11footnotetext:  Université de Genève, Section de mathématiques, Genève,
Switzerland<EMAIL_ADDRESS> Univ Rennes, INRIA
(Research team MINGuS), IRMAR (CNRS UMR 6625) and ENS Rennes, France.
<EMAIL_ADDRESS>
#  Hopf algebra structures for the backward error analysis of ergodic
stochastic differential equations

Eugen Bronasco1 and Adrien Laurent2

###### Abstract

While backward error analysis does not generalise straightforwardly to the
strong and weak approximation of stochastic differential equations, it extends
for the sampling of ergodic dynamics. The calculation of the modified equation
relies on tedious calculations and there is no expression of the modified
vector field, in opposition to the deterministic setting. We uncover in this
paper the Hopf algebra structures associated to the laws of composition and
substitution of exotic aromatic S-series, relying on the new idea of clumping.
We use these algebraic structures to provide the algebraic foundations of
stochastic numerical analysis with S-series, as well as an explicit expression
of the modified vector field as an exotic aromatic B-series.

Keywords:  geometric numerical integration, exotic aromatic series, Butcher
series, clumped forests, Hopf algebra, stochastic differential equations,
backward error analysis, invariant measure, ergodicity.

AMS subject classification (2020):  Primary: 16T05, 41A58, 60H35. Secondary:
37M25, 65L06, 70H45.

##  1 Introduction

Consider overdamped Langevin dynamics on the $d$-dimensional torus
$\mathbb{T}^{d}$ of the following form,

$dX(t)=f(X(t))dt+dW(t),$ (1)

where $f=-\nabla V$ is a gradient vector field deriving from a
$\mathcal{C}^{\infty}$ potential $V$, $W$ is a standard $d$-dimensional
Brownian motion on a probability space equipped with a filtration
$(\mathcal{F}_{t})_{t>0}$ and fulfilling the standard assumptions. There are
three main types of approximations of (1). A strong approximation focuses on
approaching $X(t)$ for a given realisation of $W$. A weak approximation
approaches averages of functionals of the solution $\mathbb{E}[\phi(X(t))]$
where $\phi\in\mathcal{C}^{\infty}(\mathbb{T}^{d},\mathbb{R})$ is a smooth
test function. An approximation for the invariant measure approximates
averages of functionals at the equilibrium for ergodic systems. This paper
develops new algebraic tools for the creation and study of integrators for the
approximation of overdamped Langevin dynamics in $\mathbb{T}^{d}$ and also on
manifolds $\mathcal{M}$, in the weak sense and for the invariant measure.

For the study of ordinary differential equations $y^{\prime}=f(y)$, backward
error analysis [41] rewrites a one-step numerical integrator
$y_{n+1}=\Phi_{h}(y_{n})$ as the exact solution of a modified problem
$\tilde{y}^{\prime}=\tilde{f}(\tilde{y})$. The modified vector field
$\tilde{f}$ typically writes as a formal series in $f$ and its partial
derivatives (a Butcher series) and the properties of the integrator (order,
preservation of invariants or measures, symplecticity,…) can be read on
$\tilde{f}$. Backward error analysis was in particular instrumental in the
understanding of the long time energy preservation property of symplectic
methods [9]. In the stochastic setting, it is known since [88] that backward
error analysis does not generalise straightforwardly for the Euler-Maruyama
method in the strong or weak sense. The approach is successful in [91] for the
Milstein scheme, but only up to order two. In [33], the idea of backward error
analysis is generalised in the context of the approximation for the invariant
measure: assume that the exact and numerical dynamics are ergodic, that is,
the solution of (1) behaves in long time according to a probability density
$\rho_{\infty}$:

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\phi(X(t))dt=\int_{\mathbb{T}^{d}}\phi(x)\rho_{\infty}(x)dx\quad\text{almost
surely},$

and similarly, the integrator follows in long time a density $\rho^{h}$, where
$h$ is the timestep. Then, using the idea of backward error analysis, the
paper [33] provides an explicit expansion of the invariant measure $\rho^{h}$
of the integrator of the form

$\rho^{h}=\rho_{\infty}+h\rho_{1}+h^{2}\rho_{2}+\dots$

The idea was generalised for the creation of modified equations in [3]: given
an equation of the form (1) and an integrator $X_{n+1}=\Phi_{h}(X_{n})$
satisfying mild assumptions, there exists a (truncated) modified vector field
$\tilde{f}^{[N]}=f+hf_{1}+\ldots+h^{N}f_{N}$ for any $N$, such that the
integrator applied to the following modified problem has order $N$ for
sampling the invariant measure of (1):

$d\tilde{X}(t)=\tilde{f}^{[N]}(\tilde{X}(t))dt+dW(t).$

The modified vector field $\tilde{f}=f+hf_{1}+\ldots$ is iteratively defined
in [3] as a formal power series in $h$ in relation to the measure $\rho^{h}$
of the integrator:

$\mathcal{L}^{*}\rho_{1}=-\operatorname{div}(f_{1}\rho_{\infty}),\quad\mathcal{L}^{*}\rho_{2}=-\operatorname{div}(f_{1}\rho_{1})-\operatorname{div}(f_{2}\rho_{\infty}),\quad\dots$

where $\mathcal{L}$ is the generator of equation (1). Note that $\tilde{f}$ is
not uniquely defined by these identities. The calculations of the modified
vector field $\tilde{f}$ are intricate, so that the algebraic tool of exotic
Butcher series was introduced in [57] to simplify the approach. While the
original approach of [3] works at any order, there is no proof in [57] that
the calculations rewrite with exotic B-series beyond order three. In addition,
there is no explicit expression of the modified vector field $\tilde{f}$. This
paper answers these problems by providing the Hopf algebra foundations of
exotic series and by using them to give a new explicit expression of the
modified vector field $\tilde{f}$ as an exotic B-series at any order.

Butcher-series were first introduced in [19, 42] (see also the textbooks [41,
20, 21] and the review [73]) for the study of order conditions for Runge-Kutta
methods in numerical analysis. They were later applied successfully to a
variety of different fields such as geometric numerical integration [41],
quantum field theory [28], rough paths [40, 43], or to stochastic numerical
analysis. We mention in particular the aromatic extension of B-series
introduced in [27, 47] for the study of volume-preserving integrators. They
allow to compute the divergence of a B-series, and were later studied in [72,
76, 11, 36, 12, 55, 54] for their algebraic, geometric, and numerical
properties. For stochastic numerical analysis, we mention in particular the
early works [16, 49, 17] that first introduced stochastic trees and B-series
for the strong convergence of SDEs, and the works by Rößler [81, 82, 83, 84]
and Debrabant and Kværnø [31, 30, 32] for the design and analysis of high
order strong and weak integrators on a finite time interval, as well as the
works [8, 52]. The different stochastic extensions of B-series were recently
formalised and unified into the exotic aromatic S-series formalism [57, 58,
53, 14] for the study of stochastic numerical analysis in the weak sense and
for the invariant measure. It was then shown in [56] that the exotic aromatic
formalism is a very natural extension of B-series and aromatic B-series as
they all satisfy similar universal geometric properties.

We present in this paper the algebraic and combinatorial structures related to
the exotic aromatic S-series and study their concrete applications in
stochastic numerical analysis. This is not straightforward as the Hopf algebra
structures used in the deterministic case [24, 26] do not extend naturally to
the stochastic case. We introduce the concept of decorated aromatic S-series,
defined using decorated aromatic forests, to simplify the study of the
algebraic structures related to the exotic aromatic S-series. Our
investigation involves the study of the D-algebra structure [77] over the
decorated aromatic forests, and we employ this structure to introduce a Hopf
algebroid [67, 75], which we refer to as the Grossman-Larson Hopf algebroid.
This Hopf algebroid is intricately linked to the composition law of decorated
aromatic S-series. The substitution law does not generalise straightforwardly
with decorated aromatic S-series and the main difficulty comes from the
aromas, as first observed in [11] in the context of aromatic forests. We
introduce the concept of clumped forests, which represent monomials of
aromatic trees and form the universal enveloping algebra of aromatic trees.
The difference between aromatic forests and clumped forests lies in the aromas
that are attached to the rooted components in the latter case:
$\forest{(b),b}\cdot\forest{b[b]}\neq\forest{b}\cdot\forest{(b),b[b]}$. These
clumped forests are essential in establishing the algebraic foundations
required for the substitution law, in the spirit of [22, 68, 69, 70, 80].
Subsequently, we proceed to construct the space of exotic aromatic forests
using the decorated aromatic forests we introduced earlier. We use our
analysis of algebraic structures over decorated aromatic forests to present
the composition [14] and substitution law for the exotic aromatic S-series.

The Hopf algebra structures related to exotic aromatic S-series allow us to
elegantly formalise stochastic numerical analysis theory. The composition law
gives the methodology to derive order conditions for arbitrarily high orders,
and to study the accuracy of the composition of different numerical
integrators and of postprocessors [90]. The substitution law allows us to
derive a new explicit expression of a modified vector field that writes as an
exotic B-series for backward error analysis and for creating modified
equations in the Euclidean case. This simplifies greatly the computation of
the modified vector field and the exotic B-series expression ensures that the
modified vector field satisfies important natural geometric properties such as
orthogonal equivariance [56].

The structure of the paper is the following. Section 2 presents a
comprehensive summary of the new algebraic structure of exotic aromatic
forests, relying on the novel idea of clumping and on two Hopf algebras of
trees. The numerical applications are gathered in Section 3 and include the
algebraic formalisation of stochastic order theory in the weak sense and for
the invariant measure, the application of the composition law to the
composition of integrators and postprocessors, and the application of the
substitution law to backward error analysis and modified equations. We then
present in Section (4) the detailed study of the algebraic structures
associated to exotic aromatic forests.

##  2 Exotic aromatic and clumped forests

We consider graphs $\pi=(V,E,S)$ where $V$ is a finite set of vertices and
$E\subset V\times V$ is a set of directed edges. The empty graph is included
and written $\mathbf{1}$. If $e=(v,w)\in E$, the edge $e$ is going from the
source $v$ to the target $w$, $v$ is a predecessor of $w$, and $w$ is a
successor of $v$. The stolons111In botany, stolons are horizontal connections
that link the base of two plants, allowing a plant to clone itself. Strawberry
plants are an example of plants with stolons. in $S\subset V\times
V/\langle(x,y)-(y,x)\rangle$ link some of the vertices without successors by
pairs, but these vertices are in at most one stolon. The vertices that do not
have a successor and are not part of stolons are called roots and the vertices
that are not in stolons and do not have predecessors are called leaves. The
graphs have two kinds of connected components: the ones that have a root,
called trees, and the other ones, called aromas. The aromas either have a
stolon or have a cycle, that is a list of vertices $\\{v_{1},\dots,v_{n}\\}$
where $v_{i+1}$ is the successor of $v_{i}$ and $v_{1}$ is the successor of
$v_{n}$. We call aromatic forests, gathered in the set $AF$, the equivalence
classes of such graphs, where two graphs are equivalent if there exists a
bijection between their sets of vertices and edges that preserve successors,
predecessors, and stolons. When drawn, by convention, the orientation of the
edges goes from top to bottom and in counterclockwise direction for cycles.

We consider the following subsets of $AF$. The set of trees $T$ contains the
connected aromatic forests with one root. The trees with up to four vertices
are

$\forest{b},\ \forest{b[b]},\ \forest{b[b,b]},\ \forest{b[b[b]]},\
\forest{b[b,b,b]},\ \forest{b[b,b[b]]},\ \forest{b[b[b,b]]},\
\forest{b[b[b[b]]]}.$

We obtain the set $F$ of forests by taking all possible unordered monomials of
trees, including the empty forest $\mathbf{1}$.

$F=\\{\mathbf{1},\ \forest{b},\ \forest{b,b},\ \forest{b[b]},\
\forest{b,b,b},\ \forest{b,b[b]},\ \forest{b[b,b]},\
\forest{b[b[b]]},\dots\\}$

The subset of aromatic forests with $n$ roots is written $AF_{n}$, which gives
us a first grading on $AF$. In particular, the set of aromas is

$A=AF_{0}=\\{\mathbf{1},\ \forest{(b)},\ \forest{b=b},\ \forest{(b,b)},\
\forest{(b[b])},\ \forest{b=b,(b)},\ \forest{b=b,b=b},\dots\\},$

with the double edge denoting a stolon, and the aromatic trees are in

$AT=AF_{1}=\\{\forest{(b,b[b,b[b]],b[b]),(b,b),b[b]},\
\forest{(b),(b),(b,b[b]),b[b[b]]},\ \forest{b[b,b]},\dots\\}.$

Note that the set $AF\simeq A\times F$ of aromatic forests is the product of
the sets of aromas and forests. The vector space spanned by a given set is
written in calligraphic font. For instance, we write
$\mathcal{AF}=\operatorname{Span}_{\mathbb{R}}(AF)$.

Note that the definition of aromas taken here is more general than in [47, 27,
11], as we choose to include stolons among the aromas. The reason is that the
combinatorial structure of stolonic and cyclic aromas is similar, and the
exotic extension of the standard aromatic forests relies in particular on
stolons.

Decorated aromatic forests are aromatic forests $\pi\in AF$ endowed with a
decoration of their vertices $\alpha:V(\pi)\to D$ for a given set $D$. A
morphism $\varphi:(\pi_{1},\alpha_{1})\to(\pi_{2},\alpha_{2})$ between
decorated aromatic forests is a graph morphism $\varphi:\pi_{1}\to\pi_{2}$
such that $\alpha_{1}=\alpha_{2}\circ\varphi$. The group of automorphisms is
denoted by $\operatorname{Aut}(\pi,\alpha)$ and
$\sigma(\pi,\alpha):=|\operatorname{Aut}(\pi,\alpha)|$. The set of decorated
aromatic forests is denoted by $AF_{D}$ and the corresponding vector space by
$\mathcal{AF}_{D}$. We will often omit writing $\alpha$ for simplicity and
will denote the elements of $AF_{D}$ by $\pi$.

The vector space $\mathcal{AF}_{D}$ of decorated aromatic forests forms a
commutative algebra $(\mathcal{AF}_{D},\cdot)$ which can be defined as the
symmetric algebra $S_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$ of decorated
aromatic trees $\mathcal{AT}_{D}$ over the ring of decorated aromas
$\mathcal{A}_{D}$. We consider the symmetric algebra
$\mathcal{CF}_{D}:=S(\mathcal{AT}_{D})$ of decorated aromatic trees
$\mathcal{AT}_{D}$ over the base field $\mathbb{R}$. The basis of
$\mathcal{CF}_{D}$ is denoted by $CF_{D}$ and its elements are called clumped
forests. We note that
$\forest{(b),b}\cdot\forest{b[b]}\neq\forest{b}\cdot\forest{(b),b[b]}$ in
$CF_{D}$. Some elements of $\mathcal{CF}_{D}$ are listed below:

$\forest{(b),b}\cdot\forest{b[b,b]},\quad\forest{b}\cdot\forest{(b),b[b,b]},\quad\forest{(b,b[b]),b[b,b]}\cdot\forest{b=b,b[b]}\cdot\forest{(b[b,b]),b=b[b],b}.$

A detailed study of the algebraic structure of decorated aromatic and clumped
forests is presented in Section 4.

###  2.1 Decorated aromatic S-series

Let $I$ be a finite set of indices. Then the elementary differential
$f^{k}_{I}$ is defined as:

$f^{k}_{I}:=\frac{\partial^{|I|}f^{k}}{\prod_{i\in I}\partial x_{i}}.$

We define the elementary differential map $F_{D}$ by

$\displaystyle F_{D}(\pi)[\phi]=\sum_{\underset{v\in
V}{i_{v}=1,\dots,d}}\delta_{I_{S}}\left(\prod_{v\in
V}f^{i_{v}}_{\alpha(v),I_{\Pi(v)}}\right)\phi_{I_{R}},$

where $V$ is the set of vertices, $R$ is the set of roots, $\Pi(v)$ is the set
of predecessors of $v$, $\delta_{I_{S}}$ identifies the indices of the
stolons, that is, $i_{x}=i_{y}$ if $(x,y)\in S$, and $f_{\alpha(v)}$ is the
vector field corresponding to the vertices decorated by $\alpha(v)$. For
example,

$F_{D}(\forest{b=w,(w),w[b,b],b[w]})[\phi]=\sum_{i,j,k,l,m,n,p=1}^{d}f^{i}g^{i}g^{j}_{j}f^{k}f^{l}g^{m}_{ij}g^{n}f^{p}_{n}\phi_{m,p}.$

where $f$ and $g$ are vector fields corresponding to $\bullet$ and $\circ$,
respectively.

Let us introduce S-series over decorated aromatic forests. Let
$\overline{\mathcal{AF}}_{D}$ be the space of formal sums of the following
form

$\sum_{\pi\in AF_{D}}a(\pi)\pi,\quad\text{with }a\in\mathcal{AF}_{D}^{*}.$

It is the completion with respect to the graduation given by the number of
vertices. We extend $F_{D}$ by linearity to $\overline{\mathcal{AF}}_{D}$.
Recall the symmetry coefficient $\sigma(\pi):=|\operatorname{Aut}(\pi)|$. Let
the map $\delta_{\sigma}:\mathcal{AF}_{D}^{*}\to\overline{\mathcal{AF}}_{D}$
be the isomorphism between the dual and the completion given by

$\delta_{\sigma}(a)=\sum_{\pi\in AF_{D}}\frac{a(\pi)}{\sigma(\pi)}\pi.$

We will abuse the notation and use $\delta_{\sigma}$ to denote analogous
isomorphisms for other spaces. The space will be made clear from the context
and will always be the domain of the functional to which $\delta_{\sigma}$ is
applied.

Let $\left|.\right|\colon AF_{D}\rightarrow\mathbb{N}$ be a map defining a
finite grading on $AF_{D}$, that we call the order on decorated aromatic
forests.

###### Definition 2.1

An S-series over decorated aromatic forests is defined for
$a\in\mathcal{AF}_{D}^{*}$ and some smooth vector fields $f_{d}$, $d\in D$, as
the following formal power series

$S_{(f_{d})_{d\in D}}^{h}(a)[\phi]:=\sum_{\pi\in
AF_{D}}h^{\left|\pi\right|}\frac{a(\pi)}{\sigma(\pi)}F_{D}(\pi)[\phi],\quad\text{for
}a\in\mathcal{AF}_{D}^{*}.$

We often omit the dependency in the $f_{d}$ and write for simplicity
$S^{h}(a)$ and $S(a)=S^{1}(a)$. For example, let $D=\\{\bullet,\circ\\}$ and
$F_{D}(\bullet)=f,F_{D}(\circ)=g$, then,

$\displaystyle S_{f,g}(a)[\phi]$
$\displaystyle=\left(1+a(\forest{(b)})\operatorname{div}(f)+a(\forest{(w)})\operatorname{div}(g)+a(\forest{b=b})\langle
f,f\rangle+\cdots\right)\phi$
$\displaystyle+\left(1+a(\forest{(b),b})\operatorname{div}(f)+a(\forest{(w),b})\operatorname{div}(g)+a(\forest{b=b,b})\langle
f,f\rangle+\cdots\right)\sum_{i=1}^{d}f^{i}\phi_{i}+\cdots,$

with $\langle{-},{-}\rangle$ the standard Euclidean scalar product.

S-series are used to represent formal sums of differential operators. We use
S-series to represent formal sums of vector fields by using the identification
between first order operators and vector fields and by requiring the
corresponding functional $a_{0}:\mathcal{AF}_{D}\to\mathbb{R}$ to satisfy

$\operatorname{supp}(a_{0})\subset\mathcal{AT}_{D}.$

Such series are called B-series, written $B^{h}(a_{0})$, and $a_{0}$ is called
an infinitesimal character. B-series are used in deterministic numerical
analysis [41] for the representation of numerical integrators
$y_{1}:=\Psi_{h}(y_{0})$ applied to $y^{\prime}=f(y)$ as
$\Psi_{h}(y_{0})=y_{0}+B^{h}(a_{0})(y_{0})$ for an infinitesimal character
$a_{0}$. This is typically done by Taylor expanding the numerical method with
respect to its time stepsize $h$. Given any smooth function
$\phi:\mathbb{R}^{d}\to\mathbb{R}$, the composition of
$\Psi_{h}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ with $\phi$ is represented by the
following exponential of B-series:

$\phi\circ\Psi_{h}=F_{D}\Big{(}\exp^{\cdot}\big{(}\delta_{\sigma}(a_{0})\big{)}\Big{)}[\phi],\quad\text{with
}\exp^{\cdot}(x):=1+\sum_{k=1}^{\infty}\frac{x^{\cdot k}}{k!}.$

We use the formalism of aromatic forests and the convolution product
$\odot:\mathcal{AF}_{D}^{*}\otimes\mathcal{AF}_{D}^{*}\to\mathcal{AF}_{D}^{*}$
defined as $a\odot b:=(a\otimes b)\circ\Delta$ with
$\Delta:\mathcal{AF}_{D}\to\mathcal{AF}_{D}\otimes\mathcal{AF}_{D}$ being the
deshuffle coproduct (not $\mathcal{A}_{D}$-linear) over decorated aromatic
forests to write the exponential of B-series as an S-series as

$\phi\circ\Psi_{h}=S^{h}(\exp^{\odot}(a_{0}))[\phi]\quad\text{with
}\exp^{\odot}(x):=1+\sum_{k=1}^{\infty}\frac{x^{\odot k}}{k!}.$ (2)

###  2.2 Exotic aromatic S-series

Exotic aromatic forests are a specific class of decorated aromatic forests
that naturally appears in the study of numerical methods for solving
stochastic differential equations with additive noise [57, 58, 53]. We use
decorations to encode paired nodes, called lianas, that allow to represent the
Laplacian of a Taylor series in a jet bundle. The exotic aromatic forests are
defined in the following way, where we mention that additional decorations
could be added, in the spirit of partitioned S-series.

###### Definition 2.2

An exotic aromatic forest is a decorated aromatic forest $(\pi,\alpha)\in
AF_{D}$ with the decorations $D=\\{\bullet\\}\cup\mathbb{N}$,
$\mathbb{N}=\\{1,2,3,\dots\\}$, that follows the following rules. All vertices
of $\pi$ are black, except the leaves that can be either black or numbered. If
a natural number is used as a decoration, then it must decorate two leaves,
that is, $|\alpha^{-1}(n)|\in\\{0,2\\}$ for any $n\in\mathbb{N}$. Two exotic
aromatic forests $(\pi_{1},\alpha_{1})$ and $(\pi_{2},\alpha_{2})$ are
considered to be identical if $\pi_{1}=\pi_{2}$ and there exists a map
$\varphi\in S_{\mathbb{N}}$ such that
$\alpha_{1}\raisebox{-2.15277pt}{$|$}_{\mathbb{N}}=\varphi\circ\alpha_{2}\raisebox{-2.15277pt}{$|$}_{\mathbb{N}}$.
The pair of numbered leaves that correspond to the same number is called a
liana, gathered in the set $L\subset V\times V/\langle(x,y)-(y,x)\rangle$. The
order of an exotic aromatic forest $\pi$ is the following, where $V_{\bullet}$
denotes the number of black vertices of $\pi$,

$\left|\pi\right|=\left|V_{\bullet}\right|+\left|L\right|-\left|S\right|,$

where $S\subset V\times V/\langle(x,y)-(y,x)\rangle$ is the set of stolons.

The exotic aromatic trees of order up to two are the following. Note that the
order does not coincide with the number of black vertices in general.

$\forest{b},\ \forest{b[b]},\ \forest{b[1,1]},\ \forest{(b),b},\
\forest{(b[1]),1},\ \forest{b=b,b},\ \forest{b=b[1],1}.$

The set of exotic aromatic forests is denoted by $EAF$. Exotic aromatic
forests with one root are called exotic aromatic trees and form a set denoted
by $EAT$. Exotic aromatic forests without a root are called exotic aromas and
form a set denoted by $EA$. The corresponding vector spaces are denoted by
$\mathcal{EAF}$, $\mathcal{EAT}$ and $\mathcal{EA}$, respectively. We refer to
Appendix A for further examples. We emphasize that the order of a forest is
never negative. Note also that the following exotic aromatic forests are
identical:

$\forest{(b[b[3],1,1]),b[b[2],b[2,3]]}=\forest{(b[b[2],3,3]),b[b[1],b[1,2]]}.$

To obtain exotic aromatic S-series, we define the elementary differential map
by

$\displaystyle F_{\mathcal{E}}(\pi)[\phi]=\sum_{\underset{v\in
V}{i_{v}=1,\dots,d}}\delta_{I_{S\cup L}}\left(\prod_{v\in
V_{\bullet}}f^{i_{v}}_{I_{\Pi(v)}}\right)\phi_{I_{R}},$

where $R$ is the set of roots, $\Pi(v)$ is the set of predecessors of $v$, and
$\delta_{I_{S\cup L}}$ identifies the indices of the lianas and stolons, that
is, $i_{x}=i_{y}$ if $(x,y)\in S\cup L$. For instance, we have

$\pi=\forest{(b[1]),b=b[2],b[1],2},\quad
F_{\mathcal{E}}(\pi)[\phi]=\sum_{i,j,s,l_{1},l_{2}=1}^{d}f^{i}_{il_{1}}f^{s}f^{s}_{l_{2}}f^{j}_{l_{1}}\phi_{jl_{2}}.$

Given a coefficient map $a\in\mathcal{EAF}^{*}$, an exotic aromatic S-series
is the following formal series (see Definition 2.1),

$S^{h}(a):=\sum_{\pi\in
EAF}h^{\left|\pi\right|}\frac{a(\pi)}{\sigma(\pi)}F_{\mathcal{E}}(\pi),\quad
S(a)=S^{1}(a),$

where the symmetry coefficient $\sigma(\pi)$ counts the number of graph
automorphisms leaving $\pi$ unchanged. The S-series formalism allows us to
rewrite the tedious combinatorics of numerical analysis in terms of simple
graph operations that do not involve the dimension of the problem.

Theorems 2.5 and 2.8 present the composition and substitution laws of exotic
aromatic S-series used in Section 3 to describe the composition of numerical
methods, the construction of the modified equation of a method, and the
backward error analysis. The proofs are derived straightforwardly from the
analysis of Section 4.

###### Definition 2.3

[14] The Butcher-Connes-Kreimer coproduct on $\mathcal{EAF}$ is defined as

$\Delta_{BCK}(\pi):=\sum_{\pi_{0}\subset\pi}(\pi\setminus\pi_{0})\otimes\pi_{0},$

where the sum runs over all rooted subforests $\pi_{0}\in EAF$ of $\pi$ such
that $\pi\setminus\pi_{0}\in EAF$ and there are no edges going from $\pi_{0}$
to $\pi\setminus\pi_{0}$ in $\pi$. We note that the paired number vertices
cannot be separated across different sides of the tensor product.

###### Example 2.4

For example,

$\displaystyle\Delta_{BCK}(\forest{(b[1]),b[1,b]})$
$\displaystyle=\mathbf{1}\otimes\forest{(b[1]),b[1,b]}+\forest{1,1}\otimes\forest{(b),b[b]}+\forest{b}\otimes\forest{(b[1]),b[1]}+\forest{(b[1]),1}\otimes\forest{b[b]}+$
$\displaystyle\quad\forest{1,1,b}\otimes\forest{(b),b}+\forest{(b[1]),1,b}\otimes\forest{b}+\forest{1,b[1,b]}\otimes\forest{(b)}+\forest{(b[1]),b[1,b]}\otimes\mathbf{1},$
$\displaystyle\Delta_{BCK}(\forest{b[1,1,2,b[2]]})$
$\displaystyle=\mathbf{1}\otimes\forest{b[1,1,2,b[2]]}+\forest{1,1}\otimes\forest{b[2,b[2]]}+\forest{2,2}\otimes\forest{b[1,1,b]}+\forest{1,1,2,2}\otimes\forest{b[b]}+$
$\displaystyle\quad\forest{2,b[2]}\otimes\forest{b[1,1]}+\forest{1,1,2,b[2]}\otimes\forest{b}+\forest{b[1,1,2,b[2]]}\otimes\mathbf{1}.$

We present in Appendix A the values of $\Delta_{BCK}$ over all connected
forests and exotic aromas up to order $3$.

######  Theorem 2.5 (Composition law)

[14] Let $S(a)$ and $S(b)$ be two exotic aromatic S-series and let $\phi$ be a
test function. Then,

$S(a)[S(b)[\phi]]=S(a\ast b)[\phi],\quad\text{with }a\ast b=(a\otimes
b)\circ\Delta_{BCK},$

with $\Delta_{BCK}$ being the Butcher-Connes-Kreimer coproduct over exotic
aromatic forests.

We define an extension ${\mathcal{CEF}^{1}}$ of the concept of decorated
clumped forests to the exotic context as
${\mathcal{CEF}^{1}}:=S(\mathcal{EAT})$. Some elements of
${\mathcal{CEF}^{1}}$ are

$\forest{(b),b[1,1]}\cdot\forest{(b[2]),b[2]},\quad\forest{(b[3],b[3]),b[b[b,4],4]}\cdot\forest{b=b,b[b]}.$

Note that exotic aromatic trees are a subset of primitive elements of
$\mathcal{EAF}$ with the full set of primitive elements being discussed in
Section 4.4.

We consider a particular case of the substitution law from Theorem 4.19 in
which the black vertices are substituted by
$\delta_{\sigma}(b_{0})\in\overline{\mathcal{EAT}}$ while the numbered
vertices are not changed, that is, a number vertex $\forest{i_{n}}$ is
substituted by $\forest{i_{n}}$ for $n\in\mathbb{N}$. This allows us to
simplify the definition of the CEM coaction.

###### Definition 2.6

Let the CEM coaction
$\Delta_{CEM}:\mathcal{EAF}\to{\mathcal{CEF}^{1}}\otimes\mathcal{EAF}$ over
exotic aromatic forests be defined as

$\Delta_{CEM}(\pi):=\sum_{p\subset\pi}p\otimes\pi/_{p},$

where the sum is over all clumped exotic subforests $p\in{\mathcal{CEF}^{1}}$
that cover all black vertices and $\pi/_{p}$ is the exotic aromatic forest
obtained by contracting the exotic aromatic trees of $p$ into black vertices.
If the forest $\pi\in\mathcal{EAF}$ doesn’t have valid subforests
$p\in{\mathcal{CEF}^{1}}$, then $\Delta_{CEM}(\pi)=\mathbf{1}\otimes\pi$. For
details see the proof of Theorem 4.23.

###### Example 2.7

The substitution on exotic aromatic forests has additional difficulties
compared to the deterministic context. While the coaction $\Delta_{CEM}$ works
with a similar idea to the partitions of trees as in [26, Thm. 3.2], it
ensures that paired vertices stay in the same component and it sums over all
possible combinations of clumped exotic forests in ${\mathcal{CEF}^{1}}$ (and
not exotic aromatic forests) in the left side of the tensor product. For
instance, we have

$\displaystyle\Delta_{CEM}(\forest{(b[1]),b[1,b]})$
$\displaystyle=\forest{(b[1]),b[1,b]}\otimes\forest{b}+\forest{(b[1]),1,b[b]}\otimes\forest{b[b]}+\forest{(b),b[b]}\otimes\forest{b[1,1]}+\forest{b,b[b]}\otimes\forest{(b[1]),b[1]}$
$\displaystyle+\forest{(b[1]),b[1],b}\otimes\forest{b[b]}+\forest{(b[1]),1,b,b}\otimes\forest{b[b,b]}+\forest{(b),b}\cdot\forest{b}\otimes\forest{b[1,1,b]}+\forest{b,b,b}\otimes\forest{(b[1]),b[1,b]},$
$\displaystyle\Delta_{CEM}(\forest{b[1,1,2,b[2]]})$
$\displaystyle=\forest{b[1,1,2,b[2]]}\otimes\forest{b}+\forest{b[b]}\otimes\forest{b[1,1,2,2]}+\forest{b[1,1,b]}\otimes\forest{b[2,2]}$
$\displaystyle+\forest{b[2,b[2]]}\otimes\forest{b[1,1]}+\forest{b[1,1],b}\otimes\forest{b[2,b[2]]}+\forest{b,b}\otimes\forest{b[1,1,2,b[2]]}.$

We present in Appendix A the values of $\Delta_{CEM}$ over the elements of
$\mathcal{EA}$ and $\operatorname{Prim}(\mathcal{EAF})$ up to order $3$.

We use the discussion from Section 4.3 to obtain Theorem 2.8.

######  Theorem 2.8 (Substitution law)

Let $a\in\mathcal{EAF}^{*},~{}$ $b_{0}\in\mathcal{EAT}^{*}$, then,

$S_{S_{f}(b_{0})}(a)=S_{f}(b_{c}\star a),\quad\text{with }b_{c}\star
a=(b_{c}\otimes a)\circ\Delta_{CEM},$

where $b_{c}$ is the character of ${\mathcal{CEF}^{1}}$ that extends $b_{0}$
and $\Delta_{CEM}$ is defined in Definition 2.6.

Theorem 4.23 provides a simplified procedure to compute the substitution law
for exotic aromatic S-series using the Hopf algebra structure of $CEF$.

##  3 Backward order analysis and modified equations for stochastic
differential equations

In the numerical analysis of ordinary differential equations, the composition
and substitution of B-series have a variety of applications, including the
derivation of order conditions, the composition of numerical methods, backward
error analysis, or high-order integration based on modified equations (see
[26] and references therein). The aromatic B-series appear naturally in the
study of volume-preserving integrators and we refer to [27, 47, 11] for
details. The stolons appear when studying projection methods on embedded
manifolds, while the lianas are used in stochastic numerical analysis (see
[53]). In this section, we focus on the applications of the composition and
substitution laws for the numerical approximation of stochastic differential
equations. As described in the original references [57, 58, 14], exotic
aromatic B-series and S-series are a crucial calculation tool for the high-
order approximation of SDEs in the weak context and for the invariant measure.
There are several major differences with the deterministic context, especially
on the concept of backward error analysis. Most importantly, our analysis
gives an explicit expression as an exotic B-series of the modified vector
field for the backward error analysis and for using modified equations for the
invariant measure in $\mathbb{R}^{d}$ (see Theorem 3.10 and Theorem 3.14).

###  3.1 Stochastic order theory with exotic aromatic S-series

We consider stochastic differential equations with additive noise on the
$d$-dimensional torus $\mathcal{M}=\mathbb{T}^{d}$ or on smooth compact
manifolds $\mathcal{M}=\\{x\in\mathbb{R}^{d},\zeta(x)=0\\}$ of codimension $1$
with a smooth constraint $\zeta\colon\mathbb{R}^{d}\to\mathbb{R}$ of the form

$dX(t)=\Pi_{\mathcal{M}}(X(t))f(X(t))dt+\Pi_{\mathcal{M}}(X(t))\circ
dW(t),\quad X(0)=x\in\mathcal{M},$ (3)

where $x$ is assumed deterministic for simplicity,
$\Pi_{\mathcal{M}}\colon\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ is the
orthogonal projection on the tangent bundle of the manifold $\mathcal{M}$, $f$
is a smooth vector field, $W$ is a standard $d$-dimensional Brownian motion on
a probability space equipped with a filtration $(\mathcal{F}_{t})_{t>0}$ and
fulfilling the standard assumptions. Dynamics of the form (3) include
overdamped Langevin dynamics [62] when $f=-\nabla V$ for $V$ a smooth
potential. If $\mathcal{M}=\mathbb{T}^{d}$, the orthogonal projection is
$\Pi_{\mathcal{M}}=I_{d}$ and equation (3) reduces to

$dX(t)=f(X(t))dt+dW(t).$

There are three main types of approximations of (3). A strong approximation
focuses on approaching $X(t)$ for a given realisation of $W$. A weak
approximation approaches averages of functionals of the solution
$\mathbb{E}[\phi(X(t))]$ where
$\phi\in\mathcal{C}^{\infty}(\mathcal{N},\mathbb{R})$ is a smooth test
function defined on an open neighbourhood $\mathcal{N}$ of $\mathcal{M}$. An
approximation for the invariant measure approximates averages of functionals
at the equilibrium for ergodic systems. In this paper, we focus on the
algebraic calculation underlying the high order approximation of (3) in the
weak sense and for the invariant measure. We give the definitions of order in
terms of S-series directly and we refer to the textbooks [74, 62, 3] for the
detailed analysis on non-compact manifolds.

The generator of equation (3) in $\mathbb{T}^{d}$ can be represented as an
exotic S-series $S(l)$ with coefficient map $l\in\mathcal{EF}^{*}$. It is
given by the following linear combination of the primitive elements of order
one:

$\delta_{\sigma}(l)=\forest{b}+\frac{1}{2}\forest{1,1},\quad\mathcal{L}\phi=S(l)[\phi]=\phi^{\prime}f+\frac{1}{2}\Delta\phi,$

where we recall that $\delta_{\sigma}$ is given by

$\delta_{\sigma}(a)=\sum_{\pi\in\mathcal{E}\mathcal{A}\mathcal{F}}\frac{a(\pi)}{\sigma(\pi)}\pi.$

In the manifold case, we add a new decoration $\circ$ to
$\mathcal{E}\mathcal{A}\mathcal{F}$ that represents the gradient of the
constraint $n=\nabla\zeta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and we
extend the elementary differential with $F_{\mathcal{E}}(\circ)=n$. The
definition of the S-series is

$S^{h}(a)=\sum_{\pi\in\mathcal{E}\mathcal{A}\mathcal{F}}h^{\left|\pi\right|}\frac{a(\pi)}{\sigma(\pi)\left|n\right|^{\left|V_{\circ}\right|}}F_{\mathcal{E}}(\pi),\quad\left|\pi\right|=\left|V_{\bullet}\right|+\frac{\left|V_{\circ}\right|}{2}+\left|L\right|-\left|S\right|,$

and the generator becomes

$\displaystyle\delta_{\sigma}(l)$
$\displaystyle=\forest{b}-\forest{w=b,w}-\frac{1}{2}\forest{(w),w}+\frac{1}{2}\forest{w=w[w],w}+\frac{1}{2}\forest{1,1}-\frac{1}{2}\forest{w,w},$
$\displaystyle\mathcal{L}\phi$
$\displaystyle=S(l)[\phi]=\phi^{\prime}f-\left|n\right|^{-2}\langle
n,f\rangle\phi^{\prime}n-\frac{1}{2}\left|n\right|^{-2}\operatorname{div}(n)\phi^{\prime}n$
$\displaystyle+\frac{1}{2}\left|n\right|^{-4}\langle
n,n^{\prime}n\rangle\phi^{\prime}n+\frac{1}{2}\Delta\phi-\frac{1}{2}\left|n\right|^{-2}\phi^{\prime\prime}(n,n).$

The quantity of interest in the weak context is
$u(t,x)=\mathbb{E}[\phi(X(t))|X(0)=x]$. It satisfies the following formal
expansion in a neighbourhood of $\mathcal{M}$, derived from the backward
Kolmogorov equation (see, for instance, [44, 35, 3, 50, 51, 53]),

$u(x,h)=\exp(h\mathcal{L})[\phi](x)=S^{h}(e)[\phi](x),\quad
e=\exp^{\ast}(l):=\sum_{n=0}^{\infty}\frac{1}{n!}l^{\ast n},$

where we recall that $\ast$ is the composition law (see Theorem 2.5). The
first terms of $e$ in $\mathbb{T}^{d}$ are

$\delta_{\sigma}(e)=\mathbf{1}+\forest{b}+\frac{1}{2}\forest{1,1}+\frac{1}{2}\forest{b[b]}+\frac{1}{2}\forest{b,b}+\frac{1}{2}\forest{b,1,1}+\frac{1}{4}\forest{b[1,1]}+\frac{1}{2}\forest{b[1],1}+\frac{1}{8}\forest{1,1,2,2}+\dots$
(4)

In opposition to the deterministic formalism of Butcher trees, the S-series of
the exact flow is not the exponential of a combination of trees, but the
exponential of a combination of forests, as the primitive elements of exotic
forests do not reduce to exotic trees:

$\mathcal{ET}\subsetneq\operatorname{Prim}(\mathcal{EF}),\quad\mathcal{T}=\operatorname{Prim}(\mathcal{F}).$

This major difference with deterministic geometric numerical integration
makes, in particular, backward error analysis much more tedious in the
stochastic case, as we rely on an additional non-trivial operation, similar to
the integration by parts, that transforms primitive elements into exotic
trees. In the manifold case, the first terms of $S(e)$ can be found in [58].
Consider now a one-step integrator for solving (3) of the form

$X_{n+1}=\Phi_{h}(X_{n}),\quad X_{0}=x\in\mathcal{M},$ (5)

where $h$ is the stepsize of the method and the dependence in random variables
is omitted for clarity. We assume that the method (5) is an exotic aromatic
S-series method, that is, the numerical analogue of $u(x,h)$ satisfies the
following formal Talay-Tubaro expansion [89] in a neighbourhood of
$\mathcal{M}$:

$\mathbb{E}[\phi(X_{1})|X_{0}=x]=S^{h}(a)[\phi](x).$

For instance, the following class of stochastic Runge-Kutta integrators
presented in [53] satisfy this assumption naturally in $\mathbb{T}^{d}$:

$\displaystyle Y_{n}^{i}$
$\displaystyle=X_{n}+h\sum\limits_{j=1}^{s}a_{i,j}f(Y_{n}^{j})+\sqrt{h}d_{i}\xi_{n},\qquad
i=1,\dots,s,$ (6) $\displaystyle X_{n+1}$
$\displaystyle=X_{n}+h\sum\limits_{i=1}^{s}b_{i}f(Y_{n}^{i})+\sqrt{h}\xi_{n},$

where we use one Gaussian vector $\xi_{n}\sim\mathcal{N}(0,I_{d})$ at each
step for simplicity. Thanks to the result [14, Prop. 4.3] (see also [41, 82]),
the exotic S-series $S(a)$ of methods of the form (6) is given by the
following coefficient map,

$\tau=\forest{b_{i}{}_{2}70[b_{j}{}_{1}80[1,2],1],2,b_{k}{}_{2}70[b_{l}{}_{9}0,b_{m}{}_{9}0]}\qquad
a(\tau)=\sum_{i,j,k,l,m=1}^{s}b_{i}a_{i,j}d_{j}^{2}d_{i}b_{k}a_{k,l}a_{k,m},$

where the decoration of $\tau$ by $\\{i,j,k,l,m\\}$ is added for illustration
purposes. The coefficient maps $a$ of the exact flow and of stochastic Runge-
Kutta methods (6) are naturally characters over $\mathcal{EF}$ equipped with
the concatenation product, that is, $a$ satisfies

$a(\pi_{1}\cdot\pi_{2})=a(\pi_{1})a(\pi_{2}),\quad\pi_{1},\pi_{2}\in\mathcal{EF}.$

We denote $\operatorname{Char}(\SS)$ the set of characters over a subset $\SS$
of $\mathcal{EF}$ equipped with the concatenation product.

The standard definition of weak order [89, 74] rewrites in the following way
with exotic aromatic S-series.

###### Definition 3.1

Let an integrator (5) with S-series $S^{h}(a)$ such that
$a-e\in\mathcal{EAF}^{*}_{\geq p+1}$, i.e., if $a$ and $e$ coincide on all
exotic aromatic forests of order up to $p$. Then, the integrator has (at
least) weak order $p$ for solving (3), that is, for $T>0$ and $h\leq h_{0}$
small enough with $Nh=T$, for the initial condition $X_{0}\in\mathcal{M}$, for
all test functions $\phi\in\mathcal{C}^{\infty}(\mathcal{N},\mathbb{R})$, the
following estimate holds

$\left|\mathbb{E}[\phi(X_{n})]-\mathbb{E}[\phi(X(nh))]\right|\leq
C(h_{0},T,\phi)h^{p},\quad n=0,\dots,N.$

An integrator of at least weak order one is called consistent.

###### Example 3.2

Consider for instance the Euler-Maruyama method in $\mathbb{T}^{d}$:

$X_{n+1}=X_{n}+hf(X_{n})+\sqrt{h}\xi_{n},\quad\xi_{n}\sim\mathcal{N}(0,I_{d}).$
(7)

The exotic aromatic S-series of the Euler-Maruyama method is given by __

$\exp^{\cdot}(\delta_{\sigma}(l))=\delta_{\sigma}(\exp^{\odot}(l))=\mathbf{1}+\forest{b}+\frac{1}{2}\forest{1,1}+\frac{1}{2}\forest{b,b}+\frac{1}{2}\forest{b,1,1}+\frac{1}{8}\forest{1,1,2,2}+\dots,$
(8)

where the convolution product is $a\odot b:=(a\otimes b)\circ\Delta$ with
$\Delta$ the deshuffle coproduct over exotic aromatic forests (see Section
2.1). Comparing the S-series (4) and (8) shows that, under the standard
regularity assumptions [89, 74, 3], the method is of weak order one.

The overdamped Langevin dynamics (3) with $f=-\nabla V$ are ergodic (see, for
instance, the works [62, 33, 3]), that is, there exists a unique invariant
measure $d\mu_{\infty}$ on $\mathcal{M}$ that has a probability density
$\rho_{\infty}$ with respect to $d\sigma_{\mathcal{M}}$, the canonical measure
on $\mathcal{M}$ induced by the Euclidean metric of $\mathbb{R}^{d}$, such
that for all test functions $\phi$,

$\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\phi(X(t))dt=\int_{\mathcal{M}}\phi(x)d\mu_{\infty}(x)\quad\text{almost
surely},\quad\rho_{\infty}(x)\propto\exp\left(-2V(x)\right).$

The invariant measure is the unique solution to
$\mathcal{L}^{*}d\mu_{\infty}=0$. Similarly, a consistent integrator (5) is
ergodic if there exists a unique probability measure $d\mu^{h}$ that is
absolutely continuous w.r.t. the measure $d\sigma_{\mathcal{M}}$ such that for
all test functions $\phi\in\mathcal{C}^{\infty}(\mathcal{N},\mathbb{R})$ and
all initial condition $X_{0}\in\mathcal{M}$,

$\lim_{N\rightarrow\infty}\frac{1}{N+1}\sum_{n=0}^{N}\phi(X_{n})=\int_{\mathcal{M}}\phi(x)d\mu^{h}(x).$

We refer for instance to [33] for appropriate assumptions to obtain the
ergodicity of the numerical scheme. We take over the equivalence relation
$\sim$ on exotic aromatic S-series and their dual defined in [57, 58, 14].
This relation is called the integration by parts as

$a\sim
b\quad\Rightarrow\quad\int_{\mathcal{M}}S^{h}(a)[\phi](x)d\mu_{\infty}(x)=\int_{\mathcal{M}}S^{h}(b)[\phi](x)d\mu_{\infty}(x),$

and involves detaching and grafting back edges. The integration by parts
allows one to transform primitive elements into trees. We give examples and
refer to [57, 58, 14] for the detailed definition:

$\forest{1,1}\sim-2\forest{b},\quad\forest{b[1],1}\sim-\forest{b[1,1]}-2\forest{b[b]},\quad\forest{b[1,b],1}\sim-\forest{b[1,1,b]}-\forest{b[1,b[1]]}-2\forest{b[b,b]},\quad\forest{b[1,2],1,2}\sim-\forest{b[1,2,2],1}-2\forest{b[1,b],1}.$

The approximation for the invariant measure is defined algebraically in the
following way. We shall see that it can also be understood as the stochastic
extension of pseudo-volume-preserving approximations.

###### Definition 3.3

Consider a consistent ergodic integrator (5) with S-series $S^{h}(a)$ and
invariant measure $d\mu^{h}$. Then, if
$a-\delta_{\mathbf{1}}\sim\mathcal{EAF}_{\geq p+1}^{*}$, the integrator has
(at least) order $p$ for sampling the invariant measure of (3), that is, for
$h\leq h_{0}$ small enough and
$\phi\in\mathcal{C}^{\infty}(\mathcal{N},\mathbb{R})$, the following estimate
holds,

$\left|\int_{\mathcal{M}}\phi(x)d\mu^{h}(x)-\int_{\mathcal{M}}\phi(x)d\mu_{\infty}(x)\right|\leq
Ch^{p}.$

An integrator of weak order $p$ immediately has at least order $p$ for the
invariant measure as $e\sim\delta_{\mathbf{1}}$, but there exist methods with
high order for the invariant measure and weak order one (see, for instance,
for the underdamped and overdamped Langevin equation [13, 59, 60, 3, 4, 57,
58]). An important result for our analysis in the $\mathbb{T}^{d}$ case with a
gradient vector field $f=-\nabla V$ is in [14, Thm. 5.8]. We mention that the
assumption of this result is naturally satisfied for the stochastic Runge-
Kutta integrators (6).

######  Proposition 3.4 ([57, 14])

There exists an algorithm that transforms
$a_{c}\in\operatorname{Char}(\mathcal{EF})$ over exotic forests into an
equivalent character $a_{c}\circ
A^{*}\in\operatorname{Char}(S(\mathcal{ET}))/{\sim}$ over the symmetric
algebra spanned by exotic trees. This defines a map $A(.)=.\circ
A^{*}\colon\operatorname{Char}(\mathcal{EF})\rightarrow\operatorname{Char}(S(\mathcal{ET}))/{\sim}$
such that $a_{c}\sim A(a_{c})$. Moreover, $A$ naturally induces a map
$A\colon\operatorname{Prim}(\mathcal{EF})^{*}\rightarrow\mathcal{ET}^{*}/{\sim}$
satisfying the identity $A|_{\mathcal{ET}^{*}}=id$.

The precise algorithm for computing $A$ is omitted for simplicity as it can be
found in [57, 14]. The operator $A$ is called IBP+ELI in [14] and it shares
similarities with the horizontal homotopy operator presented in [55] (see also
[5, 6]). We mention that the map $A$ extends to the manifold case (see [58]).

###### Remark 3.5

The output of $A$ is defined up to linear combinations of exotic trees that
vanish by integration by parts. The precise description of the kernel of the
integration by parts relation $\sim$ is an important open question for the
high order integration for the invariant measure. The first occurrence of a
non-trivial element in the kernel appears for order four. Here is one such
element:

$\displaystyle 26\forest{b[1,1,b[b]]}$
$\displaystyle-13\forest{b[1,1,2,2]}-5\forest{b[b[b[1]],1}-21\forest{b[b[b,1,1]]}+5\forest{b[b[b[1],1]]}-5\forest{b[b[1,1],b]}+10\forest{b[b[b,b]]}$
$\displaystyle+13\forest{b[b[1],1,2,2]}-13\forest{b[b[1,2,2],1]}-10\forest{b[b[b],b]}-5\forest{b[b[1],1,b]}+5\forest{b[b[b,1],1]}+13\forest{b[1,1,b[2,2]]}\sim
0$

###  3.2 The composition law in stochastic numerical analysis

The use of the composition rule of exotic aromatic S-series in stochastic
numerical analysis first appears in [57, 58, 53] (see also [32]), without the
Hopf algebra formalism.

######  Proposition 3.6 (Composition of integrators)

Consider two independent integrators $\Phi^{1}_{h}$ and $\Phi^{2}_{h}$ with
exotic S-series $S^{h}(a_{1})$ and $S^{h}(a_{2})$, then the composition of
$\Phi^{1}_{h}$ and $\Phi^{2}_{h}$ has the following S-series

$\mathbb{E}[\phi((\Phi^{2}_{h}\circ\Phi^{1}_{h})(x))]=S^{h}(a_{1}\ast
a_{2})[\phi](x).$

###### Example 3.7

Consider the explicit and implicit Euler methods in $\mathbb{T}^{d}$:

$\Phi^{1}_{h}(x)=x+hf(x)+\sqrt{h}\xi^{1},\quad\Phi^{2}_{h}(x)=x+hf(\Phi_{2}(x))+\sqrt{h}\xi^{2},$

with independent random variables $\xi_{1}$, $\xi_{2}$. Then, a calculation
yields that the composed method $\Phi$ has weak order 2 for solving equation
(3):

$\mathbb{E}[\phi(\Phi_{h}(x))]=S^{h}(\mathbf{1}+l+\frac{l^{\ast
2}}{2}+\dots)[\phi](x),\quad\Phi_{h}=\Phi^{2}_{h/2}\circ\Phi^{1}_{h/2}.$

The composed method $\Phi$ coincides in law with the stochastic trapezoidal
method

$\Phi_{h}(x)=x+h\frac{f(x)+f(\Phi_{h}(x))}{2}+\sqrt{h}\xi.$

In the spirit of effective order for ODEs [18], the postprocessing idea is
used in [90] for the high-order sampling of the invariant measure of (3) in
$\mathbb{T}^{d}$ (or $\mathbb{R}^{d}$). The approach with trees was then
introduced in [57, Sec. 5.3], where order conditions are presented for
stochastic Runge-Kutta methods and postprocessors. Consider an integrator
$\Phi$ for the invariant measure. After applying $\Phi$ through the entire
time interval, we apply a correction $\overline{\Phi}$ at the very last step.
If $\overline{\Phi}$ is chosen carefully, the postprocessed integrator has a
higher order than the original method $\Phi$. This methodology yields a
costless way to improve the order of a method.

######  Proposition 3.8 (Postprocessed integrators)

Consider an integrator with S-series $S^{h}(a)$ of order $p$ for the invariant
measure of (3), and a correction with S-series of the form

$S^{h}(\overline{a})=\mathbf{1}+h\alpha_{1}l+\dots+h^{p-1}\alpha_{p-1}l^{\ast
p-1}+\dots,$

for some constants $\alpha_{k}\in\mathbb{R}$. Then, the postprocessed method
has order $p+1$ for the invariant measure of (3) in $\mathbb{T}^{d}$ if the
following condition is satisfied:

$a+[l,\overline{a}]\sim\mathcal{EF}_{\geq p+1}^{*},\quad[a,b]:=a\ast b-b\ast
a.$

###### Example 3.9

The following scheme, first introduced in [59], has weak order one and order
two for the invariant measure,

$\Phi_{h}(x)=x+hf(x+\frac{\sqrt{h}}{2}\xi)+\sqrt{h}\xi,\quad\overline{\Phi}_{h}(x)=x+\frac{\sqrt{h}}{2}\xi.$

Further examples can be found in [90, 1, 57].

###  3.3 The substitution law in stochastic numerical analysis

The main applications of the substitution law of standard B-series are
backward error analysis and modified equations for ODEs. Adapting backward
error analysis and modified equations in the stochastic context is challenging
[88] and it is an active field of research [91, 2, 33, 3, 50, 51, 63, 39, 56].
It was in particular proven in [33, 3] that backward error analysis rewrites
nicely in the context of the invariant measure. We use the substitution law
and the integration by parts operation to show that the modified vector field
writes as an exotic S-series, for which we provide an explicit expression. For
simplicity, we work in $\mathbb{T}^{d}$ and discuss the manifold case at the
end of the subsection.

The goal of backward error analysis is to find a modified vector field written
formally as an exotic aromatic B-series,

$h\widetilde{f}=B^{h}(b)=hf+h^{2}f_{1}+h^{3}f_{2}+\dots,\quad
b\colon\mathcal{ET}\rightarrow\mathbb{R},\quad b(\bullet)=1,$

for some vector fields $f_{1}$, $f_{2}$, …that typically write as polynomials
in the coordinates of $f$ and its partial derivatives, such that the invariant
measure of the ergodic integrator (5) with S-series $S(a)$ coincides with the
invariant measure of the modified dynamics:

$d\widetilde{X}(t)=\Pi_{\mathcal{M}}(\widetilde{X}(t))\widetilde{f}(\widetilde{X}(t))dt+\Pi_{\mathcal{M}}(\widetilde{X}(t))\circ
dW(t),\quad\widetilde{X}(0)=\widetilde{X}_{0}\in\mathcal{M}.$ (9)

The coefficient map $b$ is the solution to the substitution $b_{c}\star e\sim
a$ where $\star$ is the substitution law (see Theorem 2.8).

It is known [33, 3] that there exists a modified vector field for large
classes of methods, such as stochastic Runge-Kutta methods. The calculations
are tedious and were rewritten with exotic series in [57, 14]. There is,
however, no proof that the calculations can be carried out up to any order in
these works as there is no reason in general why the modified vector field
could be written as an exotic B-series. A geometric justification of the
importance of writing the modified vector field as an exotic aromatic B-series
is given in [56]: it enforces that $\widetilde{f}$ is invariant with respect
to orthogonal changes of coordinates, which is a natural property in the
stochastic context. We provide here a simple and natural algebraic criterion,
satisfied by large classes of methods, for the description of integrators that
have a modified vector field in the form of an exotic B-series. In addition,
we give the first explicit expression of the modified vector field
$\widetilde{f}$ relying on the map $A$ defined in Proposition 3.4. This shows
in particular that exotic series are a powerful tool for the stochastic
backward error analysis.

######  Theorem 3.10 (Backward error analysis)

Consider a consistent method with the exotic S-series $S^{h}(a)$ for solving
equation (3) with $f=-\nabla V$. Assume that $a$ is a character of
$(\mathcal{EF},\cdot)$. Then, there exists a modified vector field
$h\widetilde{f}=B^{h}(b)$ that can be written as an exotic B-series with a
coefficient map $b\colon\mathcal{ET}\rightarrow\mathbb{R}$ satisfying
$b_{c}\star e\sim a$, and given by

$b=\delta_{\bullet}+A\Big{(}\sum_{k=0}^{\infty}(-1)^{k}A_{\tilde{\star}e}^{k}(a-e)\Big{)}\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}},$

where $A_{\tilde{\star}e}:\mathcal{EF}^{*}\to\mathcal{EF}^{*}$ satisfies
$A_{\tilde{\star}e}(x)=A(x)\tilde{\star}e$ and

$a\tilde{\star}e=(a\otimes
e)\circ\tilde{\Delta}_{CEM},\quad\text{and}\quad\tilde{\Delta}_{CEM}(\pi)=\Delta_{CEM}(\pi)-\bullet\otimes\pi-\pi\otimes\bullet,$

for $a\in\mathcal{EF}^{*}$, $\pi\in EF$ such that $|\pi|>1$.

  * Proof

After initializing $b_{0}=\delta_{\bullet}$, we construct recursively the
coefficient map sequence $(b_{n})$ by

$b_{n}=b_{n-1}+A(a-b_{n-1,c}\star
e)\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}},\quad b_{n-1,c}\star
e=(b_{n-1,c}\otimes e)\circ\Delta_{CEM},$

with $b_{n-1,c}:=\exp^{\odot}(b_{n-1})$ and the coproduct
$\Delta_{CEM}:\mathcal{EF}\to S(\mathcal{ET})\otimes\mathcal{EF}$. Assume
$b_{n-1}(\bullet)=1$, since $A^{*}(\bullet)=\bullet-\forest{1,1}$ and the
method associated to $a$ is consistent, we find

$b_{n}(\bullet)=b_{n-1}(\bullet)+(a-b_{n-1,c}\star
e)(\bullet-\forest{1,1})=1+a(\bullet)-a(\forest{1,1})-e(\bullet)+e(\forest{1,1})=1,$

thus we obtain that $b_{n}(\bullet)=1$ for any $n$ by induction. For all
$\tau\in\mathcal{ET}$ such that $|\tau|>1$, using the reduced coproduct yields

$\displaystyle b_{n}(\tau)$
$\displaystyle=b_{n-1}(\tau)+A(a)(\tau)-(b_{n-1}\otimes
e)(A^{*}(\tau)\otimes\bullet)$ $\displaystyle-(b_{n-1}\otimes
e)(\bullet\otimes A^{*}(\tau))-A(b_{n-1,c}\tilde{\star}e)(\tau)$
$\displaystyle=b_{n-1}(\tau)-A(b_{n-1})(\tau)+A(a-e-
b_{n-1,c}\tilde{\star}e)(\tau)$
$\displaystyle=A(a-e+b_{n-1,c}\tilde{\star}e)(\tau),$

where we recall that the only exotic tree $\tau$ for which $|\tau|\leq 1$ is
$\tau=\bullet$ and we used $A(b_{n-1})=b_{n-1}$. The first values are

$\displaystyle b_{1}$
$\displaystyle=\delta_{\bullet}+A(a-e)\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}},$
$\displaystyle b_{2}$
$\displaystyle=\delta_{\bullet}+A(a-e)\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}}-A(A(a-e)\tilde{\star}e)\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}},$
$\displaystyle\vdots$ $\displaystyle b_{n}$
$\displaystyle=\delta_{\bullet}+A\big{(}\sum_{k=0}^{n-1}(-1)^{k}A_{\tilde{\star}e}^{k}(a-e)\big{)}\raisebox{-2.15277pt}{$|$}_{\mathcal{ET}},\quad\text{where
}A_{\tilde{\star}e}(x)=A(x)\tilde{\star}e.$

Since $|A^{*}(\tau)|_{e}\leq|\tau|_{e}$ where $|\tau|_{e}$ is the number of
edges of $\tau$, and $\tilde{\Delta}_{CEM}^{n}(\tau)=0$ if $n\geq|\tau|_{e}$,
we have for all $n\geq|\tau|_{e}$,

$b_{n}(\tau)=\delta_{\bullet}+A\big{(}\sum_{k=0}^{|\tau|_{e}-1}(-1)^{k}A_{\tilde{\star}e}^{k}(a-e)\big{)}(\tau),$

so that the sequence $(b_{n})$ converges to the desired coefficient map $b$ by
stationarity. $\square$

###### Example 3.11

The first terms of the modified vector field $h\widetilde{f}=B^{h}(b)$ given
by Theorem 3.10 for the Euler-Maruyama method (7) are

$B^{h}(b)=h\forest{b}+\frac{h^{2}}{2}\forest{b[b]}+\frac{h^{2}}{4}\forest{b[1,1]}-\frac{h^{2}}{2}\forest{b[b[b]]}+\frac{h^{2}}{12}\forest{b[b,b]}-\frac{h^{2}}{4}\forest{b[b[1,1]]}-\frac{h^{2}}{12}\forest{b[b[1],1]}+\frac{h^{2}}{12}\forest{b[b,1,1]}+\frac{h^{2}}{12}\forest{b[1,1,2,2]}+\dots$

Note that removing the stochastic terms (that are, the trees with lianas) does
not yield the modified vector field for the Euler method with the standard
deterministic backward error analysis (see [41, Chap. IX]). Indeed, high order
for the invariant measure does not imply high order in the weak or strong
sense [3], hence the modified equations and order conditions in the
deterministic sense or weak sense are not the same as for the invariant
measure sampling, as highlighted in the works [3, 4, 57, 58, 87].

###### Remark 3.12

Similarly to the deterministic context [41, Chap. IX], the properties of the
scheme are observed directly on its associated modified vector field. In
particular, if $h\widetilde{f}=B^{h}(b)$ with a coefficient map $b$ satisfying
$b-\delta_{\bullet}\sim\mathcal{ET}_{>p}^{*}$, then the integrator has at
least order $p$ for the invariant measure. This gives an equivalent definition
of the order for the invariant measure based on the modified vector field, as
observed in [87, 4]. In particular, a method is exact for the invariant
measure if its modified vector field satisfies

$\operatorname{div}(\widetilde{f}-f)+\langle f,\widetilde{f}-f\rangle=0.$ (10)

Equation (10) is analogous to the one satisfied by the modified vector field
of volume-preserving aromatic B-series methods for ODEs, whose understanding
is already an important unsolved problem of deterministic geometric numerical
integration (see, for instance, the works [27, 47, 76, 11, 12, 55]).

###### Remark 3.13

Following [33], the exotic B-series $h\widetilde{f}=B^{h}(b)$ translates into
a formal expansion of the invariant measure
$d\mu^{h}=\rho^{h}d\sigma_{\mathcal{M}}$ of the integrator by solving:

$\rho^{h}=\rho_{\infty}+h\rho_{1}+h^{2}\rho_{2}+\dots,\quad\widetilde{\mathcal{L}}^{*}\rho^{h}=0,\quad\widetilde{\mathcal{L}}=\mathcal{L}+(\widetilde{f}-f)\nabla.$

For the first few terms, we find (see [33, 3] for the detailed expansion),

$\mathcal{L}^{*}\rho_{1}=-\operatorname{div}(f_{1}\rho_{\infty}),\quad\mathcal{L}^{*}\rho_{2}=-\operatorname{div}(f_{1}\rho_{1})-\operatorname{div}(f_{2}\rho_{\infty}).$

Consider now a consistent exotic aromatic B-series method with S-series
$S(a)$. Similar to backward error analysis, we are interested in finding a
modified vector field $h\widetilde{f}=B^{h}(b)$ with $b\in\mathcal{ET}^{*}$
and $b(\bullet)=1$ such that $b_{c}\star a\sim\delta_{\mathbf{1}},$ that is,
the integrator applied to the modified equation (9) is exact. This technique
allows in particular to increase the order of a numerical method when the
partial derivatives of $f$ are not costly to evaluate (see, for instance, in
the deterministic setting [23, 41, 25]). A general expansion of the modified
vector field is presented in the $\mathbb{T}^{d}$ case in [3, 57], but it is
not an exotic B-series in general and it is not unique. Following [14], we
propose a simple criterion to obtain the existence of a modified vector field
in the form of an exotic B-series in the context of $\mathbb{T}^{d}$, for
which we also provide an explicit expression.

######  Theorem 3.14 (Modified equations)

Consider a consistent method with the exotic S-series $S^{h}(a)$ for solving
equation (3) with $f=-\nabla V$. Assume that $a$ is a character of
$(\mathcal{EF},\cdot)$. Then, there exists a modified vector field
$h\widetilde{f}=B^{h}(b)$ that can be written as an exotic B-series with the
coefficient map $b\colon\mathcal{ET}\rightarrow\mathbb{R}$ satisfying
$b(\bullet)=1$, $b_{c}\star a\sim\delta_{\mathbf{1}}$, and given by

$b=\delta_{\bullet}-A\big{(}\sum_{k=0}^{\infty}(-1)^{k}A_{\tilde{\star}a}^{k}(a)\big{)}.$

To prove Theorem 3.14, we use a similar reasoning as in Theorem 3.10.

  * Proof

We introduce the sequence (in the spirit of the works [3, 57])

$b_{n}=b_{n-1}-A(b_{n-1,c}\star a),\quad b_{0}=\delta_{\bullet}.$

Similarly to the proof of Theorem 3.10, we show by induction that
$b_{n}(\bullet)=1$ and we obtain the identity
$b_{n}=-A(a+b_{n-1,c}\tilde{\star}a)$ using the reduced coproduct. The
sequence $(b_{n})$ takes the following first values

$\displaystyle b_{1}$ $\displaystyle=\delta_{\bullet}-A(a),$ $\displaystyle
b_{2}$ $\displaystyle=\delta_{\bullet}-A(a)+A(A(a)\tilde{\star}a),$
$\displaystyle\vdots$ $\displaystyle b_{n}$
$\displaystyle=\delta_{\bullet}-A\big{(}\sum_{k=0}^{n-1}(-1)^{k}A_{\tilde{\star}a}^{k}(a)\big{)},\quad\text{where
}A_{\tilde{\star}a}(x)=A(x)\tilde{\star}a,$

and $(b_{n})$ converges to
$b=\delta_{\bullet}-A\big{(}\sum_{k=0}^{\infty}(-1)^{k}A_{\tilde{\star}a}^{k}(a)\big{)}$
by stationarity. $\square$

Any stochastic Runge-Kutta method (6) has a coefficient map that is a
character, so that Theorem 3.14 applies and there exists a modified vector
field that can be written as an exotic B-series. We refer to [57, Sec. 5.1]
for examples in $\mathbb{T}^{d}$. In the manifold case, the most popular
integrators for solving constrained SDEs in the weak sense or for sampling the
invariant measure are projection methods (see, for instance, the textbook [62,
Chap. 3]). In [58], a new class of Runge-Kutta projection methods was
introduced for the high-order approximation of constrained overdamped Langevin
processes in codimension one. Theorems 3.10 and 3.14 do not extend
straightforwardly to the manifold case as there does not exist an equivalent
of Proposition 3.4 with general exotic aromatic S-series. However, one can
adapt the algorithm described in the proof of Theorem 3.14, taking over the
integration by parts relation $\sim$ described in [58]. Define the projection
operator $\Pi\colon\mathcal{EAT}\rightarrow\mathcal{EAT}$ by
$\Pi\tau=\tau-\tau\blacktriangleright_{\bullet}(\forest{w=b,w})$, with
$\blacktriangleright_{\bullet}$ the decorated insertion product (see Section
4.3), and its dual $\Pi^{*}$. Define $b_{0}=\delta_{\bullet}$. For the
induction, at the step $n$, if $A(b_{n-1,c}\star a)$ is well-defined and if
$\Pi^{*}A(b_{n-1,c}\star a)=A(b_{n-1,c}\star a)$, compute

$b_{n}=b_{n-1}-A(b_{n-1,c}\star a).$

Then, applying the integrator to the equation (9) with the modified vector
field $B_{h,f}(b_{n})$ yields a method of order $n+1$ for the invariant
measure of the original problem (3).

###### Example 3.15

For the sake of simplicity, we consider the unit sphere
$\mathcal{M}=\\{x\in\mathbb{R}^{d},\left|x\right|^{2}=1\\}$, with $n(x)=x$.
The constrained Euler scheme with implicit projection direction:

$X_{n+1}=X_{n}+hf(X_{n+1})+\sqrt{h}\xi_{n}+\lambda_{n}X_{n+1},\quad\left|X_{n+1}\right|=1.$

has order 2 for sampling the invariant measure of (3) when applied to the
modified equation (9) with the modified vector field

$\displaystyle\widetilde{f}$
$\displaystyle=f+h\bigg{[}-\frac{1}{2}f^{\prime}f-\frac{1}{4}\Delta
f+\frac{3}{4}f-\frac{1}{4}\operatorname{div}(n)f-\frac{1}{2}\langle n,f\rangle
f-\frac{1}{4}f^{\prime}n$
$\displaystyle-\frac{1}{4}\operatorname{div}(n)f^{\prime}n-\frac{1}{2}\langle
n,f\rangle f^{\prime}n-\frac{1}{4}f^{\prime\prime}(n,n)+\frac{1}{2}\langle
n,f\rangle\langle n,f^{\prime}n\rangle n$ $\displaystyle+\frac{1}{4}\langle
n,f^{\prime\prime}(n,n)\rangle
n-\frac{1}{4}\operatorname{div}(f)^{\prime}nn-\frac{1}{2}\langle
n,f^{\prime}f\rangle n+\frac{1}{4}\langle n,f^{\prime}n\rangle n$
$\displaystyle+\frac{1}{4}\operatorname{div}(n)(n,f^{\prime}n\rangle
n+\frac{1}{2}\langle n,f\rangle^{2}n-\frac{3}{4}\langle n,f\rangle
n+\frac{1}{4}\operatorname{div}(n)\langle n,f\rangle n\bigg{]}.$

##  4 Algebraic structure of decorated and exotic aromatic forests

This section is devoted to a detailed study of the algebraic structures of
decorated aromatic and exotic aromatic forests, which are used to describe the
corresponding substitution laws presented in Theorems 2.8 and 4.19. Section
4.1 defines a free D-algebra [77] over decorated aromatic forests and
describes the corresponding Grossman-Larson Hopf algebroid and pre-Hopf
algebroid, generalizing the Grossman-Larson Hopf algebra and pre-Hopf algebra
[64] structures. The Grossman-Larson Hopf algebroid is closely related to the
composition of differential operators and, consequently, to the composition
law of S-series.

Section 4.2 focuses on the algebraic structure of decorated clumped forests as
well as the relation between clumped and aromatic forests. The introduction of
decorated clumped forests is necessary due to the fact that the substitution
law is described using a homomorphism with respect to a product which is not
$\mathcal{A}_{D}$-bilinear, and therefore, we need to attach aromas to rooted
components. Section 4.3 introduces the substitution law for decorated aromatic
forests, while Section 4.4 details the algebraic results presented in this
section in the context of exotic aromatic forests.

###  4.1 D-algebra of decorated aromatic forests

In this section, we introduce algebraic structures over decorated aromatic
forests that are relevant in our description of the substitution law. Let
$\curvearrowright\,:\mathcal{AT}_{D}\otimes\mathcal{AT}_{D}\to\mathcal{AT}_{D}$
denote the grafting product over decorated aromatic trees. The grafting
product $\tau\curvearrowright\gamma$ is the sum over all ways to attach the
root of $\tau$ to a vertex of $\gamma$, for example,

$\forest{(b),b[b]}\curvearrowright\forest{(b,b),b}=2\,\forest{(b),(b,b[b[b]]),b}+\forest{(b),(b,b),b[b[b]]}.$

Divergence of an aromatic tree is defined to be a map
$d:\mathcal{AT}_{D}\to\mathcal{A}_{D}$ such that $d(\tau)$ is a sum over all
ways to attach the root of $\tau$ to one of its vertices, for example,

$d(\forest{(b),b[b,b]})=\forest{(b[b[b,b]])}+\forest{(b),(b[b,b])}+2\forest{(b),(b,b[b])}.$

Let $El_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$ be the algebra of endomorphisms
${-}\curvearrowright\tau:\mathcal{AT}_{D}\to\mathcal{AT}_{D}$. The pair
$(\mathcal{AT}_{D},\mathcal{A}_{D})$ is the tracial pre-Lie-Rinehart algebra
generated by the set $D$, that is, it satisfies the following properties:

  1. 1.

$\mathcal{A}_{D}$ is a unital commutative algebra with concatenation product
$\cdot:\mathcal{A}_{D}\otimes\mathcal{A}_{D}\to\mathcal{A}_{D}$,

  2. 2.

$\mathcal{AT}_{D}$ is an $\mathcal{A}_{D}$-module with a pre-Lie product
$\curvearrowright\,:\mathcal{AT}_{D}\otimes\mathcal{AT}_{D}\to\mathcal{AT}_{D}$,
that is,

$\tau\curvearrowright(\gamma\curvearrowright\nu)-(\tau\curvearrowright\gamma)\curvearrowright\nu=\gamma\curvearrowright(\tau\curvearrowright\nu)-(\gamma\curvearrowright\tau)\curvearrowright\nu,$

for
$\tau,\gamma\in\mathcal{AT}_{D},\nu\in\mathcal{AT}_{D}\sqcup\mathcal{A}_{D}$,

  3. 3.

for any $\tau\in\mathcal{AT}_{D}$, the map
$\tau\curvearrowright{-}:\mathcal{A}_{D}\to\mathcal{A}_{D}$ is a derivation
and the Leibniz rule holds,

$\tau\curvearrowright(\omega\cdot\nu)=(\tau\curvearrowright\omega)\cdot\nu+\omega\cdot(\tau\curvearrowright\nu),$

for $\omega\in\mathcal{A}_{D}$ and
$\nu\in\mathcal{A}_{D}\sqcup\mathcal{AT}_{D}$.

  4. 4.

there exists a map
$t:El_{\mathcal{A}_{D}}(\mathcal{AT}_{D})\to\mathcal{A}_{D}$ called a trace
that satisfies $t(\tau\curvearrowright\tilde{\gamma})=\tau\curvearrowright
t(\tilde{\gamma})$ and
$t(\tilde{\tau}\circ\tilde{\gamma})=t(\tilde{\gamma}\circ\tilde{\tau})$ with
$\tau\in\mathcal{AT}_{D},\tilde{\tau},\tilde{\gamma}\in
El_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$ and $\circ$ the composition of
endomorphisms. The divergence is then defined as
$d(\tau):=t({-}\curvearrowright\tau)$.

More details can be found in [36] where it is proven that
$(\mathcal{AT}_{D},\mathcal{A}_{D})$ is a free tracial pre-Lie-Rinehart
algebra. We extend the structure of the tracial pre-Lie-Rinehart algebra by
considering a symmetric $\mathcal{A}_{D}$-bilinear form
$\langle{-},{-}\rangle:\mathcal{AT}_{D}\otimes\mathcal{AT}_{D}\to\mathcal{A}_{D}$
with the Leibniz rule:

$\tau\curvearrowright\langle\gamma,\nu\rangle=\langle\tau\curvearrowright\gamma,\nu\rangle+\langle\gamma,\tau\curvearrowright\nu\rangle,\quad\text{for
}\tau,\gamma,\nu\in\mathcal{AT}_{D}.$

The aroma $\langle\gamma,\nu\rangle\in\mathcal{A}_{D}$ is called a stolon and
is denoted by a horizontal double edge that connects the roots of the
corresponding trees, for example,
$\langle\forest{(b),b},\forest{b[b]}\rangle=\forest{(b),b=b[b]}$.

###### Proposition 4.1

The tracial stolonic pre-Lie-Rinehart algebra
$(\mathcal{AT}_{D},\mathcal{A}_{D})$ is free.

The classical Guin-Oudom process [79] extends uniquely a pre-Lie product over
a vector space $V$ to a product over the symmetric algebra $S(V)$. We
generalize and use this process to extend uniquely the pre-Lie-Rinehart
product $\curvearrowright$ over a $\mathcal{A}_{D}$-module $\mathcal{AT}_{D}$
to the symmetric algebra
$\mathcal{AF}_{D}:=S_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$ of decorated
aromatic forests.

###### Proposition 4.2

There exists a unique extension of the $\curvearrowright$ product to
$\mathcal{AF}_{D}$ such that

  1. (i)

$\omega\mathbf{1}\curvearrowright\nu=\omega\nu,\quad$ for
$\omega\in\mathcal{A}_{D},\nu\in\mathcal{A}_{D}\sqcup\mathcal{AT}_{D}$,

  2. (ii)

$(\tau\cdot\pi)\curvearrowright\nu=\tau\curvearrowright(\pi\curvearrowright\nu)-(\tau\curvearrowright\pi)\curvearrowright\nu,\quad$
for $\tau\in\mathcal{AT}_{D},\pi\in\mathcal{AF}_{D}$,

  3. (iii)

$\pi\curvearrowright(\mu_{1}\cdot\mu_{2})=\sum_{(\pi)}(\pi_{(1)}\curvearrowright\mu_{1})\cdot(\pi_{(2)}\curvearrowright\mu_{2}),\quad$
for $\mu_{1},\mu_{2}\in\mathcal{AF}_{D}$,

with deshuffle coproduct
$\Delta_{\mathcal{A}_{D}}(\pi)=\sum_{(\pi)}\pi_{(1)}\otimes_{\mathcal{A}_{D}}\pi_{(2)}$.

We follow the structure of the proof of Proposition 2.7 of [79] which proves
an analogous statement for $S(\mathcal{AT}_{D})$. We check that the relations
$(i),(ii),(iii)$ are well-defined over
$S_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$.

  * Proof

It follows from $(i)$, $(iii)$, and the coassociativity of
$\Delta_{\mathcal{A}_{D}}$ that

$\tau\curvearrowright\mathbf{1}=0,\quad\text{and}\quad\tau\curvearrowright(\pi_{1}\cdots\pi_{n})=\sum_{k=1}^{n}\pi_{1}\cdots(\tau\curvearrowright\pi_{k})\cdots\pi_{n}.$

The relation $(ii)$ is well-defined with respect to the choice of $\tau$ using
the Lemma 2.5 of [79] which is based on induction on the length of the
monomial and the pre-Lie relation. This means that
$(\tau\cdot\pi)\curvearrowright\nu$ is well-defined for $\tau\cdot\pi\in
S(\mathcal{AT}_{D})$ using $(i)$ and $(ii)$. Let $\mathcal{J}$ be an ideal of
$S(\mathcal{AT}_{D})$

$\mathcal{J}:=\langle(\omega\tau\cdot\gamma-\tau\cdot\omega\gamma)\cdot\pi:\omega\in\mathcal{A}_{D},\tau,\gamma\in\mathcal{AT}_{D},\pi\in
S(\mathcal{AT}_{D})\rangle.$

It remains to show that $\mathcal{J}\curvearrowright\nu=0$ which follows from
the property
$(\omega\tau\cdot\pi)\curvearrowright\nu=\omega(\tau\cdot\pi\curvearrowright\nu)$,
with $\omega\in\mathcal{A}_{D}$, proved by induction on the length of the
monomial $\pi$. The initial step is shown below for
$\gamma\in\mathcal{AT}_{D}$ using the $\mathcal{A}_{D}$-linearity in the left
operand of $\curvearrowright$:

$(\omega\tau\cdot\gamma)\curvearrowright\nu=\omega\tau\curvearrowright(\gamma\curvearrowright\nu)-(\omega\tau\curvearrowright\gamma)\curvearrowright\nu=\omega(\tau\cdot\gamma\curvearrowright\nu).$

Assume the property to be true for monomials shorter than
$\pi=\tau_{1}\cdots\tau_{n}\in S(\mathcal{AT}_{D})$ and recall that

$\omega\tau\curvearrowright\pi=\sum_{k=1}^{n}\omega(\tau\curvearrowright\tau_{k})\cdot\pi_{\hat{k}},\quad\text{where
}\pi_{\hat{k}}:=\tau_{1}\cdots\tau_{k-1}\tau_{k+1}\cdots\tau_{n}.$

Then, by induction, we have,

$\displaystyle(\omega\tau\curvearrowright\pi)\curvearrowright\nu$
$\displaystyle=\big{(}\sum_{k=1}^{n}\omega(\tau\curvearrowright\tau_{k})\cdot\pi_{\hat{k}}\big{)}\curvearrowright\nu$
$\displaystyle=\omega\big{(}\sum_{k=1}^{n}(\tau\curvearrowright\tau_{k})\cdot\pi_{\hat{k}}\big{)}\curvearrowright\nu$
$\displaystyle=\omega((\tau\curvearrowright\pi)\curvearrowright\nu).$

This allows us to prove the inductive step:

$(\omega\tau\cdot\pi)\curvearrowright\nu=\omega\tau\curvearrowright(\pi\curvearrowright\nu)-(\omega\tau\curvearrowright\pi)\curvearrowright\nu=\omega(\tau\cdot\pi\curvearrowright\nu).$

Therefore, relations $(i)$ and $(ii)$ extend $\curvearrowright$ to
$S_{\mathcal{A}_{D}}(\mathcal{AT}_{D})\otimes(\mathcal{A}_{D}\oplus\mathcal{AT}_{D})\to(\mathcal{A}_{D}\oplus\mathcal{AT}_{D})$.
Due to the Leibniz rule, the property
$(\omega\pi)\curvearrowright\nu=\omega(\pi\curvearrowright\nu)$, the
cocommutativity and coassociativity of $\Delta_{\mathcal{A}_{D}}$, the
relation $(iii)$ is well-defined. Therefore, it defines $\curvearrowright$ on
$\mathcal{AF}_{D}=S_{\mathcal{A}_{D}}(\mathcal{AT}_{D})$. $\square$

Following the D-algebra structure from [77], see also [68, 69, 70], we define
the tracial commutative D-algebra $(\mathcal{AF}_{D},\curvearrowright)$ of
decorated aromatic forests graded by the number of roots.

###### Definition 4.3

Let $(A,\cdot)$ be a unital commutative graded algebra with unit $\mathbf{1}$.
Let $A$ be equipped with a non-associative product $\curvearrowright$ and let
$x\in A_{1}$ and $a,b\in A$ satisfy the relation

$x\curvearrowright(a\cdot b)=(x\curvearrowright a)\cdot
b+a\cdot(x\curvearrowright b).$

The triple $(A,\cdot,\curvearrowright)$ is a commutative D-algebra if the
following identities are satisfied

$\displaystyle\mathbf{1}\curvearrowright a$ $\displaystyle=a,$ $\displaystyle
a\curvearrowright x$ $\displaystyle\in A_{1},$ $\displaystyle(\omega\cdot
a)\curvearrowright b$ $\displaystyle=\omega\cdot(a\curvearrowright b),$
$\displaystyle(x\cdot a)\curvearrowright b$
$\displaystyle=x\curvearrowright(a\curvearrowright b)-(x\curvearrowright
a)\curvearrowright b,$

for $\omega\in A_{0},\;x\in A_{1},\;a,b\in A$. It is called tracial if there
exists a trace $t:El_{A_{0}}(A_{1})\to A_{0}$.

For a tracial commutative D-algebra $A$, the pair $(A_{1},A_{0})$ is a tracial
pre-Lie-Rinehart algebra. This implies that
$(\mathcal{AF}_{D},\curvearrowright)$ is the free tracial commutative
D-algebra since $(\mathcal{AT}_{D},\mathcal{A}_{D})$ is the free tracial pre-
Lie-Rinehart algebra and $(\mathcal{AF}_{D},\curvearrowright)$ is obtained
uniquely using the Guin-Oudom process (Proposition 4.2). We extend the
structure of the D-algebra $(\mathcal{AF}_{D},\curvearrowright)$ with the
$\mathcal{A}_{D}$-bilinear form
$\langle{-},{-}\rangle:\mathcal{AT}_{D}\otimes\mathcal{AT}_{D}\to\mathcal{A}_{D}$
and see that it remains free.

A map $\varphi:A\to A^{\prime}$ between two tracial commutative D-algebras $A$
and $A^{\prime}$ is a D-algebra morphism if $\varphi(A_{1})\subset
A^{\prime}_{1}$ and

$\displaystyle\varphi(a\cdot b)=\varphi(a)\cdot\varphi(b),$
$\displaystyle\quad\varphi(a\curvearrowright
b)=\varphi(a)\curvearrowright\varphi(b),$ $\displaystyle\varphi(\langle
x,y\rangle)=\langle\varphi(x),\varphi(y)\rangle,$
$\displaystyle\quad\varphi(t(\tilde{x}))=t(\varphi(\tilde{x})),$

for $a,b\in A$, $x,y\in A_{1}$, $\tilde{x}\in El_{A_{0}}(A_{1})$.

###### Example 4.4

Let $\mathcal{X}$ be the space of vector fields
$\mathbb{R}^{d}\to\mathbb{R}^{d}$. The symmetric algebra
$S_{\mathcal{C}^{\infty}(\mathbb{R}^{d})}(\mathcal{X})$ over the ring of
$\mathcal{C}^{\infty}(\mathbb{R}^{d})$ maps is a commutative tracial D-algebra
and represents the space of differential operators in $\mathbb{R}^{d}$. The
non-associative product is given by the covariant derivation, for example, let
$f,g,h:\mathbb{R}^{d}\to\mathbb{R}^{d}$, then,

$(fg)[h]=\sum_{i,j=1}^{d}f^{i}g^{j}h_{ij},\quad\text{where
}h_{ij}:=\frac{\partial^{2}h}{\partial x_{i}\partial x_{j}}.$

Divergence of a vector field is $d(f)=\sum_{i=1}^{d}f^{i}_{i}$ and the
bilinear product is the inner product, i.e., $\langle
f,g\rangle=\sum_{i=1}^{d}f^{i}g^{i}$.

We note that the algebra of endomorphisms generated by
$\pi\curvearrowright{-}:\mathcal{AF}_{D}\to\mathcal{AF}_{D}$ for
$\pi\in\mathcal{AF}_{D}$ is isomorphic to the algebra
$(\mathcal{AF}_{D},\diamond)$ with $\diamond$ denoting the Grossman-Larson
product defined as

$\pi_{1}\curvearrowright(\pi_{2}\curvearrowright{-})=(\pi_{1}\diamond\pi_{2})\curvearrowright{-},\quad\pi_{1},\pi_{2}\in\mathcal{AF}_{D}.$

Let us obtain a Grossman-Larson $\mathcal{A}/_{\mathbb{R}}$-bialgebra [75]
from the commutative D-algebra $\mathcal{AF}_{D}$.

###### Proposition 4.5

Let $\epsilon_{\mathcal{A}_{D}}:\mathcal{AF}_{D}\to\mathcal{A}_{D}$ be a
$\mathcal{A}_{D}$-linear map such that
$\epsilon_{\mathcal{A}_{D}}(\mathbf{1})=\mathbf{1}$ and
$\epsilon_{\mathcal{A}_{D}}(\pi)=0$ for $\pi\notin\mathcal{AF}_{D,0}$, then
$B_{GL}:=(\mathcal{AF}_{D},\diamond,\mathbf{1},\Delta_{\mathcal{A}_{D}},\epsilon_{\mathcal{A}_{D}})$
is a $\mathcal{A}/_{\mathbb{R}}$-bialgebra [75] called Grossman-Larson
$\mathcal{A}/_{\mathbb{R}}$-bialgebra. That is, it satisfies the following
properties:

  * (1)

$\epsilon_{\mathcal{A}_{D}}(\mathbf{1})=\mathbf{1}$,  (2)
$\Delta_{\mathcal{A}_{D}}(\mathbf{1})=\mathbf{1}\otimes_{\mathcal{A}_{D}}\mathbf{1}$,

  * (3)

$\epsilon_{\mathcal{A}_{D}}(\pi\diamond\mu)=\epsilon_{\mathcal{A}_{D}}(\pi\diamond\epsilon_{\mathcal{A}_{D}}(\mu))$,
(4)
$\Delta_{\mathcal{A}_{D}}(\pi\diamond\mu)=\Delta_{\mathcal{A}_{D}}(\pi)\diamond_{\otimes}\Delta_{\mathcal{A}_{D}}(\mu)$,

with
$(\pi_{(1)}\otimes_{\mathcal{A}_{D}}\pi_{(2)})\diamond_{\otimes}(\mu_{(1)}\otimes_{\mathcal{A}_{D}}\mu_{(2)})=(\pi_{(1)}\diamond\mu_{(1)})\otimes_{\mathcal{A}_{D}}(\pi_{(2)}\diamond\mu_{(2)})~{}$.

  * Proof

Since Grossman-Larson product is isomorphic to the composition of maps, it is
associative. The properties $(1)$ and $(2)$ follow from the definition of the
counit and coproduct. Property $(4)$ can be proven using the definition of the
Grossman-Larson product together with the relation $(iii)$ from Proposition
4.2. To prove the property $(3)$, we note that
$\epsilon_{\mathcal{A}_{D}}(\pi_{1}\diamond\pi_{2})$ is non-zero if and only
if $\pi_{2}\in\mathcal{A}_{D}$ and is equal to the aromas obtained by grafting
all trees of $\pi_{1}$ onto $\pi_{2}$ in all possible ways. $\square$

Proposition 4.5 can be applied to a general commutative D-algebra if the
associativity of the Grossman-Larson product is proven analogously to [79,
Lemma 2.10]. We note that if we exclude all aromas, i.e.
$\mathcal{A}_{D}=\\{\mathbf{1}\\}$, the $\mathcal{A}/_{\mathbb{R}}$-bialgebra
$B_{GL}$ becomes a graded connected bialgebra which is also a Hopf algebra. We
also note that the product $\diamond_{\otimes}$ is well-defined over
$\Delta_{\mathcal{A}_{D}}(\mathcal{AF}_{D})\subset\mathcal{AF}_{D}\otimes_{\mathcal{A}_{D}}\mathcal{AF}_{D}$,
but not over $\mathcal{AF}_{D}\otimes_{\mathcal{A}_{D}}\mathcal{AF}_{D}$ since
$\diamond$ is not $\mathcal{A}_{D}$-linear in its right operand.

###### Remark 4.6

We say that $B_{GL}$ is cocomplete since
$B_{GL}=\bigcup_{n=0}^{\infty}B_{GL,n}$ and is a free
$\mathcal{A}_{D}$-module. An analog of the Cartier-Milnor-Moore theorem is
proven in [75] which states that a cocomplete and graded projective
$\mathcal{A}/_{\mathbb{R}}$-bialgebra is the universal enveloping Lie-Rinehart
algebra of the Lie-Rinehart algebra of its primitive elements. The freeness of
the $\mathcal{A}_{D}$-module $B_{GL}$ implies its graded projectiveness,
therefore, the $\mathcal{A}/_{\mathbb{R}}$-bialgebra $B_{GL}$ is the universal
enveloping Lie-Rinehart algebra of the pre-Lie-Rinehart algebra
$(\mathcal{AT}_{D},\mathcal{A}_{D})$.

###### Remark 4.7

If we replace the coalgebra structure of $B_{GL}$ by the deshuffle coproduct
$\Delta:\mathcal{AF}_{D}\to\mathcal{AF}_{D}\otimes\mathcal{AF}_{D}$ and
$\epsilon:\mathcal{AF}_{D}\to\mathbb{R}$, then we get a Grossman-Larson Hopf
algebra dual (up to the symmetry coefficients) to the Hopf algebra mentioned
in [11, Thm. 4.4].

$\mathcal{A}/_{\mathbb{R}}$-bialgebras are also called cocommutative
bialgebroids. We show that $B_{GL}$ defines a Hopf algebroid as introduced in
[67].

###### Proposition 4.8

Let $S_{\diamond}:(\mathcal{AF}_{D},\diamond)\to(\mathcal{AF}_{D},\diamond)$
be the algebra anti-isomorphism defined as

  * (i)

$S_{\diamond}(\omega):=\omega,\quad
S_{\diamond}(\omega\pi):=S_{\diamond}(\pi)\diamond\omega,\quad$ for
$\omega\in\mathcal{A}_{D},\pi\in\mathcal{AF}_{D}$,

  * (ii)

$S_{\diamond}(\tau):=-\tau,\quad
S_{\diamond}(\tau\pi):=-S_{\diamond}(\pi)\diamond\tau-
S_{\diamond}(\tau\curvearrowright\pi),\quad$ for $\tau\in\mathcal{T}_{D}$.

Then, $H_{GL}:=(B_{GL},S_{\diamond})$ is the Grossman-Larson Hopf algebroid
with $S_{\diamond}$ being called an antipode and satisfying the following
conditions where $\pi\in\mathcal{AF}_{D}$ and
$\Delta_{\mathcal{A}_{D}}(\pi)=\sum_{(\pi)}\pi_{(1)}\otimes_{\mathcal{A}_{D}}\pi_{(2)}$,

  * (1)

$\sum_{(\pi)}S_{\diamond}(\pi_{(1)})\diamond\pi_{(2)}=\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\pi))$,

  * (2)

$\sum_{(\pi)}\hat{\pi}_{(1)}\diamond
S_{\diamond}(\hat{\pi}_{(2)})=\mathbf{1}\epsilon_{\mathcal{A}_{D}}(\pi)$, with
$\gamma(\pi_{(1)}\otimes_{\mathcal{A}_{D}}\pi_{(2)})=\hat{\pi}_{(1)}\otimes\hat{\pi}_{(2)}$,

where $\gamma$ is the section of the projection
$P:\mathcal{AF}_{D}\otimes\mathcal{AF}_{D}\to\mathcal{AF}_{D}\otimes_{\mathcal{A}_{D}}\mathcal{AF}_{D}$
that places all aromas on the left side of the tensor product, that is,
$\hat{\pi}_{(2)}$ has no aromas.

  * Proof

It can be seen that $S_{\diamond}$ defined this way is an anti-isomorphism due
to the associativity of the Grossman-Larson product and the definition which
can be rewritten as

$S_{\diamond}(\omega\diamond\pi)=S_{\diamond}(\pi)\diamond
S_{\diamond}(\omega),\quad
S_{\diamond}(\tau\diamond\pi)=S_{\diamond}(\pi)\diamond S_{\diamond}(\tau).$

Let us prove that $S_{\diamond}$ satisfies the condition $(1)$ and $(2)$. We
start with $(1)$ by noting that the operation
$S_{\diamond}({-})\diamond{-}:\mathcal{AF}_{D}\otimes_{\mathcal{A}_{D}}\mathcal{AF}_{D}\to\mathcal{AF}_{D}$
is well-defined since

$S_{\diamond}(\omega\pi_{(1)})\diamond\pi_{(2)}=S_{\diamond}(\pi_{(1)})\diamond\omega\diamond\pi_{(2)}=S_{\diamond}(\pi_{(1)})\diamond\omega\pi_{(2)}.$

We see that $(1)$ is satisfied for $\pi=\omega\in\mathcal{A}_{D}$. Let us
check that $(1)$ is satisfied for $\pi=\omega\tau\in\mathcal{AT}_{D}$:

$S_{\diamond}(\tau)\diamond\omega+S_{\diamond}(\mathbf{1})\diamond\omega\tau=-\tau\omega-\tau\curvearrowright\omega+\omega\tau=S_{\diamond}(\tau)\curvearrowright\omega=\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\omega\tau)),$

where $\tau\in\mathcal{T}_{D}$. Let
$\tau\in\mathcal{T}_{D},\pi\in\mathcal{F}_{D},\omega\in\mathcal{A}_{D}$, then,
$\tau\pi\omega=\tau\diamond\pi\omega-\tau\curvearrowright\pi\omega$. We use
induction on the length of $\pi$ and assume that the condition $(1)$ is
satisfied for $\tau\curvearrowright\pi\omega$. We check that the left-hand
side of $(1)$ applied to $\tau\diamond\pi\omega$ is $0$:

$\displaystyle\sum_{(\tau\diamond\pi\omega)}S_{\diamond}((\tau\diamond\pi\omega)_{(1)})\diamond(\tau\diamond\pi\omega)_{(2)}$
$\displaystyle=\sum_{(\tau),(\pi\omega)}S_{\diamond}(\tau_{(1)}\diamond\pi_{(1)})\diamond\tau_{(2)}\diamond\pi_{(2)}\omega$
$\displaystyle=\sum_{(\pi\omega)}S_{\diamond}(\pi_{(1)})\diamond\big{(}\sum_{(\tau)}S_{\diamond}(\tau_{(1)})\diamond\tau_{(2)}\big{)}\diamond\pi_{(2)}\omega=0,$

since $\sum_{(\tau)}S_{\diamond}(\tau_{(1)})\diamond\tau_{(2)}=0$. This
implies that left-hand side of $(1)$ applied to $\tau\pi\omega$ is

$\displaystyle\sum_{(\tau\pi\omega)}S_{\diamond}((\tau\pi\omega)_{(1)})\diamond(\tau\pi\omega)_{(2)}$
$\displaystyle=-\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\tau\curvearrowright\pi\omega))$
$\displaystyle=\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\tau\pi\omega))+\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\pi\omega)\diamond\tau)$
$\displaystyle=\mathbf{1}\epsilon_{\mathcal{A}_{D}}(S_{\diamond}(\tau\pi\omega)).$

This proves that $S_{\diamond}$ satisfies the condition $(1)$. To prove
condition $(2)$, we recall that the Grossman-Larson product is
$\mathcal{A}_{D}$-linear in its left operand and that $\gamma$ places all
aromas on the left side of the tensor product. This implies that both sides of
the condition $(2)$ are $\mathcal{A}_{D}$-linear and the condition is reduced
to the analogous condition over the Grossman-Larson Hopf algebra over
$\mathcal{F}_{D}$. This proves that $S_{\diamond}$ satisfies the condition
$(2)$. $\square$

The space of decorated aromatic forests with commutative product $\cdot$ and
deshuffle coproduct forms a Hopf algebra
$H:=(\mathcal{AF}_{D},\cdot,\mathbf{1},\Delta_{\mathcal{A}_{D}},\epsilon_{\mathcal{A}_{D}},S)$.
The Hopf algebra $H$ together with the product $\curvearrowright:H\otimes H\to
H$ forms a pre-Hopf algebroid which is a generalization of the pre-Hopf
algebra [64] that satisfies the following conditions for $\pi,\mu,\eta\in H$,
a map $\beta:H\to H$, and a section $\gamma$ of the projection $P:H\otimes
H\to H\otimes_{\mathcal{A}_{D}}H$ with
$\gamma(\pi_{(1)}\otimes_{\mathcal{A}_{D}}\pi_{(2)})=\hat{\pi}_{(1)}\otimes\hat{\pi}_{(2)}$:

  * (1)

$\pi\curvearrowright(\mu\cdot\eta)=(\pi_{(1)}\curvearrowright\mu)\cdot(\pi_{(2)}\curvearrowright\eta)$,

  * (2)

$\pi\curvearrowright(\mu\curvearrowright\eta)=(\pi_{(1)}\cdot(\pi_{(2)}\curvearrowright\mu))\curvearrowright\eta$,

  * (3)

$\hat{\pi}_{(1)}\curvearrowright(\beta(\hat{\pi}_{(2)})\curvearrowright{-})=\epsilon_{\mathcal{A}_{D}}(\pi)\operatorname{id}$,

  * (4)

$\beta(\pi_{(1)})\curvearrowright(\pi_{(2)}\curvearrowright{-})=\epsilon_{\mathcal{A}_{D}}(\beta(\pi))\operatorname{id}$.

Conditions $(1)$ and $(2)$ follow from the definition of the D-algebra and the
conditions $(3)$ and $(4)$ are satisfied for $\beta$ and $\gamma$ being the
anitipode $S_{\diamond}$ and $\gamma$ from Proposition 4.8. We present an
alternative proof for $(8)$ of [64, Lemma 2.3].

###### Lemma 4.9

For all $\pi,\mu\in H$, we have $\pi\curvearrowright
S(\mu)=S(\pi\curvearrowright\mu)$.

  * Proof

Consider $\mu=\tau\in\mathcal{AT}_{D}$, then
$\pi\curvearrowright\tau\in\mathcal{AT}_{D}$ and $\pi\curvearrowright
S(\tau)=-\pi\curvearrowright\tau=S(\pi\curvearrowright\tau)$. Assume the
statement is true for all monomials shorter than $\mu=\tau_{1}\cdots\tau_{n}$
for $\tau_{i}\in\mathcal{AT}_{D}$, then,

$\displaystyle S(\pi\curvearrowright\tau_{1}\cdots\tau_{n})$
$\displaystyle=-\sum_{(\pi)}(\pi_{(1)}\curvearrowright\tau_{1})S(\pi_{(2)}\curvearrowright\tau_{2}\cdots\tau_{n})$
$\displaystyle=-\sum_{(\pi)}(\pi_{(1)}\curvearrowright\tau_{1})(\pi_{(2)}\curvearrowright
S(\tau_{2}\cdots\tau_{n}))$
$\displaystyle=-\pi\curvearrowright(\tau_{1}S(\tau_{2}\cdots\tau_{n}))=\pi\curvearrowright
S(\mu).$

This finishes the proof. $\square$

We check that the subadjacent Hopf algebra with antipode $\hat{S}_{\diamond}$
defined in [64, Thm. 2.4] corresponds to the Grossman-Larson Hopf algebroid by
showing that the antipodes coincide.

###### Lemma 4.10

Let $\hat{S}_{\diamond}$ be the antipode defined in [64, Thm. 2.4], that is,

$\hat{S}_{\diamond}(\pi):=\sum_{(\pi)}\hat{S}_{\diamond}(\pi_{(1)})\curvearrowright
S(\pi_{(2)}),\quad\text{for }\pi\in H.$

Then, $\hat{S}_{\diamond}=S_{\diamond}$ where $S_{\diamond}$ is defined in
Proposition 4.8.

  * Proof

We see that $\hat{S}_{\diamond}(\omega)=\omega$ and
$\hat{S}_{\diamond}(\tau)=-\tau$ for $\omega\in\mathcal{A}_{D}$ and
$\tau\in\mathcal{T}_{D}$. We check by induction and using the fact that
$S_{\diamond}$ is a coalgebra homomorphism that
$\hat{S}_{\diamond}(\omega\pi)=S_{\diamond}(\omega\pi)$ for
$\pi\in\mathcal{AF}_{D}$:

$\displaystyle\hat{S}_{\diamond}(\omega\pi)$
$\displaystyle=\sum_{(\pi)}S_{\diamond}(\pi_{(1)})\curvearrowright
S(\omega\pi_{(2)})=\sum_{(\pi)^{2}}(S_{\diamond}(\pi_{(1)})\curvearrowright\omega)(S_{\diamond}(\pi_{(2)})\curvearrowright
S(\pi_{(3)}))$
$\displaystyle=\sum_{(\pi)}(S_{\diamond}(\pi_{(1)})\curvearrowright\omega)S_{\diamond}(\pi_{(2)})=S_{\diamond}(\pi)\diamond\omega.$

We use the same properties to show that
$\hat{S}_{\diamond}(\tau\pi)=S_{\diamond}(\tau\pi)$:

$\displaystyle\hat{S}_{\diamond}(\tau\pi)$
$\displaystyle=S_{\diamond}(\pi_{(1)})\curvearrowright
S(\tau\pi_{(2)})+S_{\diamond}(\tau\pi_{(1)})\curvearrowright S(\pi_{(2)})$
$\displaystyle=-S_{\diamond}(\pi_{(1)})\curvearrowright\tau
S(\pi_{(2)})-(S_{\diamond}(\pi_{(1)})\diamond\tau)\curvearrowright
S(\pi_{(2)})-S_{\diamond}(\tau\curvearrowright\pi_{(1)})\curvearrowright
S(\pi_{(2)})$ $\displaystyle=-S_{\diamond}(\pi)\diamond\tau-
S_{\diamond}(\pi_{(1)})\curvearrowright
S(\tau\curvearrowright\pi_{(2)})-S_{\diamond}(\tau\curvearrowright\pi_{(1)})\curvearrowright
S(\pi_{(2)})$ $\displaystyle=-S_{\diamond}(\pi)\diamond\tau-
S_{\diamond}(\tau\curvearrowright\pi),$

where we omit writting the sums to simplify the notation. Therefore,
$\hat{S}_{\diamond}=S_{\diamond}$ following the definition from Proposition
4.8. $\square$

Following [7, Def. 1.1], the Hopf algebra $H$ together with the Hopf algebroid
$H_{GL}$ forms a generalization of the Hopf brace, that is, the following
compatibility condition is satisfied

$\pi\diamond(\mu\cdot\eta)=\sum_{(\pi)^{2}}(\pi_{(1)}\diamond\mu)\cdot
S(\pi_{(2)})\cdot(\pi_{(3)}\diamond\eta),$

with $\pi,\mu,\eta\in\mathcal{AF}_{D}$ and
$\Delta^{2}(\pi)=\sum_{(\pi)^{2}}\pi_{(1)}\otimes\pi_{(2)}\otimes\pi_{(3)}$. A
proof can be found in [64, Thm. 2.13].

###  4.2 Decorated clumped forests

Recall that decorated clumped forests are defined as a symmetric algebra
$\mathcal{CF}_{D}=S(\mathcal{AT}_{D})$ over $\mathbb{R}$ in Section 2. We use
the Guin-Oudom process [79] to define the product $\curvearrowright$ on
$\mathcal{CF}_{D}$. The commutative D-algebra
$(\mathcal{CF}_{D},\curvearrowright)$ that we obtain in this way has
$\mathcal{CF}_{D,0}=\\{\mathbf{1}\\}$ and is in many ways similar to the
commutative D-algebra of classical forests
$(\mathcal{F}_{D},\curvearrowright)$. Decorated clumped forests have a
convenient algebraic structure described in the following result.

###### Theorem 4.11

The Grossman-Larson $\mathcal{A}/_{\mathbb{R}}$-bialgebroid of
$(\mathcal{CF}_{D},\curvearrowright)$ is a Hopf algebra dual up to the
symmetry to the Butcher-Connes-Kreimer Hopf algebra over clumped forests.

  * Proof

As $\mathcal{CF}_{D,0}=\\{\mathbf{1}\\}$, the
$\mathcal{A}/_{\mathbb{R}}$-bialgebroid structure reduces to a graded
connected bialgebra, that is, to a Hopf algebra. Its duality to the
corresponding Butcher-Connes-Kreimer Hopf algebra over clumped forests can be
seen by following the proof for classical forests from [45]. $\square$

Let us consider the following inner product for $\pi,\mu\in\mathcal{CF}_{D}$
or $\mathcal{AF}_{D}$:

$\langle\pi,\mu\rangle_{\sigma}:=\begin{cases}\sigma(\pi),&\text{if
}\pi=\mu,\\\ 0,&\text{otherwise}.\end{cases}$

We use it to obtain the following duality between concatenation product and
deshuffle coproduct.

###### Lemma 4.12

Let $\cdot$ be the concatenation product and let $a,b\in\mathcal{CF}^{*}_{D}$
or $\mathcal{AF}^{*}_{D}$, then,

$\delta_{\sigma}(a)\cdot\delta_{\sigma}(b)=\delta_{\sigma}(a\odot
b),\quad\text{where }a\odot b=(a\otimes b)\circ\Delta,$

where $\Delta:\mathcal{CF}_{D}\to\mathcal{CF}_{D}\otimes\mathcal{CF}_{D}$ or
$\Delta:\mathcal{AF}_{D}\to\mathcal{AF}_{D}\otimes\mathcal{AF}_{D}$ is the
$\mathbb{R}$-linear deshuffle coproduct.

  * Proof

We check that
$\langle\pi\cdot\mu,\eta\rangle_{\sigma}=\langle\pi\otimes\mu,\Delta(\eta)\rangle_{\sigma}$
for $\pi,\mu,\eta\in\mathcal{CF}_{D}$ or $\mathcal{AF}_{D}$. $\square$

Let the map
$\Phi:(\mathcal{CF}_{D},\curvearrowright)\to(\mathcal{AF}_{D},\curvearrowright)$
be a commutative D-algebra morphism that ”forgets” the clumping, for example,

$\Phi(\forest{(b),b}\cdot\forest{b[b]})=\Phi(\forest{b}\cdot\forest{(b),b[b]})=\forest{(b),b,b[b]}.$

We define $\Phi^{*}:\mathcal{AF}_{D}\to\mathcal{CF}_{D}$ as
$(\Phi\circ\delta_{\sigma})(a)=\delta_{\sigma}(a\circ\Phi^{*})$ with
$a\in\mathcal{CF}_{D}^{*}$, in particular, $\Phi^{*}$ is the adjoint of $\Phi$
with respect to the $\langle{-},{-}\rangle_{\sigma}$ inner product. Let us
consider the exponential maps

$\displaystyle\exp^{\cdot}:\mathcal{AT}_{D}\to\mathcal{AF}_{D},$
$\displaystyle\quad\exp^{\odot}:\mathcal{AT}_{D}^{*}\to\mathcal{AF}_{D}^{*},$
$\displaystyle\exp^{\cdot}_{C}:\mathcal{AT}_{D}\to\mathcal{CF}_{D},$
$\displaystyle\quad\exp^{\odot}_{C}:\mathcal{AT}_{D}^{*}\to\mathcal{CF}_{D}^{*}.$

Using Lemma 4.12, we obtain for $a_{0}\in\mathcal{AT}_{D}^{*}$ the following
identities

$\exp^{\cdot}(\delta_{\sigma}(a_{0}))=\delta_{\sigma}(\exp^{\odot}(a_{0})),\quad\exp^{\cdot}_{C}(\delta_{\sigma}(a_{0}))=\delta_{\sigma}(\exp^{\odot}_{C}(a_{0})),$

with the functionals $\exp^{\odot}(a_{0})$ and $\exp^{\odot}_{C}(a_{0})$ being
characterized in Propositon 4.13.

###### Proposition 4.13

Let $a_{0}\in\mathcal{AT}^{*}$ be an infinitisimal character and let

$a_{e}:=\exp^{\odot}(a_{0}),\quad a_{c}:=\exp^{\odot}_{C}(a_{0}).$

Then, $a_{c}\in\mathcal{CF}_{D}^{*}$ is a character of
$(\mathcal{CF}_{D},\cdot)$ and $a_{e}=a_{c}\circ\Phi^{*}$ with
$a_{e}\in\mathcal{AF}_{D}^{*}$.

  * Proof

We prove that $a_{c}:=\exp^{\odot}_{C}(a_{0})$ is a character of
$\mathcal{CF}_{D}$ by considering a decorated clumped forest
$\pi=\tau_{1}\cdots\tau_{n}\in\mathcal{CF}_{D}$ where
$\tau_{i}\in\mathcal{AT}_{D}$. Then,

$\displaystyle\exp_{C}^{\odot}(a_{0})(\pi)$
$\displaystyle=\frac{1}{n!}\cdot(a_{0}\otimes\cdots\otimes
a_{0})(\Delta^{n-1}(\pi))$ $\displaystyle=\frac{1}{n!}\cdot\sum_{\sigma\in
S_{n}}a_{0}(\tau_{\sigma(1)})\cdots
a_{0}(\tau_{\sigma(n)})=a_{0}(\tau_{1})\cdots a_{0}(\tau_{n}).$

We prove $a_{e}=a_{c}\circ\Phi^{*}$ by using the identity
$\exp^{\cdot}(\delta_{\sigma}(a_{0}))=\Phi\big{(}\exp^{\cdot}_{C}(\delta_{\sigma}(a_{0}))\big{)}$.
$\square$

We define $F_{D}$ from Section 2.1 over $\mathcal{CF}_{D}$ by
$F_{D}:=F_{D}\circ\Phi$ where we use the same notation for the morphism over
clumped and aromatic forests. This way, we obtain S-series over decorated
clumped forests. We note that, following the definition of $\Phi^{*}$, any
S-series $S(a)$ with $a\in\mathcal{CF}_{D}^{*}$ is identical to the S-series
over decorated aromatic forests $S(a\circ\Phi^{*})$. Moreover, given any
functional $a\in\mathcal{AF}_{D}^{*}$, there exists a functional
$a_{C}\in\mathcal{CF}_{D}^{*}$ such that $a=a_{C}\circ\Phi^{*}$, since
$\Phi^{*}$ is injective. A possible definition of $a_{C}$ is

$a_{C}(\pi)=\frac{1}{n^{m}}a(\Phi(\pi)),\quad\text{for
}\pi\in\mathcal{CF}_{D},$

where $n$ is the number of rooted components and $m$ is the number of aromas.

We note that a tree $\tau\in\mathcal{T}_{D}$ induces a map
$\tau^{\prime}:\mathcal{A}_{D}\to\mathcal{AT}_{D}$ with
$\tau^{\prime}(\omega)=\omega\tau$ for $\omega\in\mathcal{A}_{D}$. We can
extend $({-})^{\prime}$ to $\mathcal{F}_{D}$ in two possible ways, that is,
for a $\pi\in\mathcal{F}_{D}$, we have two maps,

$\pi^{\prime}:\mathcal{A}_{D}\to\mathcal{AF}_{D},\quad\pi^{\prime\prime}:\mathcal{A}_{D}\to\mathcal{CF}_{D},$

defined as

$(\pi\cdot\mu)^{\prime}(\omega)=\omega\pi\mu,\quad(\pi\cdot\mu)^{\prime\prime}(\omega)=(\pi^{\prime\prime}\odot\mu^{\prime\prime})(\omega),$

with
$\pi^{\prime\prime}\odot\mu^{\prime\prime}:=\cdot\circ(\pi^{\prime\prime}\otimes\mu^{\prime\prime})\circ\Delta$
where $\Delta$ is a deshuffle coproduct over $\mathcal{A}_{D}$. For example,

$(\forest{b,b[b]})^{\prime}(\forest{(b)})=\forest{(b),b,b[b]},\quad(\forest{b,b[b]})^{{\prime\prime}}(\forest{(b)})=\forest{(b),b}\cdot\forest{b[b]}+\forest{b}\cdot\forest{(b),b[b]}.$

The following result presents a convenient method to compute $\Phi^{*}$.

###### Proposition 4.14

Given a decorated forest $\pi\in\mathcal{F}_{D}$ and
$\omega\in\mathcal{A}_{D}$, we have the following identity

$\Phi^{*}(\pi^{\prime}(\omega))=\pi^{\prime\prime}(\omega).$

  * Proof

We take the adjoints of $\pi^{\prime}$ and $\pi^{\prime\prime}$ with respect
to the $\langle{-},{-}\rangle_{\sigma}$ inner product and denote them by
$\pi^{\prime*}:\mathcal{AF}_{D}\to\mathcal{A}_{D}$ and
$\pi^{{\prime\prime}*}:\mathcal{CF}_{D}\to\mathcal{A}_{D}$. Then,
$\pi^{\prime*}(\omega\pi)=\sigma(\pi)\omega$ where $\omega\in\mathcal{A}_{D}$
and $\pi\in\mathcal{F}_{D}$. To prove the statement, we have to show that
$\pi^{{\prime\prime}*}(\pi_{\omega})=\sigma(\pi)\omega$ where
$\pi_{\omega}\in\mathcal{CF}_{D}$ is a decorated clumped forest that occurs as
a term in $\pi^{\prime\prime}(\omega)$, that is, we need to show that
$\pi^{\prime*}(\Phi(\pi_{\omega}))=\pi^{{\prime\prime}*}(\pi_{\omega})$.

We note that for $\tau\in\mathcal{T}_{D}$,
$\tau^{{\prime\prime}*}(\omega)=\tau^{\prime*}(\omega)=\sigma(\tau)\omega$ and
the statement is true. We use an inductive assumption and Lemma 4.12 to obtain

$(\pi\cdot\mu)^{{\prime\prime}*}(\eta_{\omega})=(\cdot\circ(\pi^{{\prime\prime}*}\otimes\mu^{{\prime\prime}*})\circ\Delta)(\eta_{\omega})=\sigma(\pi)\sigma(\mu)|A|\omega,$

where $A$ is the set of $\eta_{\omega,(1)}\otimes\eta_{\omega,(2)}$ such that
the rooted components of $\eta_{\omega,(1)}$ and $\eta_{\omega,(2)}$ are
isomorphic to $\pi$ and $\mu$, respectively. We note that
$\sigma(\pi)\sigma(\mu)|A|=\sigma(\pi\cdot\mu)$ and the proof is finished.
$\square$

For example, let $\forest{(b),b=b,b,b[b]}\in\mathcal{AF}_{D}$, then,

$\Phi^{*}(\forest{(b),b=b,b,b[b]})=\forest{(b),b=b,b}\cdot\forest{b[b]}+\forest{b=b,b}\cdot\forest{(b),b[b]}+\forest{(b),b}\cdot\forest{b=b,b[b]}+\forest{b}\cdot\forest{(b),b=b,b[b]}.$

We note that Propositions 4.13 and 4.14 imply the following identity

$a_{e}(\omega\pi)=a_{c}(\pi^{{\prime\prime}}(\omega)),\quad\text{for
}\pi\in\mathcal{F}_{D},\quad\omega\in\mathcal{A}_{D},$

a version of which is presented in [11] for classical aromatic forests.

###  4.3 Substitution law for decorated aromatic S-series

We introduce the decorated insertion product
$\blacktriangleright_{d}:\mathcal{AT}_{D}\otimes\mathcal{AT}_{D}\to\mathcal{AT}_{D}$
which inserts the aromatic tree from the left operand into the vertices
decorated by $d\in D$ of the right operand in all possible ways. For example,
we recall that
$\forest{(w),b[w[w]]}=\text{div}(\circ)(\circ\curvearrowright\circ)\curvearrowright\bullet$,
then,

$\displaystyle\forest{(b),b[w]}\blacktriangleright_{\circ}\forest{(w),b[w[w]]}$
$\displaystyle=\text{div}(\forest{(b),b[w]})(\circ\curvearrowright\circ)\curvearrowright\bullet+\text{div}(\circ)(\forest{(b),b[w]}\curvearrowright\circ)\curvearrowright\bullet+\text{div}(\circ)(\circ\curvearrowright\forest{(b),b[w]})\curvearrowright\bullet$
$\displaystyle=\forest{(b[b[w]]),b[w[w]]}+\forest{(b),(b[w]),b[w[w]]}+\forest{(b),(b,w),b[w[w]]}+\forest{(b),b[w[b[w]]]}+\forest{(b),b[b[w[w]]]}+\forest{(b),b[b[w,w]]}+\forest{(b[w]),b[b[w]]}.$

The product $\blacktriangleright_{d}$ generalizes the insertion product
studied in [85, 71, 86]. Then, the family
$(\mathcal{AT}_{D},(\blacktriangleright_{d})_{d\in D})$ is a multi-pre-Lie
algebra [15, 37], that is, for $\tau,\gamma,\nu\in\mathcal{AT}_{D}$ and
$d,e\in D$ we have

$\tau\blacktriangleright_{d}(\gamma\blacktriangleright_{e}\nu)-(\tau\blacktriangleright_{d}\gamma)\blacktriangleright_{e}\nu=\gamma\blacktriangleright_{e}(\tau\blacktriangleright_{d}\nu)-(\gamma\blacktriangleright_{e}\tau)\blacktriangleright_{d}\nu.$

Let $\mathcal{AT}_{D}^{\oplus
D}:=\mathcal{AT}_{D}\otimes\mathbb{R}[D]=\bigoplus_{d\in
D}\mathcal{AT}_{D}\iota_{d}$ where $\iota_{d}$ for $d\in D$ form the basis of
$\mathbb{R}[D]$ and let us define the action
$\blacktriangleright:\mathcal{AT}_{D}^{\oplus
D}\otimes\mathcal{AT}_{D}\to\mathcal{AT}_{D}$ by

$\tau\iota_{d}\blacktriangleright\gamma=\tau\blacktriangleright_{d}\gamma.$

Let us consider $S(\mathcal{AT}_{D}^{\oplus D})$ which becomes
$\mathcal{CF}_{D}^{\otimes D}:=\bigotimes_{d\in D}\mathcal{CF}_{D}\iota_{d}$
after we assume the identity
$\pi\iota_{d}\cdot\mu\iota_{d}=(\pi\cdot\mu)\iota_{d}$. We check that the
Guin-Oudom process for multi-pre-Lie products [37, Thm. 2.4] is well-defined
and use it to define

$\blacktriangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}\to\mathcal{CF}_{D}.$

For example, let $D:=\\{\bullet,\circ\\}$, then,
$(\forest{b[b]}\iota_{\circ})\cdot(\forest{w,w}\iota_{\bullet})\blacktriangleright\forest{w[b,b]}=2\forest{b[w,w,b]}+4\forest{b[w,b[w]]}+2\forest{b[b[w,w]]}$.
We note that for $\pi,\mu\in\mathcal{CF}_{D}$, we have
$\pi\iota_{d}\blacktriangleright\mu=0$ if the number $|\pi|_{AT}$ of aromatic
trees in $\pi$ is greater than the number $|\mu|_{d}$ of vertices decorated by
$d$ in $\mu$. We define now the substitution action
$\triangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}\to\mathcal{CF}_{D}$ in the following way

$(\otimes_{d\in
D}\pi_{d}\iota_{d})\triangleright\mu:=\begin{cases}(\otimes_{d\in
D}\pi_{d}\iota_{d})\blacktriangleright\mu,&\text{if
}|\pi_{d}|_{AT}=|\mu|_{d}\text{ for all }d\in D,\\\
0,&\text{otherwise.}\end{cases}$

The substitution action substitutes all vertices in the right operand by the
aromatic trees from the left operand. We recall that $CF_{D}$ is a free
D-algebra generated by the set $D$, therefore, given a map
$\varphi:D\to\mathcal{AT}_{D}$, there exists a unique morphism
$A_{\varphi}:\mathcal{CF}_{D}\to\mathcal{CF}_{D}$. The morphism $A_{\varphi}$
can be written using the substitution action.

###### Lemma 4.15

Given a morphism $A_{\varphi}:\mathcal{CF}_{D}\to\mathcal{CF}_{D}$ that acts
on generators $D$ as $\varphi$, we have

$A_{\varphi}(\pi)=\otimes_{d\in
D}\exp^{\cdot}_{C}(\varphi(d)\iota_{d})\triangleright\pi.$

  * Proof

Assume $\varphi(d)=\tau_{d}\in\mathcal{AT}_{D}$, then,

$\otimes_{d\in
D}\exp^{\cdot}_{C}(\tau_{d}\iota_{d})\triangleright\pi=\frac{1}{\prod_{d\in
D}|\pi|_{d}!}(\otimes_{d\in
D}\tau_{d}^{|\pi|_{d}}\iota_{d})\blacktriangleright\pi=A_{\varphi}(\pi),$

where we use the definition of $\blacktriangleright$. $\square$

Let $a^{\otimes D}:=\otimes_{d\in D}a_{d}\iota_{d}$ with
$a_{d}\in\mathcal{CF}^{*}_{D}$ be a functional over $\mathcal{CF}_{D}^{\otimes
D}$ defined as

$a^{\otimes D}(\otimes_{d\in D}\pi_{d}\iota_{d})=\prod_{d\in
D}a_{d}(\pi_{d}).$

For example, let $D=\\{\bullet,\circ\\}$, then,
$(a\iota_{\bullet}b\iota_{\circ})\big{(}\forest{b[b,w[b]],w[b]}\iota_{\bullet}\cdot\forest{b,b[w]}\iota_{\circ}\big{)}=a(\forest{b[b,w[b]],w[b]})b(\forest{b,b[w]})$.
We define a coaction
$\Delta_{CEM}:\mathcal{CF}_{D}\to\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}$ and show in Proposition 4.17 that it is dual to the
substitution action up to the symmetry coefficient, that is, we have

$\delta_{\sigma}(b^{\otimes D}\star a)=\delta_{\sigma}(b^{\otimes
D})\triangleright\delta_{\sigma}(a)\,\quad\text{where }b^{\otimes D}\star
a=(b^{\otimes D}\otimes a)\circ\Delta_{CEM},$

for $a\in\mathcal{CF}_{D}^{*}$, $b^{\otimes D}\in\mathcal{CF}_{D}^{\otimes
D*}$, and where we denote $\sigma^{\otimes D}$ by $\sigma$ for simplicity. The
coaction $\Delta_{CEM}$ is a generalization of the Calaque–Ebrahimi-
Fard–Manchon (CEM) coproduct from [22].

###### Definition 4.16

Define the CEM coaction
$\Delta_{CEM}:\mathcal{CF}_{D}\to\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}$ as

$\Delta_{CEM}(\pi)=\sum_{\otimes_{d\in D}p_{d}\iota_{d}}(\otimes_{d\in
D}p_{d}\iota_{d})\otimes\pi/_{\otimes_{d\in D}p_{d}\iota_{d}},\quad\text{for
}\pi\in\mathcal{CF}_{D},$

where the sum is over all monomials of disjoint decorated clumped subforests
$p_{d}\iota_{d}$ that partition $\pi$, and $\pi/_{\otimes_{d\in
D}p_{d}\iota_{d}}$ denotes the decorated clumped forest obtained by
contracting the decorated aromatic trees of $p_{d}\iota_{d}$ into vertices
decorated by $d$.

For example, let $D=\\{\bullet,\circ\\}$, then,

$\displaystyle\Delta_{CEM}(\forest{b})$
$\displaystyle=\forest{b}\iota_{\bullet}\otimes\forest{b}+\forest{b}\iota_{\circ}\otimes\forest{w},$
$\displaystyle\Delta_{CEM}(\forest{b[w]})$
$\displaystyle=\forest{b[w]}\iota_{\bullet}\otimes\forest{b}+\forest{b[w]}\iota_{\circ}\otimes\forest{w}+\forest{b,w}\iota_{\bullet}\otimes\forest{b[b]}+\forest{b,w}\iota_{\circ}\otimes\forest{w[w]}$
$\displaystyle\quad\forest{b}\iota_{\bullet}\cdot\forest{w}\iota_{\circ}\otimes\forest{b[w]}+\forest{b}\iota_{\circ}\cdot\forest{w}\iota_{\bullet}\otimes\forest{w[b]},$
$\displaystyle\Delta_{CEM}(\forest{b[b,w]})$
$\displaystyle=\forest{b[b,w]}\iota_{\bullet}\otimes\forest{b}+\forest{b[b],w}\iota_{\bullet}\otimes\forest{b[b]}+\forest{b[w],b}\iota_{\bullet}\otimes\forest{b[b]}+\forest{b,b,w}\iota_{\bullet}\otimes\forest{b[b,b]}+$
$\displaystyle\quad\forest{b[b,w]}\iota_{\circ}\otimes\forest{w}+\forest{b[b],w}\iota_{\circ}\otimes\forest{w[w]}+\forest{b[w],b}\iota_{\circ}\otimes\forest{w[w]}+\forest{b,b,w}\iota_{\circ}\otimes\forest{w[w,w]}+$
$\displaystyle\quad\forest{b[b]}\iota_{\bullet}\cdot\forest{w}\iota_{\circ}\otimes\forest{b[w]}+\forest{b[b]}\iota_{\circ}\cdot\forest{w}\iota_{\bullet}\otimes\forest{w[b]}+\forest{b[w]}\iota_{\bullet}\cdot\forest{b}\iota_{\circ}\otimes\forest{b[w]}+\forest{b[w]}\iota_{\circ}\cdot\forest{b}\iota_{\bullet}\otimes\forest{w[b]}+$
$\displaystyle\quad\forest{b}\iota_{\circ}\cdot\forest{b,w}\iota_{\bullet}\otimes\forest{b[w,b]}+\forest{b}\iota_{\circ}\cdot\forest{b,w}\iota_{\bullet}\otimes\forest{w[b,b]}+\forest{b,b}\iota_{\bullet}\cdot\forest{w}\iota_{\circ}\otimes\forest{b[b,w]}+$
$\displaystyle\quad\forest{b}\iota_{\bullet}\cdot\forest{b,w}\iota_{\circ}\otimes\forest{w[b,w]}+\forest{b}\iota_{\bullet}\cdot\forest{b,w}\iota_{\circ}\otimes\forest{b[w,w]}+\forest{b,b}\iota_{\circ}\cdot\forest{w}\iota_{\bullet}\otimes\forest{w[w,b]}.$

###### Proposition 4.17

Let $a\in\mathcal{CF}^{*}_{D}$, $b^{\otimes D}\in\mathcal{CF}_{D}^{\otimes
D*}$, and $b^{\otimes D}\star a=(b^{\otimes D}\otimes a)\circ\Delta_{CEM}$,
then,

$\delta_{\sigma}(b^{\otimes D}\star a)=\delta_{\sigma}(b^{\otimes
D})\triangleright\delta_{\sigma}(a).$

  * Proof

We prove the statement in two steps. First, we prove the statement over the
space of ordered clumped forests. Ordered clumped forest are forests in which
all vertices are totally ordered. Next, we describe the relationships between
the product and coproduct over ordered and non-ordered clumped forests and use
them to finish the proof.

1. The space $\mathcal{CF}_{DO}$ is defined by assigning every decorated clumped forest $\pi$ a total order over the vertices. We note that a forest $\pi\in\mathcal{CF}_{D}$ corresponds to $|\pi|!$ ordered forests in $\mathcal{CF}_{DO}$. The symmetry of any element of $\mathcal{CF}_{DO}$ is equal to $1$. Let $\mathcal{CF}_{DO}^{\otimes D}:=\bigotimes_{d\in D}\mathcal{CF}_{DO}\iota_{d}$.

We define $\triangleright^{O}:\mathcal{CF}_{DO}^{\otimes
D}\otimes\mathcal{CF}_{D}\to\mathcal{CF}_{DO}$ and
$\Delta_{CEM}^{O}:\mathcal{CF}_{DO}\to\mathcal{CF}_{DO}^{\otimes
D}\otimes\mathcal{CF}_{D}$ to be the natural extensions of $\triangleright$
and $\Delta_{CEM}$. The map $\triangleright^{O}$ substitutes the vertices
decorated by $d$ by the trees of $\pi_{d}\iota_{d}$ and chooses a total order
between the vertices of $\pi_{d}$ and $\pi_{l}$ with $d\neq l$ in all possible
ways. We note that there are $\big{(}\sum_{d\in
D}|\pi_{d}|\big{)}!/\prod_{d\in D}|\pi_{d}|!$ ways to choose a total order
between the vertices of $\pi_{d}$ and $\pi_{l}$ with $d\neq l$. Let
$\pi\in\mathcal{CF}_{DO}^{\otimes D}$ and $\mu\in\mathcal{CF}_{D}$, then,

$\pi\triangleright^{O}\mu=\sum_{\eta\in\mathcal{CF}_{DO}}N(\pi,\mu,\eta)\eta,$

where $N(\pi,\mu,\eta)$ is the number of ways to substitute the vertices of
$\mu$ by trees in $\pi$ to obtain $\eta$. We note that there are
$|\operatorname{Aut}(\mu)|$ ways to substitute the vertices of $\mu$ to obtain
the same ordered forest $\eta$, therefore, $N(\pi,\mu,\eta)=\sigma(\mu)$. The
symmetry is $1$ for an ordered clumped forest, and, since all terms in
$\Delta_{CEM}^{O}(\pi)$ have coefficient $1$, we obtain

$\delta_{\sigma}^{O}(b^{\otimes
D})\triangleright^{O}\delta_{\sigma}(a)=\delta_{\sigma}^{O}(b^{\otimes
D}\star^{O}a),\quad\text{where }b^{\otimes D}\star^{O}a=(b^{\otimes D}\otimes
a)\circ\Delta_{CEM}^{O},$

for $b^{\otimes D}\in\mathcal{CF}_{DO}^{\otimes D*}$ and
$a\in\mathcal{CF}_{D}^{*}$. The image of $\delta_{\sigma}^{O}$ is a formal sum
over ordered clumped forests.

2. We define the map $\varphi$ that forgets the ordering of the vertices and let

$\hat{\varphi}(\pi):=\frac{\varphi(\pi)}{|\pi|!},\quad\varphi^{\otimes
D}(\otimes_{d\in D}\pi_{d}\iota_{d})=\otimes_{d\in
D}\varphi(\pi_{d})\iota_{d},$

where $\varphi,\hat{\varphi}:\mathcal{CF}_{DO}\to\mathcal{CF}_{D}$ and
$\varphi^{\otimes D},\hat{\varphi}^{\otimes D}:\mathcal{CF}_{DO}^{\otimes
D}\to\mathcal{CF}_{D}^{\otimes D}$. We have the following properties

$\hat{\varphi}\circ\triangleright^{O}=\triangleright\circ(\hat{\varphi}^{\otimes
D}\otimes\operatorname{id}),\quad(\varphi^{\otimes
D}\otimes\operatorname{id})\circ\Delta_{CEM}^{O}=\Delta_{CEM}\circ\varphi.$

We also note that for a functional $b^{\otimes D}\in\mathcal{CF}_{D}^{\otimes
D*}$, $\delta_{\sigma}(b^{\otimes D})=\hat{\varphi}^{\otimes
D}(\delta_{\sigma}^{O}(b^{\otimes D}\circ\varphi^{\otimes D}))$, therefore,

$\displaystyle\delta_{\sigma}(b^{\otimes D})\triangleright\delta_{\sigma}(a)$
$\displaystyle=\hat{\varphi}^{\otimes D}\big{(}\delta_{\sigma}^{O}(b^{\otimes
D}\circ\varphi^{\otimes D})\big{)}\triangleright\delta_{\sigma}(a)$
$\displaystyle=\hat{\varphi}\big{(}\delta_{\sigma}^{O}(b^{\otimes
D}\circ\varphi^{\otimes D})\triangleright^{O}\delta_{\sigma}(a)\big{)}$
$\displaystyle=\hat{\varphi}\big{(}\delta_{\sigma}^{O}((b^{\otimes
D}\circ\varphi^{\otimes D})\star^{O}a)\big{)}$
$\displaystyle=\hat{\varphi}\big{(}\delta_{\sigma}^{O}((b^{\otimes D}\star
a)\circ\varphi)\big{)}=\delta_{\sigma}(b^{\otimes D}\star a).$

This proves that $\delta_{\sigma}$ is an algebra morphism. $\square$

We now extend the insertion action
$\blacktriangleright:\mathcal{AT}_{D}^{\oplus
D}\otimes\mathcal{AT}_{D}\to\mathcal{AT}_{D}$ into the insertion product
$\blacktriangleright:\mathcal{AT}_{D}^{\oplus
D}\otimes\mathcal{AT}_{D}^{\oplus D}\to\mathcal{AT}_{D}^{\oplus D}$ by

$\tau\iota_{d}\blacktriangleright\gamma\iota_{l}=(\tau\blacktriangleright_{d}\gamma)\iota_{l}.$

We recall that $\mathcal{CF}_{D}^{\otimes D}=S(\mathcal{AT}_{D}^{\oplus D})$
which allows us to use the Guin-Oudom process and define
$\blacktriangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}^{\otimes D}\to\mathcal{CF}_{D}^{\otimes D}$ and the
substitution product $\triangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{CF}_{D}^{\otimes D}\to\mathcal{CF}_{D}^{\otimes D}$. The
algebra $(\mathcal{CF}_{D}^{\otimes D},\triangleright)$ is endowed with the
deshuffle coproduct $\Delta$ of $S(\mathcal{AT}_{D}^{\oplus D})$. We present a
generalization of the result obtained in [22] for classical forests.

###### Theorem 4.18

The algebra $B_{\triangleright}:=(\mathcal{CF}_{D}^{\otimes
D},\triangleright)$ endowed with the deshuffle coproduct $\Delta$ is a
bialgebra dual to the CEM bialgebra $B_{CEM}:=(\mathcal{CF}_{D}^{\otimes
D},\cdot,\Delta_{CEM})$ with respect to the $\langle{-},{-}\rangle_{\sigma}$
inner product. Moreover,
$H_{CEM}:=B_{CEM}/\langle(1-\forest{b_{d}{}_{9}0})\iota_{d}\,:d\in D\rangle$
is a Hopf algebra dual to the Grossman-Larson Hopf algebra
$H_{\triangleright}$, which is an appropriate sub-bialgebra of
$B_{\triangleright}$ and a universal enveloping algebra of
$(\mathcal{AT}_{D}^{\oplus D},\blacktriangleright)$.

  * Proof

Let us start by proving that $B_{\triangleright}$ is a bialgebra. The unit is
given by $u:=\otimes_{d\in D}\exp^{\cdot}_{C}(\forest{b_{d}{}_{9}0}\iota_{d})$
and counit is given by $\epsilon:=\mathbf{1}^{*}\in\mathcal{CF}_{D}^{\otimes
D*}$. It is straightforward to check that the counit is compatible with the
product and unit. We use the property
$\Delta(\exp^{\cdot}_{C}(\tau))=\exp_{C}^{\cdot}(\tau)\otimes\exp_{C}^{\cdot}(\tau)$
to check that the unit is compatible with the coproduct, that is,
$\Delta(u)=u\otimes u$. We use

$\pi\triangleright({-}\cdot{-})=\sum_{(\pi)}(\pi_{(1)}\triangleright{-})\cdot(\pi_{(2)}\triangleright{-})$

and the associativity of $\triangleright$ to obtain the identity

$\sum_{(\pi),(\mu)}(\pi_{(1)}\triangleright\mu_{(1)}\triangleright{-})(\pi_{(2)}\triangleright\mu_{(2)}\triangleright{-})=\sum_{(\pi\triangleright\mu)}((\pi\triangleright\mu)_{(1)}\triangleright{-})((\pi\triangleright\mu)_{(2)}\triangleright{-}).$

Therefore, the compatibility of the product and coproduct is necessary. This
proves that $B_{\triangleright}$ is a bialgebra. Lemma 4.12 and Proposition
4.17 show that $B_{CEM}$ is the dual bialgebra of $B_{\triangleright}$.

Similarly to [22], we obtain a Hopf algebra once we take the quotient of
$B_{CEM}$ by the ideal
$\langle(\mathbf{1}-\forest{b_{d}{}_{9}0})\iota_{d}\,:d\in D\rangle$. The Hopf
algebra $H_{CEM}$ thus obtained is dual to the Hopf algebra
$H_{\triangleright}$ which is isomorphic to the Grossman-Larson Hopf algebra
obtained using Guin-Oudom process applied to the insertion pre-Lie product
$\blacktriangleright$. The elements of $H_{\triangleright}$ have the form

$\otimes_{d\in
D}\exp^{\cdot}_{C}(\forest{b_{d}{}_{9}0}\iota_{d})\cdot\pi,\quad\text{for
}\pi\in\mathcal{CF}_{D}^{\otimes D}.$

Following Guin-Oudom, the Grossman-Larson algebra is a universal enveloping
algebra of the respective pre-Lie algebra. $\square$

######  Theorem 4.19 (Substitution law)

Let $a\in\mathcal{CF}_{D}^{*}$ and let
$\\{b_{0,d}\in\mathcal{AT}_{D}^{*}\;:\;d\in D\\}$ be a set of infinitesimal
characters. Let us consider a map $\varphi:D\to\mathcal{AT}_{D}$ with
$\varphi(d)=\delta_{\sigma}(b_{0,d})$, let $F_{D}(d)=f_{d}$ for $d\in D$, and
let $S_{\varphi}=F_{D}\circ A_{\varphi}\circ\delta_{\sigma}$, then,

$S_{\varphi}(a)=S(b_{c}^{\otimes D}\star a),$

where $b_{c}^{\otimes D}=\otimes_{d\in D}b_{c,d}\iota_{d}$ with $b_{c,d}$
being a character of $\mathcal{CF}_{D}$ that extends $b_{0,d}$.

  * Proof

We use Lemma 4.15 and Proposition 4.13 to write $A_{\varphi}$ as

$A_{\varphi}(\pi)=\delta_{\sigma}(b_{c}^{\otimes
D})\triangleright\pi,\quad\text{where }b_{c}^{\otimes D}=\otimes_{d\in
D}b_{c,d}\iota_{d},$

with $b_{c,d}$ being a character of $\mathcal{CF}_{D}$ that extends $b_{0,d}$.
We use the fact that $S_{\varphi}=F_{D}\circ A_{\varphi}\circ\delta_{\sigma}$
and Proposition 4.17 to obtain

$S_{\varphi}(a)=F_{D}\big{(}\delta_{\sigma}(b_{c}^{\otimes
D})\triangleright\delta_{\sigma}(a)\big{)}=F_{D}\big{(}\delta_{\sigma}(b_{c}^{\otimes
D}\star a)\big{)}=S(b_{c}^{\otimes D}\star a).$

This finishes the proof. $\square$

Analogously to [69], due to the relation between the homomorphism
$A_{\varphi}:\mathcal{CF}_{D}\to\mathcal{CF}_{D}$ of D-algebra and the
corresponding map $b_{c}^{\otimes
D}\star:\mathcal{CF}_{D}^{*}\to\mathcal{CF}_{D}^{*}$, we obtain Corollary
4.20.

######  Corollary 4.20 (Cointeraction)

There is the following cointeraction between the substitution law and the
composition law:

$a_{c}^{\otimes D}\star(b\ast c)=(a_{c}^{\otimes D}\star b)\ast(a_{c}^{\otimes
D}\star c),$

for $a_{c}^{\otimes D}=\otimes_{d\in D}a_{c,d}\iota_{d}~{}$ where $a_{c,d}$
are characters of $\mathcal{CF}_{D}$ and $b,c\in\mathcal{CF}_{D}^{*}$.

  * Proof

Let $\varphi(d)=\delta_{\sigma}(a_{0,d})$, then we have

$\displaystyle\delta_{\sigma}(a_{c}^{\otimes D}\star(b\ast c))$
$\displaystyle=A_{\varphi}(\delta_{\sigma}(b\ast c))$
$\displaystyle=A_{\varphi}(\delta_{\sigma}(b))\diamond
A_{\varphi}(\delta_{\sigma}(c))$
$\displaystyle=\delta_{\sigma}((a_{c}^{\otimes D}\star b)\ast(a_{c}^{\otimes
D}\star c)).$

Since $\delta_{\sigma}$ is an isomorphism, the statement is proved. $\square$

We extend the definition of $\blacktriangleright$ to decorated aromatic
forests as $\blacktriangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{AF}_{D}\to\mathcal{AF}_{D}$ and note that $\Phi$ is a
$\mathcal{CF}_{D}^{\otimes D}$-module morphism, that is,

$\Phi(\pi\blacktriangleright\mu)=\pi\blacktriangleright\Phi(\mu).$

We extend the substitution product $\triangleright:\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{AF}_{D}\to\mathcal{AF}_{D}$ and the coproduct
$\Delta_{CEM}:\mathcal{AF}_{D}\to\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{AF}_{D}$ and use Proposition 4.17 to see that $\Phi^{*}$ is
a $\mathcal{CF}_{D}^{\otimes D}$-comodule morphism, that is,

$\Delta_{CEM}\circ\Phi^{*}=(\operatorname{id}\otimes\Phi^{*})\circ\Delta_{CEM}.$

Using the freeness of the tracial commutative D-algebra
$(\mathcal{AF}_{D},\curvearrowright)$, we note that for every map
$\varphi:D\to\mathcal{AT}_{D}$, there exists a corresponding
$A_{\varphi}:\mathcal{AF}_{D}\to\mathcal{AF}_{D}$ with the following property

$\Phi\circ A_{\varphi}=A_{\varphi}\circ\Phi,$

where we use the same notation for the $A_{\varphi}$ over decorated clumped
and aromatic forests. We describe the substitution law for S-series over
decorated aromatic forests by using the fact that any functional over
decorated aromatic forests $a\in\mathcal{AF}_{D}^{*}$ can be written as an
image of the map $\Phi^{*}$, that is, there exists a functional
$a_{C}\in\mathcal{CF}_{D}^{*}$ such that $a=a_{C}\circ\Phi^{*}$.

###### Theorem 4.21

Using the notation from Theorem 4.19 and given $a\in\mathcal{AF}_{D}$, we have

$S_{\varphi}(a)=S(b_{c}^{\otimes D}\star a),$

with convolution product $\star$ with respect to
$\Delta_{CEM}:\mathcal{AF}_{D}\to\mathcal{CF}_{D}^{\otimes
D}\otimes\mathcal{AF}_{D}$.

  * Proof

We note that given a functional $a\in\mathcal{AF}_{D}^{*}$, we can define a
functional $a_{C}\in\mathcal{CF}_{D}^{*}$ such that $a=a_{C}\circ\Phi^{*}$,
moreover, $S(a)=S(a_{C})$. The statement follows from the following
identities,

$S_{\varphi}(a)=S_{\varphi}(a_{C})=S(b_{c}^{\otimes D}\star
a_{C})=S((b_{c}^{\otimes D}\star a_{C})\circ\Phi^{*})=S(b_{c}^{\otimes
D}\star(a_{C}\circ\Phi^{*}))=S(b_{c}^{\otimes D}\star a),$

with $b_{c}^{\otimes D}:=\otimes_{d\in D}b_{c,d}\iota_{d}$. $\square$

Corollary 4.20 is extended to functionals over decorated aromatic forests in a
similar fashion.

###  4.4 Algebraic structures of exotic aromatic forests

We extend the results of Section 4 to the space $\mathcal{EAF}$ of exotic
aromatic forests defined in Definition 2.2. To do so, we build the commutative
tracial D-algebra $(\mathcal{EAF},\curvearrowright)$ of exotic aromatic
forests using a commutative tracial D-algebra
$(\mathcal{AF}_{\bullet\mathbb{N}},\curvearrowright)$ of decorated aromatic
forests.

We define the D-algebra $(\mathcal{AF}_{\bullet\mathbb{N}},\curvearrowright)$
to be a vector space spanned by aromatic forests with black vertices and
vertices decorated by natural numbers $\mathbb{N}$. The decoration is denoted
by $\alpha:V(\pi)\to\\{\bullet\\}\cup\mathbb{N}$ and we assume that every
number decorates an even number of vertices, that is, $\alpha^{-1}(n)$ is even
for $n\in\mathbb{N}$, and a vertex decorated by a number cannot have an
incoming edge. We note that our assumptions on the structure of
$\pi\in\mathcal{AF}_{\bullet\mathbb{N}}$ are compatible with the commutative
tracial D-algebra structure. For example,

$\forest{b[1,1,2,b[1]],b[2,1]}\cdot\forest{b[1,1,2,2]}=\forest{b[1,1,2,b[1]],b[2,1],b[1,1,2,2]},\quad\forest{b[1,1]}\curvearrowright\forest{b[b[1],1]}=\forest{b[b[1,1],b[1],1]}+\forest{b[b[b[1,1],1],1]}.$

We can obtain the D-algebra
$(\mathcal{AF}_{\bullet\mathbb{N}},\curvearrowright)$ by choosing an
appropriate sub-D-algebra and taking the quotient over an appropriate ideal of
the free D-algebra. We define the commutative tracial D-algebra of exotic
forests $(\mathcal{EAF},\curvearrowright)$ as a sub-D-algebra of
$(\overline{\mathcal{AF}}_{\bullet\mathbb{N}},\curvearrowright)$ spanned by
the elements

$(\pi,\alpha_{e}):=\sum_{\alpha\in P(\alpha_{e})}(\pi,\alpha)\ \in\
\overline{\mathcal{AF}}_{\bullet\mathbb{N}},$ (11)

with $|\alpha_{e}^{-1}(n)|\in\\{0,2\\}$ for $n\in\mathbb{N}$. $P(\alpha_{e})$
is the set of decorations $\alpha$ with
$\alpha^{-1}(\bullet)=\alpha_{e}^{-1}(\bullet)$ and

$\alpha(v_{1})=\alpha(v_{2})\quad\text{if
}\quad\alpha_{e}(v_{1})=\alpha_{e}(v_{2}),\quad\text{for }v_{1},v_{2}\in
V(\pi).$

We note that the exotic aromatic forests constructed in this way agree with
Definition 2.2. The D-algebra $(\mathcal{EAF},\curvearrowright)$ of exotic
aromatic forests is graded by the number of roots. The algebraic structures
related to D-algebras are studied in Section 4.1.

An exotic aromatic forest is connected if it cannot be written as a
concatenation of non-trivial exotic aromatic forests. This notion of
connectedness coincides with the one found in [56]. We note that due to the
pairings of the number vertices (that are also called lianas), exotic aromatic
forests can contain connected components which have more than one root, which
is a major difference with the standard Butcher trees and forests. For
example, the following exotic aromatic forests are connected:

$\forest{1,1},\quad\forest{b[b,1],b[b,b[1]]},\quad\forest{b[1],b[1,2],b[2,3],3}.$

We consider the coalgebra $(\mathcal{EAF},\Delta_{\mathcal{EA}})$ with
$\mathcal{EA}$-linear deshuffle coproduct $\Delta_{\mathcal{EA}}$ whose
$\mathcal{EA}$-module of primitive elements
$\operatorname{Prim}(\mathcal{EAF},\Delta_{\mathcal{EA}})$ is spanned by the
connected exotic aromatic forests, for example,

$\Delta_{\mathcal{EA}}(\forest{(b[1,1]),b[2],b[2,3],3})=\forest{(b[1,1]),b[2],b[2,3],3}\otimes\mathbf{1}+\mathbf{1}\otimes\forest{(b[1,1]),b[2],b[2,3],3}.$

We note that the space of exotic aromatic forests $\mathcal{EAF}$ can be
defined as the symmetric algebra
$S_{\mathcal{EA}}(\operatorname{Prim}(\mathcal{EAF}))$ over the ring of exotic
aromas. Analogously to Section 4.2, we define two possible extensions of the
concept of clumped forests to the exotic context,

$\mathcal{CEF}:=S(\operatorname{Prim}(\mathcal{EAF})),\quad{\mathcal{CEF}^{1}}:=S(\mathcal{EAT}).$

We note that both symmetric algebras are over the base field $\mathbb{R}$,
meaning that the exotic aromas are attached to the rooted components. We
recall that $\mathcal{EAT}$ is the space of exotic aromatic forests with one
root and ${\mathcal{CEF}^{1}}\subset\mathcal{CEF}$. Some elements of
$\mathcal{CEF}$ are

$\forest{(b),b[1],1}\cdot\forest{b[b,b[2]],b[2]}\cdot\forest{b[3],b[3,4],4},\quad\forest{b[1]=b[1],b[b,2],2}.$

We note that
$\forest{(b),b[1,1]}\cdot\forest{(b[2]),b[2]}\neq\forest{b[1,1]}\cdot\forest{(b),(b[2]),b[2]}$
in $\mathcal{CEF}$.

Let $\Phi:\mathcal{CEF}\to\mathcal{EAF}$ denote the map that forgets the
”clumping”, that is, it satisfies
$\ker{\Phi}=\langle\omega\tau\cdot\gamma\cdot\pi-\tau\cdot\omega\gamma\cdot\pi\;:\;\omega\in\mathcal{EA},\tau,\gamma\in\mathcal{EAT},\pi\in\mathcal{CEF}\rangle$.
The results of Section 2 yield the following structure on exotic aromatic and
exotic clumped forests.

###### Theorem 4.22

The space of exotic aromatic forests forms a Grossman-Larson Hopf algebroid

$(\mathcal{EAF},\mathbf{1},\diamond,\epsilon_{\mathcal{EA}},\Delta_{\mathcal{EA}},S_{\diamond}),$

and the space of clumped exotic forests forms a Grossman-Larson Hopf algebra

$(\mathcal{CEF},\mathbf{1},\diamond,\epsilon,\Delta,S_{\diamond}^{C}).$

Moreover, $\Phi:\mathcal{CEF}\to\mathcal{EAF}$ is a surjective algebra
morphism.

  * Proof

We note that, according to the definition (11), $\mathcal{EAF}$ is obtained by
taking a sub-D-algebra of $\overline{\mathcal{AF}}_{\bullet\mathbb{N}}$,
therefore, we can use the analysis from Section 4.1 to build a
$\mathcal{A}/_{\mathbb{R}}$-bialgebra
$(\mathcal{EAF},\mathbf{1},\diamond,\epsilon_{\mathcal{EA}},\Delta_{\mathcal{EA}})$.
We obtain a bialgebra $(\mathcal{CEF},\mathbf{1},\diamond,\epsilon,\Delta)$ by
noting that $\mathcal{CEF}$ can be obtained by taking a sub-D-algebra of
D-algebra $\overline{\mathcal{CF}}_{\bullet\mathbb{N}}$ which is a special
case of $\mathcal{CF}_{D}$ (Section 4.2).

Following [66], the primitive elements $\operatorname{Prim}(\mathcal{EAF})$
are endowed with a pre-Lie product:

$\pi_{1}\mathbin{\tilde{\curvearrowright}}\pi_{2}:=\pi_{1}\diamond\pi_{2}-\pi_{1}\cdot\pi_{2},\quad\text{for
}\pi_{1},\pi_{2}\in\operatorname{Prim}(\mathcal{EAF}).$

We use the pre-Lie product $\mathbin{\tilde{\curvearrowright}}$ to define the
antipode $S_{\diamond}$ for $\mathcal{EAF}$ by replacing all instances of
$\curvearrowright$ in Proposition 4.8 by the product
$\mathbin{\tilde{\curvearrowright}}$, all instances of
$\tau\in\mathcal{T}_{D}$ by $\tau\in\operatorname{Prim}(\mathcal{EF},\Delta)$,
and $\omega\in\mathcal{A}_{D}$ by $\omega\in\mathcal{EA}$ where $\mathcal{EF}$
denotes the space of exotic forests, that is, exotic aromatic forests without
aromas. The antipode $S_{\diamond}^{C}$ for $\mathcal{CEF}$ is obtained in a
similar way, but the trees $\tau\in\mathcal{T}_{D}$ are replaced by
$\tau\in\operatorname{Prim}(\mathcal{EAF})$ and the identities $(i)$ are
ignored. This proves that we have a Hopf algebroid and Hopf algebra structures
over $\mathcal{EAF}$ and $\mathcal{CEF}$, respectively. We note that
$\Phi(\mathbf{1})=\mathbf{1}$ and
$\Phi(\pi\diamond\mu)=\Phi(\pi)\diamond\Phi(\mu)$, so $\Phi$ is a surjective
algebra morphism. $\square$

###### Theorem 4.23

The space of clumped exotic forests $\mathcal{CEF}$ forms a Hopf algebra

$(\mathcal{CEF},\mathbf{1},\cdot,\mathbf{1}^{*},\Delta_{CEM},S),$

where $\Delta_{CEM}$ is the coproduct extended from
$\operatorname{Prim}(\mathcal{EAF})$ to $\mathcal{CEF}$ by respecting the
concatenation product. Moreover, $\Phi^{*}:\mathcal{EAF}\to\mathcal{CEF}$ is a
${\mathcal{CEF}^{1}}$-comodule morphism where $\Phi^{*}$ is the adjoint of
$\Phi$, that is, $\Phi\circ\delta_{\sigma}=\delta_{\sigma}\circ\Phi^{*}$.

  * Proof

The space $\mathcal{CEF}$ can be defined analogously to $\mathcal{EAF}$ using
(11) such that $\mathcal{CEF}\subset\overline{\mathcal{CF}}_{D}$ with
$D=\\{\bullet\\}\cup\mathbb{N}$. We recall that, following Theorem 4.18,
$B_{CEM}:=(\mathcal{CF}_{D}^{\otimes D},\cdot,\Delta_{CEM})$ is a bialgebra
which becomes a Hopf algebra $H_{CEM}:=(\mathcal{CF}_{D},\cdot,\Delta_{CEM})$
once we take the quotient of $B_{CEM}$ by the ideal
$\langle(\mathbf{1}-\forest{b})\iota_{\bullet}\rangle+\mathcal{J}$ defined as

$\mathcal{J}:=\langle(\mathbf{1}-\forest{i_{k}})\iota_{\forest{i_{k}}},\pi\iota_{\forest{i_{k}}}\;:\;\pi\notin\\{\mathbf{1},\forest{i_{k}}\\},k\in\mathbb{N}\rangle.$

The obtained coproduct of $H_{CEM}$ can now be described as

$\Delta_{CEM}(\pi)=\sum_{p\subset\pi}p\otimes\pi/_{p},$

where the sum is over all clumped subforests $p\in\mathcal{CF}_{D}$ that cover
all black vertices and $\pi/_{p}$ is the clumped forest obtained by
contracting the aromatic trees of $p$ into black vertices. If the forest
$\pi\in\mathcal{CF}_{D}$ doesn’t have valid subforests $p\in\mathcal{CF}_{D}$,
then $\Delta_{CEM}(\pi)=\mathbf{1}\otimes\pi$. We note that the construction
of $\mathcal{CEF}$ with the CEM Hopf algebra structure as a quotient and
subspace of $\overline{\mathcal{CF}}_{D}$ must be dual to the one seen in
(11). We obtain the Hopf algebra

$(\mathcal{CEF},\mathbf{1},\cdot,\mathbf{1}^{*},\Delta_{CEM},S),$

with $\Delta_{CEM}:\mathcal{CEF}\to\mathcal{CEF}\otimes\mathcal{CEF}$. We
recall the analysis of Section 4.3 and note that since
$\exp^{\odot}_{C}(b_{0})$ is a character of ${\mathcal{CEF}^{1}}$, we consider
$\Delta_{CEM}:\mathcal{CEF}\to{\mathcal{CEF}^{1}}\otimes\mathcal{CEF}$. We
refer to the discussion from Section 4.3 for showing that $\Phi^{*}$ is a
${\mathcal{CEF}^{1}}$-comodule morphism. $\square$

We can simplify the computation of the substitution law for exotic aromatic
S-series using the map $\Phi$ and Theorem 4.23. Given a functional
$a\in\mathcal{EAF}^{*}$, we use the discussion from Section 4.2 to define a
functional $a_{C}\in\mathcal{CEF}$ over clumped exotic forests as

$a_{C}(\pi):=\frac{1}{n^{m}}a(\Phi(\pi)),$

where $n$ is the number of rooted components and $m$ is the number of aromas.
We have the property $a=a_{C}\circ\Phi^{*}$ which we use to compute the
substitution law as follows

$b_{c}\star a=b_{c}\star(a_{C}\circ\Phi^{*})=(b_{c}\star a_{C})\circ\Phi^{*}.$

The substitution law $b_{c}\star a_{C}$ over $\mathcal{CEF}$ is easier to
compute since $\Delta_{CEM}$ over $\mathcal{CEF}$ respects concatenation.
Therefore, computing the values of $\Delta_{CEM}$ on
$\operatorname{Prim}(\mathcal{EAF})$ is enough to obtain its values over all
clumped exotic forests.

###### Example 4.24

Let us compute the CEM coproduct over $\mathcal{EAF}$ using the comodule
morphism $\Phi^{*}$.

$\displaystyle(\Delta_{CEM}\circ\Phi^{*})(\forest{(b),b[1],1,b})$
$\displaystyle=\Delta_{CEM}(\forest{(b),b[1],1}\cdot\forest{b})+\Delta_{CEM}(\forest{b[1],1}\cdot\forest{(b),b})$
$\displaystyle=\Delta_{CEM}(\forest{(b),b[1],1})\hat{\cdot}\Delta_{CEM}(\forest{b})+\Delta_{CEM}(\forest{b[1],1})\hat{\cdot}\Delta_{CEM}(\forest{(b),b}),$

where
$(\pi_{(1)}\otimes\pi_{(2)})\hat{\cdot}(\mu_{(1)}\otimes\mu_{(2)})=(\pi_{(1)}\cdot\mu_{(1)})\otimes(\pi_{(2)}\cdot\mu_{(2)})$
and

$\displaystyle\Delta_{CEM}(\forest{(b),b[1],1})$
$\displaystyle=\forest{b,b}\otimes\forest{(b),b[1],1}+\forest{(b),b}\otimes\forest{b[1],1},$
$\displaystyle\Delta_{CEM}(\forest{(b),b})$
$\displaystyle=\forest{b,b}\otimes\forest{(b),b}+\forest{(b),b}\otimes\forest{b},$

$\displaystyle\Delta_{CEM}(\forest{b})$
$\displaystyle=\forest{b}\otimes\forest{b},$
$\displaystyle\Delta_{CEM}(\forest{b[1],1})$
$\displaystyle=\forest{b}\otimes\forest{b[1],1}.$

Therefore, we have

$\displaystyle(\Delta_{CEM}\circ\Phi^{*})(\forest{(b),b[1],1,b})$
$\displaystyle=\forest{b,b,b}\otimes\forest{(b),b[1],1}\cdot\forest{b}+\forest{(b),b}\cdot\forest{b}\otimes\forest{b[1],1,b}$
$\displaystyle+\forest{b,b,b}\otimes\forest{b[1],1}\cdot\forest{(b),b}+\forest{(b),b}\cdot\forest{b}\otimes\forest{b[1],1,b}$
$\displaystyle=\forest{b,b,b}\otimes\Phi^{*}(\forest{(b),b[1],1,b})+2\forest{(b),b}\cdot\forest{b}\otimes\Phi^{*}(\forest{b[1],1,b}),$

which agrees with a direct computation that gives

$\Delta_{CEM}(\forest{(b),b[1],1,b})=\forest{b,b,b}\otimes\forest{(b),b[1],1,b}+2\forest{(b),b}\cdot\forest{b}\otimes\forest{b[1],1,b}.$

##  5 Conclusion

In this paper, we uncover the fundamental algebraic structures that govern the
approximations in the weak sense and for the invariant measure of stochastic
dynamics with additive noise in both Euclidean and manifold settings. We
present in particular the novel concept of clumping for the Hopf algebra
structures associated with substitution using aromas. Additionally, we
introduce the free commutative tracial D-algebra of decorated aromatic
forests, along with the related Grossman-Larson Hopf algebroid and pre-Hopf
algebroid. This algebraic study enables us to describe the substitution law
for decorated aromatic forests by linking substitution to D-algebra
homomorphisms. Understanding the substitution law for exotic aromatic S-series
provides an elegant algebraic framework for the backward error analysis of