Contributed talks

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1. Zachary P. Adams (Max Planck Institute for Mathematics in the Sciences)
   Title: Asymptotic frequencies of stochastic oscillators
   Abstract: We study stochastic perturbations of ODE with stable limit cycles -- referred to as stochastic oscillators -- and investigate the response of the asymptotic (in time) frequency of oscillations to changing noise amplitude.
   Unlike previous studies, we do not restrict our attention to the small noise limit, and account for the fact that large deviation events may push the system out of its oscillatory regime by using the theory of quasi-ergodic measures.
   Our discussion recovers and improves upon previous results on stochastic oscillators.  In particular, existing results imply that the asymptotic frequency of a stochastic oscillator depends quadratically on the noise amplitude.
   We describe scenarios where this prediction holds, though we also show that it is not true in general -- even for small noise.  (Manuscript at https://arxiv.org/abs/2108.03728)
   Slide of Presentation: download
2. Aishah Albarakati (Clarkson University)
   Title: Projected data assimilation using dynamic mode decomposition
   Abstract: Data Assimilation (DA) is a technique that has been used to investigate the atmosphere and ocean phenomena. DA combines observations with model output, including their uncertainty, to produce an estimate of the state of a physical system. Some challenges in data assimilation include dealing with nonlinearity, non-Gaussian error behavior, and high dimensionality of the physical system. To overcome these obstacles, we develop a projected Optimal Proposal Particle Filter (PROJ-OP-PF) based on reduced-order physical and data models. Dynamic Mode Decomposition (DMD) is a recent order reduction technique that extracts the relevant information and captures the coherent structure from the snapshot dynamic. DMD is employed to derive both reduced-order models. Projected DA and DMD techniques can be applied to a variety of physical models from discretized PDEs to medium scale ocean models. We test the efficacy of these techniques on the Lorenz'96 model (L96) and Shallow Water Equations (SWE) which are high dimensional nonlinear systems. Links to relevant manuscript: https://arxiv.org/abs/2101.09252
   https://www.sciencedirect.com/science/article/abs/pii/S0898122121002121
   Slide of Presentation: download
3. Matheus Manzatto de Castro (Imperial College London)
   Title: Existence and Uniqueness of Quasi-stationary and Quasi-ergodic Measures for Absorbed Markov Processes.
   Abstract: We motivate and establish the existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed-time Markov processes under mild conditions on evolution
   Slide of Presentation: download
4. Andrea Clini (University of Oxford)
   Title: Nonlinear diffusion equations with nonlinear, conservative noise
   Abstract: We motivate and establish the pathwise well-posedness of stochastic porous media and fast diffusion equations with nonlinear conservative noise. As a consequence, the generation of a random dynamical system is obtained.
   The results are based on recasting the equation in its kinetic form, a weak formulation of the PDE that allows the noise to be handled in a linear fashion, and on rough path theory. This gives rise to the central notion of 'pathwise kinetic solution'.
   This type of stochastic equations arises, for example, as a continuum limit of mean field stochastic differential equations, as an approximative model for the fluctuating hydrodynamics of the zero-range particle process about its hydrodynamic limit, and as an approximation to the Dean-Kawasaki equation arising in fluctuating fluid dynamics.
   Slide of Presentation: download
5. Andrea Clini (University of Oxford)
   Title: Mean-field like neural models with reflecting boundary conditions
   Abstract: Even in the absence of external sensory cues, foraging rodents maintain an estimate of their position, allowing them to return home in roughly straight lines. This computation is known as dead reckoning or path integration. Recently, a specific region of the neural cortex has been identified as the location in the rat's brain where this computation is performed, and specific mean-field type neural models have been proposed to mimic the activity of the relevant neurons in the brain.
   On the side of the mathematics, these models consist of systems of SDEs describing the activity level of MN neurons stacked along N columns with M neurons each. To prevent the noise from driving the activity level of some neurons to be negative, which is clearly not desirable from the point of view of the modelling, reflecting boundary conditions are added at the SDE level. When investigating the limiting behavior, these boundary conditions persist in the associated McKean-Vlasov equation and in turn translate into no-flux boundary conditions for the corresponding Fokker-Planck PDE. The combination of the spatial interaction and the interaction along columns further complicates the picture, reducing the usual properties of mutual independence of the limiting particles.
   We discuss and answer classical questions in the mean-field theory setting: well-posedness of the relevant systems and equations, limiting behavior, sharp estimates for the rate of convergence of empirical measures.
   Slide of Presentation: download
6. Viet Hung Pham (Institute of Mathematics, VAST)
   Title: Persistence probability of random polynomials
   Abstract: The persistence probability of a stochastic process X is defined as the probability that the process X remains positive for a long interval. We will give a brief introduction on the study on the persistence probability of celebrated random algebraic polynomials: Kac, elliptic, Weyl, Bernstein (evolutionary game theory). We recall the seminal result on Kac model by Dembo et al, predictions by Scher and Majumdar, and a powerful method given by Dembo and Mukherjee. Our main result is providing the logarithmic scale behavior of persistence probability of Weyl and Bernstein polynomials. In joint work with Van-Hao Can and Manh-Hong Duong:
   https://www.cambridge.org/core/journals/journal-of-applied-probability/article/abs/persistence-probability-of-a-random-polynomial-arising-from-evolutionary-game-theory/83E0D1B3EC7EADD56B4933AF7CF7FE26
   https://link.springer.com/article/10.1007/s10955-019-02298-0.
   Slide of Presentation: download
7. Thanh Hong Phan (Thang Long University and Institute of Mathematics, VAST)
   Title: Lyapunov spectrum of non-autonomous linear SDEs driven by fractional Brownian motions
   Abstract: We show that a linear SDE driven by a fBm generates a stochastic two-parameter flow which satisfies the integrability condition, thus the notions of Lyapunov spectrum is well-defined. The spectrum can be computed using the discretized flow and is nonrandom for triangular systems which are regular in the sense of Lyapunov.
   Finally, we prove a Millionshchikov theorem stating that almost all, in the sense of an invariant measure, systems are Lyapunov regular.
   This is a joint work with N.D.Cong and L.H.Duc.
   Slide of Presentation: download
8. Paul Platzer (Ifremer - LOPS)
   Title: Finding analogues of dynamical systems
   Abstract: Analogues (i.e. nearest neighbours) have been used in several climatic and atmospheric applications including dimensionality estimation, downscaling, interpolation and forecasting, sometimes combined with data assimilation. The issue of “how longs must we wait to find a good analogue?” is fundamental and has been tackled since the 1990’s. Here I will present some recently published work on this topic (see published version at
   https://doi.org/10.1175/JAS-D-20-0382.1 and available preprint at https://arxiv.org/abs/2101.10640).
   Slide of Presentation: download
9. Van Quyet Nguyen (Institute of Mathematics, VAST)
   Title: Partial universality of the super concentration in the Sherrington-Kirkpatrick’s spin glass model
   Abstract: Consider the Sherrington-Kirkpatrick's spin glass model on complete graphs with general environments. In this talk, we will present a partial universality of the super concentration phenomenon.  Precisely, we will show that the variance of the free energy grows sublinearly in the size of its expectation when: (i)  the disordered random variable, say y, has the first four moments matching to those of the standard normal distribution; or (ii) y is a smooth Gaussian functional having the symmetric law. Additionally, we also study the universality of first and second moments of the free energy of S-K models on general graphs. This is joint work with V. H. Can and H. S. Vu.
   Slide of Presentation: download
10. Julian Sieber (Imperial College London)
    Title: Geometric ergodicity and averaging of fractional SDEs
    Abstract: Consider the SDE dX_t=b(X_t)dt+dW_t driven by a standard Wiener process W. It is very well known that, if b is contractive outside of a compact set, this equation has a unique invariant measure and the law converges exponentially fast in both Wasserstein and total variation distances. In this talk I will present an analogous result for W replaced by a fractional Brownian motion. This improves  subgeometric rates obtained obtained in previous works. As an application of the result, I will present an averaging principle for slow-fast systems with fractional noise both in the system and the environment.
    Slide of Presentation: download
11. Qiwen Sun (Nagoya University)
    Title: Controllability of Extreme Events with the Lorenz-96 Model
    Abstract: The successful development of numerical weather prediction (NWP) helps better preparedness for extreme weather events. Weather modifications have also been explored, for example, when enhancing rainfalls by cloud seeding [1]. However, it is generally believed that the tremendous energy involved in extreme events prevents any attempt of human interventions to avoid or to control their occurrences. In this study, we investigate the controllability of chaotic dynamical systems by using small perturbations to generate powerful effects and prevent extreme events. The high sensitivity to initial conditions would ultimately lead to modifications of extreme weather events with infinitesimal perturbations. We also study the efficiency of the control as a function of: the amplitude of the perturbation signal, the forecast length, the localization of the perturbation signal, and the total energy. It is expected that this control method can be applied to more complicated weather systems and to other chaotic dynamical systems not limited to NWP.
    References
    [1.] Flossmann, A. I., et al., 2019: Bull. Amer. Meteorol. Soc., 100, 1465-1480.
    Slide of Presentation: download
12. Jonathan Tam (University of Oxford)
    Title: Controlled Markov chains with observation cost
    Abstract:We present a framework for a controlled Markov chain problem with observation costs. Realisations of the chain are only given at chosen observation times. We show through dynamic programming that the value function satisfies a system of quasi-variational inequalities (QVIs). We provide analysis on this class of QVIs by proving a comparison principle and constructing solutions via approximation with penalty methods. A Bayesian parametric extension to the problem is also considered. Finally, we demonstrate the numerical performances of the approximation schemes on a range of applications.
    Slide of Presentation: download
13. Jin Yan (Queen Mary University of London)
    Title: Transition to anomalous dynamics in random systems
    Abstract: In this talk we consider a random dynamical system that samples between a contracting and a chaotic map with a certain probability p in time. We first study analytically its invariant density and Lyapunov exponents. A time-discrete Langevin equation is then generated by sums of iterates of the random system. We investigate numerically two-point correlation functions, with emphasis on the transition between an exponential decay (at $p = 1$) and a power-law decay (when p approaches 1/2).  This is a joint work with R. Klages, Y. Sato, S. Ruffo, M. Majumdar and C. Beck.
    Slide of Presentation: download