We identify the Baker-Campbell-Hausdorff recursion driven by a weight $lambda=1$ Rota--Baxter operator with the Magnus expansion relative to the post-Lie structure naturally associated to the corresponding Rota--Baxter algebra.
We consider the group (G,*) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where * denotes the convolution operation. We introduce a larger group (Gtilde,*) of unitized functions from the same incidence alge
bra, which satisfy a weaker condition of being semi-multiplicative. The natural action of Gtilde 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 Gtilde 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 G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that Gtilde can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of G in Gtilde turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.
The amalgamated $T$-transform of a non-commutative distribution was introduced by K.~Dykema. It provides a fundamental tool for computing distributions of random variables in Voiculescus free probability theory. The $T$-transform factorizes in a rath
er non-trivial way over a product of free random variables. In this article, we present a simple graphical proof of this property, followed by a more conceptual one, using the abstract setting of an operad with multiplication.
Given an additive network of input-output systems where each node of the network is modeled by a locally convergent Chen-Fliess series, two basic properties of the network are established. First, it is shown that every input-output map between a give
n pair of nodes has a locally convergent Chen-Fliess series representation. Second, sufficient conditions are given under which the input-output map between a pair of nodes has a well defined relative degree as defined by its generating series. This analysis leads to the conclusion that this relative degree property is generic in a certain sense.
We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kiraly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the
tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.
Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of
(real-valued) time series that are not easily available using existing signature-type objects.
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.
Hairers regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation proced
ures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimers formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs as well as the proposal for resonance based numerical schemes for certain partial differential equations in numerical analysis introduced in a recent work by the first author.
This chapter is divided into two parts. The first is largely expository and builds on Karandikars axiomatisation of It{^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{^o
} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{^o} stochastic exponential of matrix-valued semimartingales.
Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the
Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of Grassmann algebra. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the Grassmann algebra over the base field. We also consider the infinitesimal analog of the Boolean Bercovici--Pata map.
