problems. We show how these problems can be solved efficiently using a recent primal-dual splitting algorithm with rigorous convergence guarantees. We illustrate the method with numerical examples for nonlinear PDEs and Wasserstein geodesics. We conclude by outlining extensions to more general nonlinear mobilities and transport costs, highlighting the flexibility of the approach and its potential for a wide range of applications.

Well-posedness and optimal velocity control of a Brinkman--Cahn--Hilliard system with curvature effects

Langenbach-Seminar at WIAS Berlin, Erhard-Schmidt lecture room, 2:15 PM

Abstract: Diffuse-interface models for multiphase flows have attracted considerable interest due to their ability to describe complex interfacial dynamics, including curvature effects, within a unified and energetically consistent framework. In this talk, we present joint results for a Brinkman–Cahn–Hilliard system coupling a sixth-order phase-field evolution with a Brinkman-type momentum equation with variable shear viscosity. The Cahn–Hilliard equation includes a nonconservative source term modeling mass exchange, while the momentum equation involves a forcing term that is not divergence-free. We prove the existence of weak solutions in a divergence-free variational framework and, in the case of constant mobility and shear viscosity, establish uniqueness and continuous dependence on the forcing. We also analyze the Darcy limit and obtain existence results for the corresponding reduced system. Finally, we consider an optimal control problem with distributed velocity control, prove the existence of optimal controls, show the Fr´echet differentiability of the control-to-state operator, and derive first-order necessary optimality conditions via an adjoint system.

Anderson Hamiltonians with correlated Gaussian potentials

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: Anderson Hamiltonians, which are random Schrödinger operators, model the evolution of electrons or quantum states in a disordered system. Philip Warren Anderson (1958) showed that if there is too much disorder in the system, instead of seeing a diffusive behaviour for the electron, they get trapped. A similar localisation effect takes place in the parabolic Anderson model, which is the parabolic problem or Cauchy problem related to the Anderson Hamiltonian. The spectral properties of the Hamiltonian determine this localisation behaviour. Many such models have been studied, but often with a potential field that is i.i.d. We study these Hamiltonians with a correlated Gaussian potential and consider it eigenvalue order statistics.

This is joint work with Giuseppe Cannizzaro and Cyril Labbé.

Invariant variational problems and cohomology

Analysis Seminar Potsdam Campus Golm, Building 9, Room 2.22 and via Zoom, 11:00 AM

Abstract: The interplay of inverse variational problems with invariance properties is exploited. We focus on symmetries of globally defined locally variational field equations and find conditions for the variation of local Noether strong currents to be conserved and variationally equivalent to a global conserved current.

Information geometry of the Otto metric

Oberseminar Analysis-Probability at MPI, E1 05 (Leibniz-Saal), 3:15 PM

Abstract: In this talk, I present a novel extension of information geometry to the setting of Wasserstein geometry and highlight its applications in machine learning. The central structure of information geometry consists of a Riemannian manifold together with a pair of dual affine connections. Classically, this structure arises from a statistical model endowed with the Fisher–Rao metric, the mixture connection, and its dual, the exponential connection. While this framework captures fundamental statistical properties, it does not account for the metric structure of the underlying sample space.

Motivated by this limitation, I propose an extension to the Wasserstein setting. Within this framework, I introduce the dual of the mixture connection with respect to the Otto metric, yielding a novel form of exponential connection. This leads to a new dual structure comprising the mixture connection, the Otto metric as the Riemannian metric, and the newly defined exponential connection.

I derive the geodesic equation associated with this exponential connection and show that it coincides with the Kolmogorov forward equation of a gradient flow, also known as the continuity equation in Wasserstein geometry. Furthermore, I construct the canonical contrast function of the proposed dual structure, which I term the Wasserstein–Kullback–Leibler divergence, and demonstrate how it addresses limitations of the classical Kullback–Leibler divergence.

Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 3:00 PM

Abstract: We prove that the post-Lie algebraic structure on the decorated trees which appears spontaneously in Regularity Structures is a post-Lie deformation of a pre-Lie algebra. This post-Lie structure is the conerstone of the algebraic foundations of the theory of Regularity Structures, a theory that solves a large class of subcritical singular stochastic partial differential equations which were out of reach without a proper algebraic structure. This deformation allows to reduce the origin of these structures to a simpler object. We use the formalism of deformation theory in order to understand the nature of this deformation. This leads us to construct the differential graded Lie algebra that governs post-Lie deformations of a pre-Lie algebra, to develop the post-Lie cohomology theory for a pre-Lie algebra, by which we classify infinitesimal post-Lie deformations of a pre-Lie algebra using the second cohomology group. The rigidity of such kind of deformations is also characterized using the second cohomology group. This is a joint work with Yvain Bruned and Yunhe Sheng.

Precise large deviations for S(P)DEs: theory, numerics, applications

Oberseminar Analysis-Probability at MPI, G3 10 (Lecture hall), 3:15 PM

Abstract: In this talk, I will first review the classical theory of Laplace asymptotics for small-noise diffusions in R^n and extensions to stochastic partial differential equations (SPDEs), with particular emphasis on sharp asymptotics. Going beyond the usual log-asymptotics of large deviation theory, we focus on prefactors and thus Hessian contributions that are essential for accurate probability estimation.

I will then describe how these theoretical results can be turned into a practical computational framework for estimating rare event probabilities without sampling. The approach is designed to scale to high-dimensional state spaces, as required for SPDEs, and involves solving ODE/PDE-constrained optimization problems to identify optimal paths, as well as computing Hessian determinants around minimizers. Modern tools, such as automatic differentiation, make these steps tractable.

As an application, I will present results for intermittency in the stochastic 1D viscous Burgers equation, as a canonical toy model of fluid turbulence. Time permitting, I will conclude with an outlook on future improvements based on ideas from the functional renormalization group.

Based on joint work with Tobias Grafke, Rainer Grauer, Georg Stadler, Shanyin Tong, in Stat. Comput. 33(6), 137 (2023), arXiv:2502.20114, and arXiv:2512.03841.

Generalised graphical representations

Seminar "Interacting Random Systems" at WIAS Berlin, Room 405-406, 11:30 AM

Abstract: Numerous dynamic spatial models feature interactions between particles that are based on waiting times; the most prominent examples being first passage percolation, where a fluid spreads from a source, and the contact process, where an infection spreads on a graph with the possibility of recovery. However, waiting times are unnatural for certain models. One can argue that people have a daily routine and meet each other on a relatively fixed schedule, rather than a random waiting time after having met someone else. With the help of graphical representations, we develop a framework that moves away from waiting times towards so called contact times. As an application, we present results on a first contact percolation model, where people follow a random daily meeting routine, as well as contact processes with periodic recoveries.

Quasilinear parabolic equations in the Bessel scale

Langenbach-Seminar at WIAS, Erhard-Schmidt lecture room, 2:15 PM

Abstract: In this talk, I will discuss and analyze scalar quasilinear parabolic equations posed on nonsmooth domains with generally irregular input data. The lat ter refers to merely measurable and bounded coefficients, mixed boundary conditions, possibly inhomogeneous Robin- or Neumann boundary data, and also to nonlinear forcing functions which may act on the gradient of the solution. The driving force for the analysis will be maximal parabolic regu larity of the differential operators in the scale of (duals of) Bessel potential spaces incorporating mixed boundary conditions, with the scale parameter depending on the smoothness of the solution within the quasilinear operator. These depend on permanence principles for the elliptic counterparts which are leveraged from bilinear interpolation. A careful analysis allows to close the loop for the quasilinear equation and thus obtain local-in-time solutions under minimal assumptions. I further mention some sufficient criteria for solutions to exist globally, and how these can be leveraged in the framework of optimal control problems to achieve a satisfying theory for nonsmooth problems with strong nonlinearities. 

On the global well-posedness of the stochastic Yang-Mills-Higgs equations in two dimensions

Oberseminar Analysis-Probability at MPI, E2 10 (Leon-Lichtenstein), 3:15 PM

Abstract: The Yang-Mills-Higgs (YMH) model plays a central role in several areas of mathematics, including analysis, geometry, and probability theory. In this talk, we implement the stochastic quantization procedure for the YMH model in two dimensions. That is, we construct the two-dimensional YMH measure by proving global well-posedness and uniform-in-time bounds for the two-dimensional stochastic YMH equations. As part of our proof, we first discuss covariant stochastic objects and a covariant para-controlled Ansatz. Then, we discuss a decay mechanism for the stochastic YMH equations near unstable Yang-Mills connections. This is joint work with S. Cao, M. Hairer, and W. Zhao.

Oriented matroids of planar point configurations

Algebraic and Combinatorial Perspectives in the Mathematical Sciences via Zoom, 3:00 PM

Abstract: Oriented matroids are purely combinatorial objects that were devised, independently, by Bland to study the simplex algorithm, by Folkman and Lawrence to study face lattices of polytopes, and by Las Vergnas to study questions in graphs and combinatorics. In this talk, I will discuss the oriented matroids that arise from finite, planar point configurations. We will consider them from the point of views of computational complexity and discrete geometry, with a particular emphasis on their symmetries and typical properties.