Table Of ContentDECAY RATE OF ITERATED INTEGRALS OF BRANCHED
ROUGH PATHS
5 HORATIOBOEDIHARDJO
1
0
Abstract. Iterated integrals of paths arise frequently in the study of the
2
Taylor’s expansion for controlled differential equations. We will prove a fac-
v torial decay estimate, conjectured by M. Gubinelli, for the iterated integrals
o ofnon-geometricroughpaths. Wewillexplain, withacounterexample, why
N theconventionalapproachofusingtheneoclassicalinequalityfails. Ourproof
involvesaconcavityestimateforsumsoverrootedtreesandanon-trivialex-
1 tension of T. Lyons’ proof in 1994 for the factorial decay of iterated Young’s
2 integrals.
]
R
P
1. Introduction
.
h
The iterated integrals of a path arise naturally from the Taylor’s expansion of a
t
a
controlled differential equation driven by the path and play a fundamental role in
m
the theory of rough paths [7]. Given a path x, we are interested in the behaviour
[ of the iterated integral
ˆ
3
(1.1) Xn = dx ⊗...⊗dx
v 0,1 s1 sn
1 0<s1<...<sn<1
4 as n varies. The solution for a linear controlled differential equation has a series
6 expansion that is linear in these iterated integrals. The convergence of the series
5 expansion is often studied using the decay of these iterated integrals. The sim-
0 plest example is the case when x takes value in Rd and has an almost everywhere
.
1 derivative in L∞, in which case
0
|x˙|n
5 (cid:107)Xn (cid:107)≤ L∞.
1 0,1 n!
: The problem becomes much harder when x does not have a derivative, such as in
v
i the case when the iterated integral (1.1) is defined in terms of Young’s integration.
X
It was proved by T. Lyons [6] that if x is γ-Hölder, γ > 1, and (cid:107)x(cid:107) denotes the
2 γ
r
γ-Hölder norm of x, then
a
(cid:107)x(cid:107)n
(1.2) (cid:107)Xn (cid:107)≤(1+ζ(2γ))n−1 γ,
0,1 n!γ
where ζ is the classical Riemann Zeta function. For 0 < γ ≤ 1 and N = (cid:98)γ−1(cid:99), a
2
γ -Hölder geometric rough path takes value in the unital tensor algebra
T(N)(cid:0)Rd∗(cid:1)=1⊕Rd∗...⊕(cid:0)Rd∗(cid:1)⊗N
Keywordsandphrases. Branchedroughpaths;non-geometricroughpaths;iteratedintegrals.
AsubstantialpartofthisworkwascarriedoutwhiletheauthorwasatOxford-ManInstitute,
OxfordUniversity,UK.WearegratefultothesupportofERCgrantEsig(agreementno. 291244
) during that period. We would also like to thank M. Gubinelli and D. Yang for the useful
discussions.
1
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 2
whereRd∗ denotesthedualofRd. T.Lyons[7]showedthataγ -Hölderroughpath
x can be extended uniquely to a γ -Hölder path X in T(n)(cid:0)Rd∗(cid:1) for any n ≥ N.
He defined the n-th order iterated integrals of x up to time 1 as the n-th tensor
component of this extended path at time 1, which we will denote for latter use as
Xn , and showed that
0,1
(cid:13)(cid:13)Xn (cid:13)(cid:13)≤γ−n(cid:16)1+2(N+1)γζ((N +1)γ)(cid:17)n (cid:107)x(cid:107)nγ ,
0,1 Γ(nγ+1)
where Γ is the Gamma function and (cid:107)x(cid:107) denotes the γ-Hölder norm of the rough
γ
path x.
RecentlyM.Gubinelli[4]proposedanon-geometric theoryofroughpath,known
as the branched rough paths. The phrase “non-geometric” here refers to that the
calculus with respect to branched rough paths does not have to satisfy the chain
rule
d(XY)=YdX+XdY
which Lyons’ geometric rough paths must satisfy. For the Brownian motion B, the
rough path (almost surely defined)
ˆ ˆ ˆ
(cid:18) t t u1 (cid:19)
(s,t)→ 1, dB , dB ⊗dB
s1 s1 s2
s s s
is geometric if the integration is defined in the sense of Stratonovich and non-
geometric if the integration is defined in the sense of Itô. Branched rough paths
areindexedbytheConnes-KremierHopfalgebraoflabelledrootedtrees, whichwe
will denote by H and will recall in Section 2. The multiplication of trees in H
L L
corresponds to the multiplication of the coordinate components of the path, while
the operation of joining forests to a single root corresponds to integrating against
the path. The theory of branched rough paths is a rough path analogue Butcher’s
tree-indexed series expansions of solutions to differential equations and has also
been motivated by expansions in stochastic partial differential equations. We now
recall an equivalent definition of branched rough path due to Hairer-Kelly [3].
Definition 1. ([4], [3])Let 0 < γ ≤ 1. Let (H ,·,(cid:52),S) be the Connes-Kremier
L
Hopf algebra of rooted trees labelled by a finite set L. Let (H∗,(cid:63),δ,s) be the dual
L
Hopfalgebraof(H ,·,(cid:52),S). Aγ-branchedroughpathisamapX :[0,1]×[0,1]→
L
H∗ such that
L
1. for all s≤t and all h ,h ∈H ,
1 2 L
(1.3) (cid:104)X ,h (cid:105)(cid:104)X ,h (cid:105)=(cid:104)X ,h ·h (cid:105).
s,t 1 s,t 2 s,t 1 2
2. for all u≤s≤t,
(1.4) X (cid:63)X =X .
u,s s,t s,t
3.for all labelled rooted tree τ, if |τ| denote the number of vertices in τ, then
|(cid:104)X ,τ(cid:105)|
(1.5) (cid:107)X(cid:107) :=sup s,t <∞.
γ,τ s(cid:54)=t |t−s|γ|τ|
Remark 2. Hairer-Kelly [3] pointed out that the product (cid:63) in H∗ is induced by
L
the coproduct (cid:52) on H in the following sense: If h ∈ H is a rooted tree and
L L
(cid:52)h=(cid:80)h(1)⊗h(2), then for X,Y ∈H∗,
L
(cid:88)
(cid:104)X(cid:63)Y,h(cid:105)= (cid:104)X,h(1)(cid:105)(cid:104)Y,h(2)(cid:105).
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 3
Hairer-Kelly also realised that condition 1. in the Definition 1 of branched rough
pathisequivalenttoX takingvalueinthegroupofcharactersoftheHopfalgebra,
known as the Butcher group, analogous to the nilpotent Lie group in the geometric
case.
Example 3. Let 1 <γ ≤1, and x=(cid:0)x1,...,xd(cid:1):[0,1]→Rd be a γ-Hölder path
2
in the sense that
(cid:107)x −x (cid:107)
sup t s <∞,
|t−s|γ
s(cid:54)=t
then,accordingtoGubinelli[4],wemayliftxtoaγ-branchedroughpathX inthe
following way:
Let • be a vertex labelled by i ∈ {1,...,d}. Let τ ,...,τ be rooted trees la-
i 1 n
belledby{1,...,d},and[τ ,...,τ ] denotethelabelledtreeobtainedbyconnect-
1 n •i
ing the roots of τ ,...,τ to the labelled vertex • . Then X is defined inductively
1 n i
by (cid:104)X ,• (cid:105)=xi−xi and
s,t i t s
ˆ
t
(1.6) (cid:104)X ,[τ ,...,τ ] (cid:105)= Πn (cid:104)X ,τ (cid:105)dxi.
s,t 1 n •i j=1 s,u j u
s
Moregenerally,in[3],Hairer-Kellygaveanexplicitwayofextendingaγ-geometric
rough path to a γ-branched rough path.
For general 0 < γ ≤ 1, let N = (cid:98)γ−1(cid:99), then by Theorem 7.3 in [4], given a
family of real-valued functions ((cid:104)X ,τ(cid:105)) on [0,1]×[0,1] satisfying the
·,· τ∈HL,|τ|≤N
conditions (1),(2) and (3) in Definition 1 of branched rough path, there is a unique
way of extending ((cid:104)X ,τ(cid:105)) to a γ-branched rough path ((cid:104)X ,τ(cid:105)) .
·,· |τ|≤N ·,· τ∈HL,|τ|≥0
Gubinelli [4] conjectured that this extension, which can be interpreted as the iter-
ated integrals of the truncated branched rough path ((cid:104)X ,τ(cid:105)) has a tree
·,· τ∈H,|τ|≤N
factorial decay. Our main result, stated below, is a proof of this conjecture.
Theorem 4. Let 0 < γ ≤ 1 and N = (cid:98)γ−1(cid:99). Let X be a γ−branched rough path.
For all rooted trees τ and all s≤t,
c|τ|(t−s)γ|τ|
(1.7) |(cid:104)X ,τ(cid:105)|≤ N .
s,t τ!γ
where
(cid:32) N+1 (cid:33)
cN = 6exp 7 (cid:88) (N +1)i+1 (cid:12)(cid:12)TN(cid:12)(cid:12)2−2γ2(N+1)γζ((N +1)γ)N!γ max (cid:107)X(cid:107)|γσ,σ|−1
1≤|σ|≤N
i=0
and TN is the set of unlabelled rooted trees with at most N vertices.
Remark 5. For γ =1, Gubinelli [4] showed that the decay rate in (1.7) is attained
for the identity path X, defined for all rooted trees τ by
(t−s)|τ|
(cid:104)X ,τ(cid:105)= .
s,t τ!
Remark 6. M.Gubinelli[5]usedasimilartypeoffactorialdecayestimatetoprove
the convergence of his series expansion for the solution of the three-dimensional
Navier-Stokes equation for sufficiently small initial data.
Theorem 4, together with the Hairer-Kelly result [3] that geometric rough paths
are branched rough paths, gives another proof for the factorial decay for geometric
roughpaths. Insomecases,ourmainresultgivesasharperestimatefortheshuffled
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 4
sum of the iterated integrals than the one derived using shuffle product and the
factorial decay for geometric rough paths, as the following example demonstrates.
Example 7. Letxandy berealvaluedγ−Hölderpathson[0,1], 1 <γ ≤1. Then
2
we may estimate absolute value of the integral
ˆ
1
xndy
s s
0
by the “geometric method”
ˆ ˆ ˆ ˆ
(cid:12)(cid:12)(cid:12)(cid:12) 1xnsdys(cid:12)(cid:12)(cid:12)(cid:12) = n!(cid:12)(cid:12)(cid:12)(cid:12) 1 s... s2dxs1...dxsndys(cid:12)(cid:12)(cid:12)(cid:12)
0 0 0 0
(cid:107)(x,y)(cid:107)n+1
(1.8) ≤ n!(1+ζ(2γ))n−1 γ
(n+1)!γ
where the inequality follows from Lyons’ factorial decay estimate (1.2).
ByExample3, wemayextendthetwo-dimensionalpath(x,y)toaγ−branched
rough path X. Let σ be the labelled forest defined inductively by σ = • and
n 1 1
σ = σ ·• where the operation · is the formal multiplication of rooted trees.
n n−1 1
Let τ =[σ ] . Graphically, τ takes the following form:
n n •2 n
Then ˆ
1
(cid:104)X,τ (cid:105)= xndy
n s s
0
and our main result Theorem 4 gives the estimate
ˆ
(1.9) (cid:12)(cid:12)(cid:12)(cid:12) 1xnsdys(cid:12)(cid:12)(cid:12)(cid:12) ≤ cτn1+!γ1 = (nc+n1+11)γ,
0 n
where c is the constant (independent of n) appearing in our main result Theorem
1
4. As n → ∞, this growth rate is a much better rate than the factorial growth
given by the geometric estimate (1.8).
1.1. Thestrategyofproof. T.Lyons[7]provedthefactorialdecayforγ-geometric
rough paths using the following inductive definition of Xn
n−1
(cid:88) (cid:88)
(1.10) Xn = lim Xn−kXk .
u,t |P|→0 u,ti ti,ti+1
P⊂[u,t]ti∈Pk=1
This approach requires the use of a highly non-trivial binomial-type inequality,
known as the neoclassical inequality, of the form
(cid:88)n aiγb(n−i)γ (a+b)nγ
(1.11) ≤γ−2
Γ(iγ+1)Γ((n−i)γ+1) Γ(nγ+1)
i=0
which is proved Lyons’ 98 paper [7]. A sharp version of this inequality latter
appeared in the work of K. Hara and M. Hino [2]. Gubinelli showed in [4] that a
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 5
sufficientconditionforthefactorialdecayforbranchedroughpathsisaneoclassical
inequalityforrootedtrees. Unfortunately,weareabletogiveacounterexamplefor
such inequality (see Lemma 10 in Section 3). Lyons’ 94 approach [6], which proved
the factorial decay for the γ > 1 case, did not use the neoclassical inequality and
2
use instead the equivalent definition
N
(cid:88) (cid:88)
(1.12) Xn = lim Xn−kXk
u,t |P|→0 u,ti ti,ti+1
P⊂[u,t]ti∈Pk=1
where N = (cid:98)γ−1(cid:99). This approach also has its own difficulty, due to the fact that
the function (s,t)→ω (s,t) defined by
u
(cid:32) (cid:88)m (s−u)m−k(t−s)k(cid:33)N1+1
ω (s,t)=
u (m−k)!k!
k=N+1
is not a control in the sense that ω (s,v)+ω (v,t)(cid:2)ω (s,t). The control prop-
u u u
erty is essential in the use of Young’s method of estimating (1.12) by successively
removing partition points from the partition. Lyons’ 94 approach [6] gets around
thisproblembyusingacontrolfunction(s,t)→R (s,t)whichdominatesω (s,t)
u u
and satisfies some binomial properties similar to that of ω (s,t). A key difficulty
u
in this paper is to find the right function R in the case of branched rough paths.
Our strategy consists of:
(1) Proving a bound for the multiplication operator (cid:63) with respect to some
norm, analogous to the following bound of tensor product
(cid:107)a⊗b(cid:107)≤(cid:107)a(cid:107)(cid:107)b(cid:107)
for a∈V⊗m and b∈V⊗n in the geometric case.
(2) Prove that our function R is compatible with the tree multiplication.
(3) ProvethatourfunctionRiscompatiblewiththeoperationofjoiningforests
to a single root.
2. Branched Rough paths: Notation and Terminology
A rooted tree is a connected, rooted graph such that for every vertex in the
graph, there exists a unique path from the root to the vertex. Let T denote the
set of rooted trees. The empty tree will be denoted by 1. A forest is a finite set of
rooted trees. The set of forests will be denoted by F. We will identify two trees τ
1
and τ if the forests obtained by removing the respective roots from τ and τ are
2 1 2
equal. We define a commutative multiplication · on F by
x·y =x∪y.
Let•denotetherootedtreeconsistedofasinglevertex. Wewilluseaboldsymbol
(e.g. τ)todenoteaforestwhileusingthenormalsymbol(e.g. τ)todenotearooted
tree. For σ = {τ ,...,τ } ∈ F, where τ ,...,τ are rooted non-empty trees, let
1 n 1 n
[σ] denote the rooted tree obtained by joining the roots of τ ,...,τ to the vertex
• 1 n
•. NotethatF isthesetfreelygeneratedbyelementsoftheform{•}, throughthe
operations of · and σ → [σ] . These two operations in fact correspond to the two
•
fundamental operations in rough path theory, namely the multiplication between
path components and the integration against a path.
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 6
Tosimplifyournotation,wewilldenotetheelement{τ ,...,τ }inF simplyby
1 n
τ ...τ . We will let H denote the formal vector space spanned by F over R. For a
1 n
forest τ, c(τ) will denote the number of non-empty trees in τ and |τ| denote the
total number of vertices in the forest. For each tree τ, the tree factorial is defined
inductively as
•! = 1,
[τ ,...,τ ] ! = |[τ ,...,τ ] |τ !...τ !.
1 n • 1 n • 1 n
The factorial of a forest τ ...τ is defined to be τ !...τ !.
1 n 1 n
FollowingHairer-Kelly[3],acoproductofrootedtreescanbeinductivelydefined
as (cid:52):H→H⊗H,
(cid:52)1 = 1⊗1;
(2.1) (cid:52)[τ ...τ ] = [τ ...τ ] ⊗1+(cid:88)τ(1)...τ(1)⊗(cid:104)τ(2)...τ(2)(cid:105) ;
1 n • 1 n • 1 n 1 n
•
(cid:52)(τ ...τ ) = (cid:52)τ ...(cid:52)τ .
1 n 1 n
where the sum in (2.1) denotes summing over all terms τ(1) and τ(2) in (cid:52)τ =
i i i
(cid:80)τ(1) ⊗τ(2). Here we define the product · on H⊗H by extending linearly the
i i
relation
(a⊗b)·(c⊗d)=(a·c)⊗(b·d).
The coproduct operator (cid:52) is coassociative. In Connes-Kreimer’s original work
[1], an antipode operator S has been constructed explicitly for H, so that the
bialgebra (H,·,(cid:52),S) becomes a Hopf algebra. This Hopf algebra is called the
Connes-Kreimer Hopf algebra.
Example 8. The following are all non-empty rooted trees with 3 or less vertices
The coproduct (cid:52) also has an interpretation in terms of cuts. A cut of a rooted
tree is a set of edges in a rooted tree. A cut is admissible for a rooted tree τ if for
any vertex in τ, the path from the root to the vertex passes through at most one
element in the cut. For each admissible cuts c, let τ(1)and τ(2)denote, respectively
c c
the components in τ\c that is disconnected from the the root and the component
that is connected to the root. Then
(cid:88)
(2.2) (cid:52)τ = τ(1)⊗τ(2).
c c
Admissible cuts c
Givenaforestτ =τ ...τ andσ(1),σ(2) inF,wewilldefinethecountingfunction
1 n
c(cid:0)τ,σ(1),σ(2)(cid:1) to be the number of times σ(1)⊗σ(2) appears in the sum (2.2).
We will follow the notation of Gubinelli [4] and use
(cid:88) (cid:16) (cid:17)
f τ,τ(1),τ(2)
to denote the summation over all τ(1) and τ(2) which appears in the sum (2.2).
Remark 9. Although the definition of branched rough paths requires the rooted
trees to be labelled, the assumption and conclusion of our main result Theorem 4
are uniform estimates across all labellings. Therefore we can forget that there is
any labelling and deal with only unlabelled rooted trees or forests. We will let Tn
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 7
andFn denote,respectively,thesetofall(unlabelled)rootedtreesandforestswith
n vertices.
3. Counter example to the tree neoclassical inequality
Wenowgiveacounterexampleforaweakerversionoftheneoclassicalinequality,
which would have been sufficient in proving the factorial decay for the iterated
integrals of branched rough paths. The notation •n will denote the forest
••...•.
(cid:124) (cid:123)(cid:122) (cid:125)
n
Lemma 10. Let τ be the tree [•n] . Then for all 0 ≤ γ < 1, for all β > 0, there
n •
exists a,b>0 such that as n→∞,
(cid:32) (cid:33)γ
(3.1) (a+b)−γ|τn|(cid:88) τn! 1 aγ|τn(1)|bγ|τn(2)| →∞.
τn(1)!τn(2)! βc(τn(1))+c(τn(2))
Proof. By definition, τ ! = n+1. Observe that by the definition of coproduct (cid:52)
n
(see (2.1)),
n (cid:18) (cid:19)
(cid:88) n
(cid:52)τ = •l⊗τ +τ ⊗1,
n l n−l n
l=0
where (cid:0)n(cid:1) denotes the binomial coefficient n! . Therefore,
l (n−l)!l!
(a+b)−γ|τn|(cid:88)(cid:16) τn! (cid:17)γ 1 aγ|τn(1)|bγ|τn(2)|
τn(1)!τn(2)! βc(τn(1))+c(τn(2))
≥ (a+b)−γ(n+1)(cid:88)n (cid:16) n+1 (cid:17)γ(cid:18)n(cid:19) 1 aγlbγ(n+1−l)
n+1−l l βl+1
l=0
1 (cid:18)aγ (cid:19)n
(3.2) ≥ (a+b)−γ(n+1)bγ +bγ ,
β β
where in the last line we used that for l ≤ n, (n+1)/(n+1−l) ≥ 1 and the
binomial theorem. Since 0≤γ <1 and a≥0,
(1+a)γ ≤1+γa.
1
Therefore, for all 0<a<(βγ)γ−1,
aγ
(1+a)γ <1+ .
β
Hence for a<(βγ)γ−11 and b=1, (3.2) diverges as n tends to infinity. (cid:3)
4. Bound for the multiplication operator (cid:63)
The multiplication (cid:63) in the Hopf algebra H plays the role of the tensor product
⊗ in the theory ofgeometric rough paths. In that case, one of the key assumptions
made about the tensor norms is that for all Xn ∈V⊗n and Yk ∈V⊗k,
(cid:13)(cid:13)Xn⊗Yk(cid:13)(cid:13) ≤(cid:107)Xn(cid:107) (cid:13)(cid:13)Yk(cid:13)(cid:13)
V⊗(n+k) V⊗n V⊗k
sothatthetensormultiplicationhasnorm1. Wemighthopethatthemultiplication
withrespectto(cid:63)wouldalsohavenorm1. Unfortunately,givenanynumbersnand
k, rooted trees in general has more than one way of being cut into two components
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 8
of sizes n and k respectively. This causes the multiplication operation to have a
norm that potentially depends on n and k. Fortunately, and it is a key observation
in our proof, the norm can be bounded by a function of k, independently of n. Let
us first describe the norm that we require.
Let X ∈H∗. Define a linear functional Xk ∈H∗ by
(cid:40)
(cid:10)Xk,τ(cid:11)= (cid:104)X,τ(cid:105), |τ|=k;
0, |τ|=(cid:54) k.
We define Xn(cid:63)Yk such that for all forests τ,
(cid:10)Xn(cid:63)Yk,τ(cid:11)=(cid:10)Xn⊗Yk,(cid:52)τ(cid:11).
Let
(cid:13)(cid:13)Xk(cid:13)(cid:13) = max |(cid:104)X,τ(cid:105)|βc(τ)τ!γ,
T,γ,β |τ|=k,τ trees |τ|!γ
and
(cid:13)(cid:13)Xk(cid:13)(cid:13) = max |(cid:104)X,τ(cid:105)|βc(τ)τ!γ.
F,γ,β |τ|=k,τ forests |τ|!γ
In this section we will prove a bound on the norm of the multiplication (cid:63) with
respectto(cid:107)·(cid:107) ,whichisthefirstofthreemainstepsinprovingourmainresult.
T,γ,β
Lemma 11. (Multiplication is bounded in tree norm)Let γ ≤1 and that
k
c :=exp(cid:2)(cid:88)ki(1−γ)(cid:3), β ≥c .
k k
i=1
Let X,Y ∈H∗. Then for n≥1,
(cid:13)(cid:13)Xn(cid:63)Yk(cid:13)(cid:13)T,γ,β ≤ck(cid:12)(cid:12)Tk(cid:12)(cid:12)1−γβ−1(cid:107)Xn(cid:107)F,γ,β(cid:13)(cid:13)Yk(cid:13)(cid:13)T,γ,β,
where Tk denotes the set of rooted trees with k vertices.
Theproofwillrequireaseriesofpreliminarylemmasinvolvingthecombinatorics
(cid:80)
of rooted trees. We will use to denote the sum over all admissible cuts c
τ(2)=σ
such that τ(2) = σ. The following combinatorial lemma is crucial to proving our
c
desired lemma by induction.
Lemma12. Letτ =[τ ...τ ] ,whereτ ,...,τ arenon-empty. Letσ =[σ ...σ ] (cid:54)=
1 n • 1 n 1 n •
1 be a rooted tree. Let ∼ be a relation on the permutation group S on {1,...,n}
σ n
definedsothatπ ∼π ifσ =σ foralli. LetP bethesetof∼ -equivalent
1 2 π1(i) π2(i) σ σ
classes in the permutation group S . Then for all β,γ >0,
n
(cid:88) β−c(τ(1)) = (cid:88) Πn (cid:104) (cid:88) β−c(τi(1))(cid:105).
τ(1)!γ i=1 τ(1)!γ
τ(2)=σ π∈Pσ τi(2)=σπ(i) i
Proof. Note that by the definition of (cid:52),
(4.1) (cid:88)τ(1)⊗τ(2) = τ ⊗1+(cid:88)τ(1)...τ(1)⊗(cid:104)τ(2)...τ(2)(cid:105) .
1 n 1 n
•
We define a linear functional σ such that for each tree a,
σ(a) = 1, if a=σ,
= 0, if a(cid:54)=σ.
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 9
Let f be a linear functional defined such that for each forest τ,
βc(τ)
f(τ) = .
τ!γ
Note that f is a tree multiplication homomorphism. Define
f ⊗σ(a⊗b)=f(a)σ(b).
By applying f ⊗σ to (4.1),
(cid:88) β−c(τ(1)) (cid:88) β−c(τ1(1))−...−c(τn(1))
= .
τ(1)!γ τ(1)!γ...τ(1)!γ
τ(2)=σ [τ1(2)...τn(2)]•=σ 1 n
As [τ(2),...,τ(2)] = [σ(2),...,σ(2)] if and only if there exists π ∈ P such that
1 n • 1 n • σ
τ(2) =σ for all i,
i π(i)
(cid:88) β−c(τ(1)) = (cid:88) Πn (cid:88) β−c(τi(1)).
τ(1)!γ i=1 τ(1)!γ
τ(2)=σ π∈Pσ τi(2)=σπ(i) i
(cid:3)
Corollary 13. Let τ = [τ ...τ ] , where τ ,...,τ are non-empty. Let σ =
1 n • 1 n
[σ ...σ ] (cid:54)= 1 be a rooted tree. Let P(cid:48) denote the set of all π ∈ P such that
1 n • τ,σ σ
σ ⊆τ for all i. Then for all β,γ >0,
π(i) i
(cid:88) β−c(τ(1)) (cid:88) (cid:104) (cid:88) β−c(τi(1))(cid:105)
τ(1)!γ = Πi:σπ(i)(cid:40)τi τ(1)!γ .
τ(2)=σ π∈P(cid:48)τ,σ τi(2)=σπ(i) i
Proof. We have just shown in Lemma 12 that
(4.2) (cid:88) β−c(τ(1)) = (cid:88) Πn (cid:88) β−c(τi(1)).
τ(1)!γ i=1 τ(1)!γ
τ(2)=σ π∈Pσ τi(2)=σπ(i) i
For π ∈P , if an index i is such that σ is not a subtree of τ , then
σ π(i) i
(cid:88) β−c(τi(1))
=0.
τ(1)!γ
τi(2)=σπ(i) i
Therefore,in(4.2),summingoverP isequivalenttosummingoverP(cid:48) . Further-
σ τ,σ
more, as τ(1) =1 if τ(2) =τ ,
i i i
(cid:88) β−c(τ(1)) (cid:88) (cid:104) (cid:88) β−c(τi(1))(cid:105)
(4.3) τ(1)!γ = Πi:σπ(i)(cid:40)τi τ(1)!γ .
τ(2)=σ π∈P(cid:48)τ,σ τi(2)=σπ(i) i
(cid:3)
A key step in most factorial decay estimates for rough paths is to take the
fractionalpowerγ outsideasum. Inthegeometriccase,thejobisdonebytheneo-
classical inequality. We need the following concavity estimate in the non-geometric
case.
DECAY RATE OF ITERATED INTEGRALS OF BRANCHED ROUGH PATHS 10
Lemma 14. (Concavity estimate)Let γ ≤1. For any rooted tree σ, let
|σ|
(cid:104)(cid:88) (cid:105)
(4.4) c =exp |σ|i(1−γ) andβ ≥c .
|σ| |σ|
i=1
For all rooted trees τ and σ (cid:40)τ, we have
(cid:88) β−c(τ(1)) (cid:16) (cid:88) 1 (cid:17)γ
≤ c β−1 .
τ(1)!γ |σ| τ(1)!
τ(2)=σ τ(2)=σ
Remark 15. Thekeypointisthattheconstantwelosebytakingthepowerγoutside
the sum, c , depends only on |σ| but not |τ|. To achieve this, the conventional
|σ|
estimate for sums
n n
(4.5) (cid:88)aγ ≤n1−γ(cid:0)(cid:88)a (cid:1)γ
i i
i=1 i=1
is insufficient by itself. We must use the tree-multiplicative property of the tree-
factorial.
To prove the concavity estimate, Lemma 14, we first need a counting lemma.
Lemma 16. Let σ = [σ ,...,σ ] and τ = [τ ,...,τ ] be rooted trees such that
1 n • 1 n •
τi (cid:54)=1 for all i. Let kτ,σ =minπ∈P(cid:48) (cid:12)(cid:12){i:σπ(i) (cid:40)τi}(cid:12)(cid:12). Then kτ,σ ≥1 and
τ,σ
|P(cid:48) |≤exp(|σ|2k ).
τ,σ τ,σ
Proof. As τ (cid:54)= σ there does not exist permutation π such that σ = τ for all i.
π(i) i
In particular, we have k ≥1, which proves the first part of the lemma.
τ,σ
Let m be defined by m = |{i : σ = 1}|. As σ = σ for all j,l ∈ {i : σ = 1},
i j l i
each equivalence class in P must contain at least m! elements. Therefore,
σ
n!
|P(cid:48) |≤|P |≤ =n(n−1)...(m+1).
τ,σ σ m!
Since
(cid:12) (cid:12)
n=m+(cid:12){i:σi (cid:54)=1}(cid:12)≤m+|σ|,
we have
(4.6) |P(cid:48) | ≤ (m+|σ|)|σ|.
τ,σ
Note that as τ is assumed to be non-empty for all i,
i
(4.7) m=(cid:12)(cid:12){i:σi =1}(cid:12)(cid:12)≤ min (cid:12)(cid:12){i:σπ(i) (cid:40)τi}(cid:12)(cid:12)=kτ,σ.
π∈P(cid:48)
τ,σ
Using that for x≥1 and b∈N∪{0},
(x+b)b ≤exp(cid:0)b2x(cid:1)
in combination with the estimates (4.6) and (4.7) earlier in this proof,
(cid:12)(cid:12)P(cid:48)τ,σ(cid:12)(cid:12)≤(m+|σ|)|σ| ≤(kτ,σ+|σ|)|σ| ≤exp(cid:0)|σ|2kτ,σ(cid:1).
(cid:3)
