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The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of $B$-series. For this purpose, we present a so-called arborification of the Hoffman--Ihara theory of quasi-shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi-shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.

Given a single-input, single-output (SISO) system with a Chen-Fliess series representation whose generating series has a well defined relative degree, it is shown that there is a notion of universal zero dynamics that describes a set of dynamics evolving on a locally convex (infinite dimensional) Lie group so as to render the system's output exactly zero. Minimum phase in this setting is defined in terms of the boundedness of the applied input which zeros the output. As an application, it is shown that one can design a zero dynamics attack on cyber-infrastructure using only an estimate of the plant's generating series. That is, detailed knowledge of the plant's internal dynamics is not needed.

We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where ``$*$'' denotes the convolution operation. We introduce a larger group $(\widetilde{\mathcal{G}},*)$ of unitized functions from the same incidence algebra, which satisfy a weaker condition of being ``semi-multiplicative''. The natural action of $\widetilde{\mathcal{G}}$ on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of $\widetilde{\mathcal{G}}$ in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of $t$-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko and Wysoczanski. It is known that the group $\mathcal{G}$ can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that $\widetilde{\mathcal{G}}$ can also be identified as group of characters of a Hopf algebra $\mathcal{T}$, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of $\mathcal{G}$ into $\widetilde{\mathcal{G}}$ turns out to be the dual of a natural bialgebra homomorphism from $\mathcal{T}$ onto Sym., Comment: Version 2: Small corrections and some added material in Sections 11 and 12

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of M\"obius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions., Comment: 41 pages. v4: Fixed a small mistake on p.23; it does not affect the rest of the paper. Final version, to appear in Adv. Math

Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.

Calculating multiple zeta values at arguments of any sign in a way that is compatible with both the quasi-shuffle product as well as meromorphic continuation, is commonly referred to as the renormalisation problem for multiple zeta values. We consider the set of all solutions to this problem and provide a framework for comparing its elements in terms of a free and transitive action of a particular subgroup of the group of characters of the quasi-shuffle Hopf algebra. In particular, this provides a transparent way of relating different solutions at non-positive values, which answers an open question in the recent literature., For abbreviated version of the manuscript see arXiv:1510.09159 [math.NT]

In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the properties of group like-elements belonging to (suitable completions of) those Hopf algebras. Of particular interest is the case of post-Lie algebras defined in terms of solutions of modified classical Yang–Baxter equations. In this setting we will study factorization properties of the aforementioned group-like elements.

Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $\epsilon$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $\epsilon$-cumulants are introduced and shown to characterize the notion of $\epsilon$-independence.

In this work we extend the recently introduced group-theoretical approach to moment-cumulant relations in non-commutative probability theory to the notion of conditionally free cumulants. This approach is based on a particular combinatorial Hopf algebra which may be characterised as a non-cocommutative generalisation of the classical unshuffle Hopf algebra. Central to our work is the resulting non-commutative shuffle algebra structure on the graded dual. It implies an extension of the classical relation between the group of Hopf algebra characters and its Lie algebra of infinitesimal characters and, among others, the appearance of new forms of adjoint actions of the group on its Lie algebra which happens to play a key role in the new algebraic understanding of conditionally free cumulants., Comment: 24 pages

The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen–Fliess series with a rational generating series. The idea is to realize the corresponding Chen–Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.

Consider a set of single-input, single-output nonlinear systems whose input-output maps are described only in terms of convergent Chen-Fliess series without any assumption that finite dimensional state space models are available. It is shown that any additive or multiplicative interconnection of such systems always has a Chen-Fliess series representation that can be computed explicitly in terms of iterated formal Lie derivatives., Comment: Added and updated references, fixed typos

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q-analogs of multiple zeta values. Regarding the former we identify the derivation relation of order two with a Hoffman-Ohno type relation. Then we describe relations between the Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin q-multiple zeta values., revised version, accepted for publication in Advances in Mathematics

In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given., Comment: revised and updated

The file supplement.pdf contains an overview of the algebraic concepts and additional numerical simulations.

In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, as a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants. We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties., 18 pages, 1 figure