Computation and Combinatorics in Dynamics, Stochastics and Control

Frontmatter

Facilitated Exclusion Process

Abstract

We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant which is exactly solvable through Pfaffian Schur processes. Our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE). We also discuss a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of the first particles when the rate of the first one varies. In terms of half-space last passage percolation, this corresponds to last passage times close to the boundary when the size of the boundary weights is simultaneously scaled close to the critical point.

Jinho Baik, Guillaume Barraquand, Ivan Corwin, Toufic Suidan

Stochastic Functional Differential Equations and Sensitivity to Their Initial Path

Abstract

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of randomisation of the initial condition.

D. R. Baños, G. Di Nunno, H. H. Haferkorn, F. Proske

Grassmannian Flows and Applications to Nonlinear Partial Differential Equations

Abstract

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher–Kolmogorov–Petrovskii–Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.

Margaret Beck, Anastasia Doikou, Simon J. A. Malham, Ioannis Stylianidis

Gog and Magog Triangles

Abstract

We survey the problem of finding an explicit bijection between Gog and Magog triangles, a combinatorial problem which has been open since the 1980s. We give some of the ideas behind a recent approach to this question and also prove some properties of the distribution of inversions and coinversions in Gog triangles.

Philippe Biane

The Clebsch Representation in Optimal Control and Low Rank Integrable Systems

Abstract

Certain kinematic optimal control problems (the Clebsch problems) and their connection to classical integrable systems are considered. In particular, the rigid body problem and its rank 2k counterparts, the geodesic flows on Stiefel manifolds and their connection with the work of Moser, flows on symmetric matrices, and the Toda flows are studied.

Anthony M. Bloch, François Gay-Balmaz, Tudor S. Ratiu

The Geometry of Characters of Hopf Algebras

Abstract

Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coalgebra that is also equipped with an antiautomorphism known as the antipode, satisfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra morphisms from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups.

Geir Bogfjellmo, Alexander Schmeding

Shape Analysis on Homogeneous Spaces: A Generalised SRVT Framework

Abstract

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.

Elena Celledoni, Sølve Eidnes, Alexander Schmeding

Universality in Numerical Computation with Random Data: Case Studies, Analytical Results and Some Speculations

Abstract

We discuss various universality aspects of numerical computations using standard algorithms. These aspects include empirical observations and rigorous results. We also make various speculations about computation in a broader sense.

Percy Deift, Thomas Trogdon

BSDEs with Default Jump

Abstract

We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λ _t). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

Roxana Dumitrescu, Miryana Grigorova, Marie-Claire Quenez, Agnès Sulem

The Faà di Bruno Hopf Algebra for Multivariable Feedback Recursions in the Center Problem for Higher Order Abel Equations

Abstract

Poincaré’s center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a convenient coordinate system. In 1990, Devlin described an algebraic approach for constructing sufficient conditions for a center using a linear recursion for the generating series of the solution to the Abel equation. Subsequent work by the authors linked this recursion to feedback structures in control theory and combinatorial Hopf algebras, but only for the lowest degree case. The present work introduces what turns out to be the nontrivial multivariable generalization of this connection between the center problem, feedback control, and combinatorial Hopf algebras. Once the picture is completed, it is possible to provide generalizations of some known identities involving the Abel generating series. A linear recursion for the antipode of this new Hopf algebra is also developed using coderivations. Finally, the results are used to further explore what is called the composition condition for the center problem.

Kurusch Ebrahimi-Fard, W. Steven Gray

Continuous-Time Autoregressive Moving-Average Processes in Hilbert Space

Abstract

We introduce the class of continuous-time autoregressive moving-average (CARMA) processes in Hilbert spaces. As driving noises of these processes we consider Lévy processes in Hilbert space. We provide the basic definitions, show relevant properties of these processes and establish the equivalents of CARMA processes on the real line. Finally, CARMA processes in Hilbert space are linked to the stochastic wave equation and functional autoregressive processes.

Fred Espen Benth, André Süss

Pre- and Post-Lie Algebras: The Algebro-Geometric View

Abstract

We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras and MKW Hopf algebras are then coordinate rings of the infinite-dimensional affine varieties consisting of series of trees, resp. Lie series of ordered trees. Furthermore we describe the Hopf algebras which are coordinate rings of the automorphism groups of these varieties, which govern the substitution law in pre- and post-Lie algebras.

Gunnar Fløystad, Hans Munthe-Kaas

Extension of the Product of a Post-Lie Algebra and Application to the SISO Feedback Transformation Group

Abstract

We describe both post- and pre-Lie algebra \(\mathfrak {g}_{SISO}\) associated to the affine SISO feedback transformation group. We show that it is a member of a family of post-Lie algebras associated to representations of a particular solvable Lie algebra. We first construct the extension of the magmatic product of a post-Lie algebra to its enveloping algebra, which allows to describe free post-Lie algebras and is widely used to obtain the enveloping of \(\mathfrak {g}_{SISO}\) and its dual.

Loïc Foissy

Infinite Dimensional Rough Dynamics

Abstract

We review recent results about the analysis of controlled or stochastic differential systems via local expansions in the time variable. This point of view has its origin in Lyons’ theory of rough paths and has been vastly generalised in Hairer’s theory of regularity structures. Here our concern is to understand this local expansions when they feature genuinely infinite dimensional objects like distributions in the space variable. Our analysis starts reviewing the simple situation of linear controlled rough equations in finite dimensions, then we introduce unbounded operators in such linear equations by looking at linear rough transport equations. Loss of derivatives in the estimates requires the introduction of new ideas, specific to this infinite dimensional setting. Subsequently we discuss how the analysis can be extended to systems which are not intrinsically rough but for which local expansion allows to highlight other phenomena: in our case, regularisation by noise in linear transport. Finally we comment about other application of these ideas to fully-nonlinear conservations laws and other PDEs.

Massimiliano Gubinelli

Heavy Tailed Random Matrices: How They Differ from the GOE, and Open Problems

Abstract

Since the pioneering works of Wishart and Wigner on random matrices, matrices with independent entries with finite moments have been intensively studied. Not only it was shown that their spectral measure converges to the semi-circle law, but fluctuations both global and local were analyzed in fine details. More recently, the domain of universality of these results was investigated, in particular by Erdos-Yau et al and Tao-Vu et al. This survey article takes the opposite point of view by considering matrices which are not in the domain of universality of Wigner matrices: they have independent entries but with heavy tails. We discuss the properties of these matrices. They are very different from Wigner matrices: the limit law of the spectral measure is not the semi-circle distribution anymore, the global fluctuations are stronger and the local fluctuations may undergo a transition and remain rather mysterious.

Alice Guionnet

An Analyst’s Take on the BPHZ Theorem

Abstract

We provide a self-contained formulation of the BPHZ theorem in the Euclidean context, which yields a systematic procedure to “renormalise” otherwise divergent integrals appearing in generalised convolutions of functions with a singularity of prescribed order at their origin. We hope that the formulation given in this article will appeal to an analytically minded audience and that it will help to clarify to what extent such renormalisations are arbitrary (or not). In particular, we do not assume any background whatsoever in quantum field theory and we stay away from any discussion of the physical context in which such problems typically arise.

Martin Hairer

Parabolic Anderson Model with Rough Dependence in Space

Abstract

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter \(H \in (\frac {1}{4}, \frac {1}{2})\) in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the nth moment of the solution.

Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart, Samy Tindel

Perturbation of Conservation Laws and Averaging on Manifolds

Abstract

We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator \({\mathscr L}_x\) for which we obtain a quantitative locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter x. These results are obtained under the assumption that \({\mathscr L}_x\) satisfies Hörmander’s bracket conditions, or more generally \({\mathscr L}_x\) is a family of Fredholm operators with sub-elliptic estimates. For stochastic systems in which the slow and the fast variable are not separate, conservation laws are essential ingredients for separating the scales in singular perturbation problems we demonstrate this by a number of motivating examples, from mathematical physics and from geometry, where conservation laws taking values in non-linear spaces are used to deduce slow-fast systems of stochastic differential equations.

Xue-Mei Li

Free Probability, Random Matrices, and Representations of Non-commutative Rational Functions

Abstract

A fundamental problem in free probability theory is to understand distributions of “non-commutative functions” in freely independent variables. Due to the asymptotic freeness phenomenon, which occurs for many types of independent random matrices, such distributions can describe the asymptotic eigenvalue distribution of corresponding random matrix models when their dimension tends to infinity. For non-commutative polynomials and rational functions, an algorithmic solution to this problem is presented. It relies on suitable representations for these functions.

Tobias Mai, Roland Speicher

A Review on Comodule-Bialgebras

Abstract

We review some recent applications of the notion of comodule-bialgebra in several domains such as Combinatorics, Analysis and Quantum Field Theory.

Dominique Manchon

Renormalization: A Quasi-shuffle Approach

Abstract

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semigroup (different in nature from the Connes-Marcolli “cosmical Galois group”). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov’s preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process.

Frédéric Menous, Frédéric Patras

Hopf Algebra Techniques to Handle Dynamical Systems and Numerical Integrators

Abstract

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators. Given a specific problem, those techniques construct an abstract, universal version of it which is solved algebraically; then, the results are transferred to the original problem with the help of a suitable morphism. In earlier contributions, the abstract problem is formulated either in the dual of the shuffle Hopf algebra or in the dual of the Connes-Kreimer Hopf algebra. In the present contribution we extend these techniques to more general Hopf algebras, which in some cases lead to more efficient computations.

Ander Murua, Jesús M. Sanz-Serna

Quantitative Limit Theorems for Local Functionals of Arithmetic Random Waves

Abstract

We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz et al. (Ann Inst Fourier (Grenoble) 58(1):299–335, 2008), Krishnapur et al. (Ann Math 177(2):699–737, 2013), and Marinucci et al. (Geom Funct Anal 26(3):926–960, 2016). Our techniques involve Wiener-Itô chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula.

Giovanni Peccati, Maurizia Rossi

Combinatorics on Words and the Theory of Markoff

Abstract

This is a survey on the theory of Markoff, in its two aspects: quadratic forms (the original point of view of Markoff), approximation of reals. A link wih combinatorics on words is shown, through the notion of Christoffel words and special palindromes, called central words. Markoff triples may be characterized, by using some linear representation of the free monoid, restricted to these words, and Fricke relations. A double iterated palindromization allows to construct all Markoff numbers and to reformulate the Markoff numbers injectivity conjecture (Frobenius, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin 26:458–487, 1913).

Christophe Reutenauer

An Algebraic Approach to Integration of Geometric Rough Paths

Abstract

We build a connection between rough path theory and a non-commutative algebra, and interpret the integration of geometric rough paths as an example of a non-abelian Young integration. We identify a class of slowly-varying one-forms, and prove that the class is stable under basic operations.

Danyu Yang