|
This chapter is taken from the paper [Y. Bruned, M. Hairer, L. Zambotti, Invent. Math. 215 (2019), no. 3, 1039–1156].
A rooted tree is a finite connected simple
graph without circles with a distinguished vertex
, called the root. We assume that
our trees are combinatorial, i.e. there is no particular order
imposed on edges leaving any given vertex.
Vertices of is also called nodes. The
set of nodes of the tree
is denoted by
, and set of edges the tree
is denoted by
.
We endow
with the partial order
where
iff
is on the unique path connecting
to the
root, and we orient edges in
so that if
, then
. In this way, we can always view a tree as
a directed graph.
Given a sequence of vector space ,
denote by
called a bigraded space. Given two bigraded spaces
and
, define
is s.t.
unless
(i.e.
&
). Given two bigraded spaces
and
, we
define
s.t.
can be view as a bigraded space with
Let . Define
. Denote by
the free vector space generated by
.
we are given a (finite) collection
of
subforests
of
s.t.
. Define
by
Define the linear functional by
. If
is equal to all subforests
of
containing
,
then
is a coalgebra. Since the inclusion
endows the set of typed forests with
partial order,
is an example of an
incidence coalgebra.
Given a typed forest ,
consider a several disjoint subforest
,
,
,
of
. A natural way to code
is to use a coloured forest
where
Then we have for
and
.
Assumption 1. Let .
For each colour forest
,
we are given a collection
of subforests of
s.t.
&
∀
,
,
connected component
of
, one has either
or
.
We also assume that is compatible with
forest isomorphisms described above in the sense that
In this chapter, we consider the equation in the
full sub-critical regime. The equation is formally given by
where the term represents the renormalisation
(it can be quadratic in
) and
the noise
for some
.
This chapter is a note taken from the paper [A. Chandra, A. Moinat, H.
Weber, Arch. Ration. Mech. Anal. 247 (2023), no. 3, Paper No. 48].
We introduce the following ingredients.
An operator to represent an abstract Duhamel
operator for heat equation.
Let be the smallest set containing the
symbols
s.t. “
”.
. We use
to indicate elements in
.
We call the elements in
trees. In a given
tree,
occur in that tree are called leaves
in the tree.
.
Define functions which counts, on any given
tree, the number of occurrences of
,
and
as leaves in the tree. Denote also by
.
.
,
,
,
.
,
consists of
s.t.
and
.
.
.
.
.
This note is extended from the a course called MAGIC 109 given by Dr Y. Bazkov, adding some materials from the following resources:
M. E. Sweedler. Hopf algebras, Math. Lecture Note Ser. W. A. Benjamin, Inc., New York, 1969, vii+336 pp.
We intend to study the intrinsic geometric property of the real line
by looking at how functions on the real line can
transform. For more algebraic setting, we only consider polynomials over
real variable. Let us look at the following transforms.
Given a real number and
is a function on the real line, we define the following translation
by
:
Define also the reflection
and the derivative
The translation, reflection, derivative are not independent from each other. In fact, there are some relations:
![]() |
(A.1.1) |
![]() |
(A.1.2) |
![]() |
(A.1.3) |
Proof of (A.1.1).
Also,
![]() |
(A.1.4) |
![]() |
(A.1.5) |
![]() |
(A.1.6) |
Now we write down the relation (A.1.1)-(A.1.2)-(A.1.3)-(A.1.4)-(A.1.5)-(A.1.6)
for ,
, and
,
so these transformations live their own lifes without the mention of the
functions they act on. But the algebraic structure given by (A.1.1)-(A.1.2)-(A.1.3)-(A.1.4)-(A.1.5)-(A.1.6) is too “blank”, it does not contain enough
information for us to effectively work with these abstract symbols as
symmetries of the function space on the real line. What is missing here
is how the operators
,
, and
behave when we apply them to a product of functions. Given functions
and
,
![]() |
(A.1.7) |
To get rid of the mention of and
in (A.1.7), we introduce a new symbolic
notation coproduct
,
satisfying
Likewise, we also require
and the Leibniz rule
The meaning of will become clear later in this
note.
A Hopf algebra , roughly
speeking, is a collection of operators which has
rules for multiplication (i.e., how to compose operators);
rules for coproduct (i.e., how an operator acts on a product of functions);
rules for The Antipode (roughly an analogue of inverse in group).
In the example we study in Subsection A.1.1, we say that
the operators ,
, and
generate some
Hopf algebra. However, these operators are known even from 19 centrary,
what is the point to rename them? Hopf algebras gain an advantage over
groups when we need symmetries of a noncommutative space. The
noncommuting objects upon which we want to act can be thought of as
“functions” on a mythical “quantum group”.
We consider vector space over field
(sometimes we might use
or
). Recall that a linearly
independent set
is such that for any
,
,
implies
.
Given a subset of
, say
, the span of
is defined by
. A basis
of
is linearly independent
subset of
such that
. By using Zorn's lemma, one can prove that every
vector space has a basis. All bases of
are of
the same cardinality which is denoted by
.
We can also argue that every set is a basis of some vector space.
Indeed, let
be a set, and let
Moreover, we can identify in
with an element in
, which is
, and one can check that
under this identification the set
becomes a
basis of
. This fits a
philosophy we have to adhere in this note: everything is vector space;
everything map is a linear map.
It is time to state the first result in this note, a result that we will used later.
Proposition be a basis of vector space
,
a vector space, and
a function. Then there exists a unique linear map
such that
.
We will need a number of constructions which allows us to create new
vector spaces from the existing ones. For example, direct product and
direct sum of vector spaces means
and
respectively. Let us note that for any ,
is a subspace of
.
Let us also remark that direct product and direct sum can be defined for
an uncountable family of vector spaces. However, we only consider
countable familys in this note.