CSH Workshop: “Stochastic dynamics for complex systems”


Sep 30, 2021Oct 01, 2021

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This workshop is organized by Christian Kühn, CSH External Faculty and TU Munich as well as Maximilian Engel, FU Berlin and Alexandra Neamtu, University of Konstanz.

 

The workshop will take place as an online event, a follow-up workshop will be organized in June 2022.

 

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?

 

If you are interested in participating, please sign up with office@csh.ac.at.

 

To see the full agenda, please click here.

Speakers

Sara Merino Aceituno: “Nematic alignment of self-propelled particles in the macroscopic regime”


We consider a model for collective dynamics where particles move at a constant speed and change their direction of motion to align with the direction of motion of their neighbours, up to some noise. In particular, particles may align in the same or opposite orientation; this is called nematic alignment. Starting from this particle model we derive a macroscopic model for the particle density and mean direction of motion which corresponds to a cross-diffusion system. The derivation is carried over by means of a Hilbert expansion. This cross diffusion system poses many new challenging questions.
This is a joint work with Pierre Degond (Imperial College London, UK).

Bernat Corominas Murtra: “Randomness, collective dynamics and the definition of stem cell”


Although several molecular markers have been described to predict stem cell potential, whether there exists a general definition of stemness in terms of biochemical properties of cells remains an open question. Recent results point to the interesting hypothesis that collective cell dynamics itself, far for being a consequence of individual cell properties, could play a key role in the emergence of the stem cell region in an organ. In that frame, stemness would be a context dependent property emerging from the global dynamics of the tissue. In this talk I will show a general framework based on stochastic dynamics of competition for space from which one can predict the robust emergence of a region made of functional stem cells, as well as give simple predictions on lineage-survival probability. The mathematical frame is generic and not linked to a specific organ. The results have been tested against data obtained from intravital live-imaging experiments in mammary gland development, kidney development, and from the self-renewal dynamics of the intestinal crypt. The proposed framework identifies key differences in terms of functionality that are not visible using standard approach based on molecular marker identification, and makes predictions for organs where the approach based on molecular markers does not seem yet possible.

Jan Korbel: “Nonequilibrium thermodynamics of uncertain stochastic processes”


In the real world, one almost never knows the parameters of a thermodynamic process to infinite precision. Rather there is always nonzero uncertainty in the precise values of temperatures / chemical potentials of the reservoirs, in the energy spectrum, in the control protocol, maybe even uncertainty in the number of reservoirs, etc. In general, one would expect that due to such uncertainty,
the experimentally observed versions of the laws of stochastic thermodynamics (the second law, the fluctuation theorems, the thermodynamic uncertainty relations, etc.,) will all differ from their textbook versions in which all thermodynamic parameters have zero uncertainty. Here we investigate how such uncertainty affects the experimentally observed laws of stochastic thermodynamics.

Benjamin Gess: “Synchronization for the stochastic quantisation equation”


The stochastic quantisation equation (also known as the stochastic Allen-Cahn equation) in dimensions two and three is a singular SPDE and renormalization is required to give it meaning. Its deterministic analogon is known to have finitely many unstable directions. In this talk I will discuss how the presence of noise implies uniform synchronisation/stabilization, that is, any two trajectories approach each other with speed of convergence uniform in the initial condition. More precisely, we will show how a combination of “coming down from infinity” estimates and order-preservation can be used to obtain uniform synchronisation with rates. This will be a special case of a more general framework which implies quantified synchronisation by noise for white noise stochastic semi-flows taking values in Hölder spaces of negative exponent.

Alex Blumenthal: “Quenched decay of correlations for Lagrangian flow of the stochastic Navier-Stokes equations”


Lagrangian motion refers to the trajectory traced out by a passive tracer (e.g., a molecule of solute) in an incompressible, time-varying fluid. When the fluid itself is subjected to driving / stirring, one often expects Lagrangian motion to have chaotic properties, e.g., sensitivity w.r.t. initial conditions and exponential decay of correlations. However, verifying such chaotic properties is a notoriously challenging problem, even for toy models of Lagrangian flow such as the Chirikov standard map. Remarkably, however, verifying these chaotic properties for stochastically driven systems is far more tractable. In a series of joint works with Sam Punshon-Smith and Jacob Bedrossian, we confirm these properties (and more) for the Lagrangian flow corresponding to a fluid evolving according to the 2d Navier-Stokes equations on the periodic box subjected to white-in-time, spatially Sobolev forcing and other fluids models.

Tommaso Rosati: “Positive Lyapunov exponents in a slow random environment”


We will study the positivity of the Lyapunov exponent of an Anderson model with “slowly varying” noise. The proofs build on a perturbation of Furstenberg–Khasminskii type formulas, on tools from singular stochastic PDEs and on a contraction property for positive operators on a projective space. We will also discuss some extensions of these results in a joint work in progress.

Sam Punshon-Smith: “A regularity approach to lower bounds on Lyapunov exponents for SDEs: with applications to Galerkin-Navier-Stokes”


In this talk, I will discuss a new technique, developed jointly with Alex Blumenthal and Jacob Bedrossian, for establishing quantitative lower bounds on the top Lyapunov exponent for stochastic differential equations via the regularity of the stationary measure associated to the process lifted to the sphere bundle. We apply this method to a class of ``Euler-like`` models that share structural similarities with the stochastic Navier Stokes equations and show that, under a general Lie bracket spanning condition on the projective dynamics, the top Lyapunov exponent is positive as long as the dissipation is taken small enough. This spanning condition has recently been verified in a joint work with Jacob Bedrossian for arbitrarily large Galerkin truncations of the 2d Navier-Stokes equations on the Torus of arbitrary aspect ratio via a computer assisted proof using methods from computational algebraic geometry, thereby rigorously proving a strong form of chaos for this model.

Nikolas Nüsken: “Estimating hidden parameters in stochastic multiscale systems using McKean-Vlasov dynamics and rough paths”


Motivated by the challenge of incorporating data into misspecified and multiscale dynamical mod-els, we study a McKean-Vlasov equation that contains the data stream as a common driving rough path. This setting allows us to prove well-posedness as well as continuity with respect to the driver in an appropriate rough-path topology. The latter property is key in our subsequent development of a robust data assimilation methodology: We establish propagation of chaos for the associated interacting particle system, which in turn is suggestive of a numerical scheme that can be viewed as an extension of the ensemble Kalman filter to a rough-path framework.
Finally, we discuss a data-driven method based on subsampling to construct suitable rough path lifts and demonstrate the robustness of our scheme in a number of numerical experiments related to parameter estimation problems in multiscale contexts.

Iulian Cimpean: “Ergodicity of Markov semigroups and applications to singular SDEs on Hilbert spaces”


The common principle of proving the existence of an invariant distribution for a given Markov semigroup is to show that the semigroup exhibits some regularity and some tightness properties. One aim of this talk is to explain how the problem of existence of an invariant distribution can be tackled from a purely measure theoretic perspective. More precisely, we shall characterize those finite measures $m$ for which there exists a density $ ho$ such that $ ho cdot m$ is an invariant distribution for the given semigroup. If time allows, we shall use the above mentioned characterization to prove ergodicity of a class of singular SDEs on Hilbert spaces.

Denis Villemonais: “Almost sure convergence of Measure Valued Polya Processes”


After introducing the concept of Measure Valued Polya Processes (MVPP), I will present a criterion obtained in collaboration with Cécile Mailler, which ensures the almost-dure convergence of an MVPP. An application to the convergence of reinforced processes will be used to illustrate the convergence result when the MVPP is unbalanced and with random kernels. Another application to the convergence of a reinforced diffusion process, developed in collaboration with Michel Benaim and Nicolas Champagnat, will be presented.

Mazyar Varzaneh: “Oseledets splitting on fields of Banach spaces”


The Multiplicative Ergodic Theorem (MET) is a powerful tool with various applications in different fields of mathematics, including analysis, probability theory, and geometry, and a cornerstone in smooth ergodic theory. Oseledets first proved it for matrix cocycles; since then, the theorem attracted many researchers to provide new proofs and formulations with increasing generality.
In this talk, motivated by our models in stochastic delay equations and stochastic partial differential equations (SPDE), we will present a version of MET for stationary compositions on a (possibly random) field of (potentially distinct) Banach spaces, depending on the random sample.
(Joint work with Sebastian Riedel)

Luigi Amedeo Bianchi: “Impact of white noise on a bifurcation in Swift–Hohenberg equation - Supercritical case”


We consider the impact of additive Gaussian white noise on a supercritical pitchfork bifurcation in an unbounded domain. As an example we focus on the stochastic Swift–Hohenberg equation with polynomial nonlinearity. Here we identify the order where small noise first impacts the bifurcation.
For this we will study the reduction of the essential dynamics close to the bifurcation via amplitude or modulation equations. Surprisingly, and in contrast to the strong nonlinear interaction of finitely many Fourier modes, in all our results the additive noise does not add any additional terms to the modulation equation, its nonlinear interaction always disappears via averaging effects and it just shows up as an additive forcing in the amplitude equation.
(Joint work with D.Blömker)

Jacob Bedrossian: “Qualitative and quantitative properties of stationary measures for stochastic differential equations arising from fluid models”


Understanding the stationary statistics of fluid models subjected to random forcing in singular limits is a long-standing open problem in mathematics and physics. However, even finite dimensional related problems, such as those formulated on Galerkin truncations of the incompressible Navier-Stokes equations (GNS), remain unanswered. In this talk I will discuss two works which are part of a larger program. The first work I will discuss obtains uniform estimates on stationary measures and sharp, quantitative estimates on the spectral gap of the Markov semigroup in the fluctuation-dissipation limit of GNS and similar models. The second work (still in progress) regards obtaining examples in which one can prove the existence of a stationary measure for models such that the damping has a nullspace (i.e. it does not act on every degree of freedom). Both are joint works with Kyle Liss.

Details

Start
Sep 30, 2021
End
Oct 01, 2021

Organizer

Complexity Science Hub Vienna
Phone
+43 1 59991 600
Email
office@csh.ac.at
Website
www.csh.ac.at