CSH Workshop: “Stochastic dynamics for complex systems”

Jun 01, 2022—Jun 03, 2022

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This workshop is organized by Christian Kühn (CSH External Faculty & TU Munich) together with Maximilian Engel (FU Berlin) and Alexandra Neamtu (University of Konstanz).

The workshop takes place at the Complexity Science Hub Vienna.

 

In many complex systems modeling at a full-scale detail is impossible. Either, not all relevant processes are known precisely, or even if they are, there are uncertainties in parameters and external influences. Furthermore, microscopic models often become far too complicated for analytical or even numerical treatment. In all these cases, stochastic methods have proven to be highly beneficial to understand complex dynamical systems. Although stochastic approaches have been extremely successful, e.g., in the context of stochastic differential equations, data analysis, information processing, or dynamical systems, the methodological approaches are often surprisingly disconnected at first sight.

 

The main goal of this workshop is to foster the interplay between different approaches and to explore new unifying directions and methods to tackle challenges in complex systems via stochastic dynamics. Is there a unification of stochastic methods in dynamics?

 

 

Please find the workshop agenda here.

 

Speakers

Stefanie Sonner: Random attractors for stochastic parabolic evolution equations via pathwise mild solutions

We study the longtime behavior of stochastic parabolic partial differential equations with additive noise. The differential operators in the equation depend on time and the underlying probability space. We prove the existence of global and exponential random attractors and derive estimates on their fractal dimension. To apply the framework of random dynamical systems we use the concept of pathwise mild solutions which yields pathwise representation formulas for solutions. Our approach to prove the existence of random attractors is different from the classical one where stochastic evolution equations are transformed into partial differential equations with random coefficients via the stationary Ornstein-Uhlenbeck process. This is joint work with C. Kuehn and A. Neamtu.

Jan Korbel: Thermodynamics of structure-forming systems

Structure-forming systems are ubiquitous in nature, ranging from atoms building molecules to self-assembly of colloidal amphibolic particles. The understanding of the underlying thermodynamics of such systems remains an important problem. Here, we derive the entropy for structure-forming systems that differs from Boltzmann-Gibbs entropy by a term that explicitly captures clustered states. For large systems and low concentrations, the approach is equivalent to the grand-canonical ensemble; for small systems, we find significant deviations. We derive the detailed fluctuation theorem and Crooks’ work fluctuation theorem for structure-forming systems. The connection to the theory of particle self-assembly is discussed.
We apply the results to several systems. We present the phase diagram for patchy particles described by the Kern-Frenkel potential. We show that the Curie-Weiss model with molecule structures exhibits a first-order phase transition. Finally, we show how the self-assembly of spin glass can be used for the prediction of group-size distribution in a society with homophilic interactions.

Péter Koltai: Collective variables in complex systems: from molecular dynamics to agent-based models and fluid dynamics

The identification of persistent forecastable structures in complicated or high-dimensional dynamics is vital for a robust prediction (or manipulation) of such systems in a potentially sparse-data setting. Such structures can be intimately related to so-called collective variables known for instance from statistical physics. We have recently developed a first data-driven technique to find provably good collective variables in molecular systems. Here we will show that these generalize to other applications as well, such as fluid dynamics and social dynamics.

Tommaso Rosati: Spectral gap for projective processes of hyperviscous SDPEs

We consider a linear hyperviscous SPDE with multiplicative noise as a toy model for an SPDE that does not satisfy order preservation and study the finiteness and uniqueness of its Lyapunov exponent. Under mild conditions on the noise, we can construct a Lyapunov functional for the associated projective process, based on the analysis of energy level sets. Under stronger conditions we obtain ergodicity and a spectral gap for the projective process (and hence uniqueness and Furstenberg-Khasminskii formulas for the Lyapunov exponent). This extends previous results, which in the infinite dimensional setting were essentially restricted to order preserving cases. Joint work with Martin Hairer.

Sebastian Riedel: Random dynamical systems on fields of Banach spaces

We present a framework that allows studying evolution equations on time-evolving Banach spaces with random dynamical systems. Our prime example is a stochastic delay differential equation for which the evolving Banach spaces are the spaces of controlled paths, known from rough paths theory. We formulate a multiplicative ergodic theorem and use it to prove the existence of local invariant manifolds. Eventually, we discuss further applications of our theory. Joint work with Mazyar Ghani Varzaneh (Hagen).

Alex Blumenthal: Lyapunov exponents on scales of Banach spaces

Lyapunov exponents describe the exponential rate at which trajectories of a dynamical system diverge, and have resulted in a successful abstract theory for describing the asymptotic regimes of chaotic dynamics in finite dimensions. However, many problems of interest are posed instead on infinite dimensional spaces, e.g. evolution equations, for which a variety of nonequivalent norms are often available. This leaves open the possibility that Lyapunov exponents depend on the norm. In this talk I will discuss these issues and circumstances when Lyapunov exponents are independent of the norm. Our results apply to a variety of fluid mechanics models, including incompressible passive scalar advection and linearization along trajectories of the Navier Stokes equations with either noisy or time periodic forcing.
Joint work with Sam Punshon-Smith.

Iulian Cimpean: On the path-continuity of Markov processes

Suppose that we are given a general second order integro-differential operator defined merely on a class of test functions, which corresponds to a cadlag Markov process, e.g. through the martingale problem. The aim is to present a general result which claims that if the class of test functions is rich enough (yet not necessarily a core), and if G is an open domain on which the generator has the local property expressed in a suitable way, then the Markov process has continuous paths when it passes through G. In fact, because the class of test functions is not necessarily a core, the aforementioned result holds for any Markov extension of the operator. The approach is potential theoretic and covers (possibly time-dependent) operators defined on domains in Hilbert spaces or on spaces of measures. This is a joint work with L. Beznea and M. Roeckner.

Samuel Punshon-Smith: Using regularity to estimate Lyapunov exponents

I will discuss a new method for estimating Lyapunov exponents for hypoelliptic diffusions from below using local regularity of stationary measures on the projective bundle. For damped driven truncations of nonlinear conservative SPDE in fluctuation dissipation scaling, I will outline a general strategy for proving positivity of the top Lyapunov exponent that can be applied to Galerkin truncations of the stochastic Navier Stokes equations on T^2. I will also discuss ongoing work extending this method to the Grassmannian bundle to give information on the sum of the first k Lyapunov exponents, giving a possible strategy for estimating growth of attractor dimension from below.

Florian Huber: Existence of global solutions to stochastic cross-diffusion systems

Reaction diffusion phenomena arise in many applications in physics, chemistry and biology, describing the evolution of densities or the concentrations in multicomponent systems. It often happens that the concentration of one component in a system influences the diffusion of the others. This phenomenon is known as cross-diffusion. Such cross-diffusion equations can be generally regarded as strongly coupled degenerate (stochastic) parabolic equations. Their degeneracy excludes standard methods used for for evolution equations to obtain global-in-time existence and physically relevant features of solutions. We propose a solution theory based on the entropy structure of such systems. A transformation and regularization related to this entropy allow us to obtain features like positivity and boundedness of global-in-time solutions. This talk is based on joint work with Marcel Braukhoff and Ansgar Jüngel.

Chengcheng Ling: RANDOM DYNAMIC SYSTEMS GENERATED BY THE SOLUTIONS
TO SINGULAR STOCHASTIC DIFFERENTIAL EQUATIONS

We first provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system. Then we show that this result can be applied to several classes of stochastic differential equations such as
equations with singular drift. For these examples it was previously unknown whether they generate a (local) random dynamical system or not.
After getting such random dynamical system, by chaining techniques we further show that it has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant β outside a large ball, where β depends on the singular part of the drift. This talk is based on the joint works with M. Scheutzow (TU Berlin) and I. Vorkastner (TU Berlin).

Bernat Corominas-Murtra: Randomness, geometry and stemness

What defines the number and dynamics of the stem cells that generate and renew biological tissues?
Several molecular markers have been described to predict stem cell potential with great success, e.g., in tissues like the blood, where a clear hierarchy of functional cell types can be identified. In other tissues, neutral competitive dynamics has been reported to remove a large fraction of biochemically identifiable stem cells, thereby creating a severe mismatch between cell-specific biochemical identity and the actual role of them as functional stem cells within the collective. In the case of the intestinal epithelium, this mismatch goes even further: In spite the number of biochemically identifiable (LGR5+) stem cells is the same in the crypts of both large and small intestines, lineage tracing techniques show that the number of them that actually play the role of stem cell displays remarkable differences. In that context, a natural question arises: How to determine the functional number and location of stem cells? To solve this problem, I will revise a recent approach that mathematically describes “stemness” as an emergent property arising only from the stochastic dynamics of cell movement and geometrical considerations of the underlying organ. Within this framework, one can accurately predict the robust emergence of a region made of functional stem cells and the actual number of them, as well as the lineage-survival probabilities, among other observables. The presented approach does not neglect the key role of biomarkers: Instead, it points towards the existence of complementary regulatory mechanisms --based on collective phenomena, stochastic dynamics and organ geometry--, playing an active role in determining the emergence and location of different cell functionalities.

Nikolas Nüsken: The Stein Geometry in Machine Learning: Gradient Flows, Optimal Transport and Large Deviations

Sampling or approximating high-dimensional probability distributions is a key challenge in computational statistics and machine learning. This talk will present connections to gradient flow PDEs, optimal transport and interacting particle systems, focusing on the recently introduced Stein variational gradient descent methodology and some variations. The construction induces a novel geometrical structure on the set of probability distributions related to a positive definite kernel function. We discuss the corresponding geodesic equations, infinitesimal optimal transport maps, as well as large deviation functionals. This is joint work with A. Duncan (Imperial College London), L. Szpruch (University of Edinburgh) and M. Renger (Weierstrass Institute Berlin).

Benjamin Fehrman: Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift

In this talk, we will introduce a continuum model that replicates the far-from-equilibrium behavior of certain interacting particle systems. We will show that, along appropriate scaling limits, the solutions to certain stochastic PDEs with conservative noise correctly describe the particle process in terms of a law of large numbers, central limit theorem, and large deviations principle---three concepts that we will first introduce in the simpler context of a random walk, which is a random process generated by flipping a coin. The associated SPDE is a generalized version of the Dean--Kawasaki equation. We will explain its derivation and well-posedness, and show that the solutions correctly capture the random fluctuations in the particle system through the detailed analysis of the associated skeleton equation. This analysis provides the second primary tool of this work, and allows to make rigorous the formal connection above through applications to the identification of l.s.c. envelopes of restricted rate functions, to zero noise large deviations for conservative (singular) SPDE, and to the Gamma-convergence of rate functions.

Benjamin Gess: Lyapunov exponents and synchronization by noise for systems of SPDEs

The stochastic quantisation of Euclidean quantum field theory in (Parisi, Wu, 1981) was motivated in parts by Markov Chain Monte Carlo sampling methods, used to generate samples from the $Phi^4$-measure. Extrapolation methods lead to the analysis of coupled diffusions, and thereby to the analysis of the long-time behavior of the two-point motion. The variance of the resulting sampling method can be controlled by proving synchronization/stabilization by noise. Since the Higgs standard model of quantum field theory is vector-valued, this poses the problem of synchronization by noise for systems of SPDEs. In this talk we present an approach to quantitative estimates for the corresponding Lyapunov exponents.

Details

Start
Jun 01, 2022
End
Jun 03, 2022

Venue

Complexity Science Hub Vienna
Josefstaedter Straße 39
Vienna,1080Austria+ Google Map