Rough path metrics on a Besov-Nikolskii type scale

Abstract

It is known, since the seminal work [T. Lyons: Differential equations driven by rough signals, In: Rev. Mat. Iberoam. (1998)], that the It\^{o}-map is locally Lipschitz continuous in q-variation resp. 1/q-H\"{o}lder type (rough path) metrics, for any regularity 1/q∈(0,1]1/q \in (0,1]. We extend this to a new class of Besov-type (Nikolskii) metrics, with arbitrary regularity 1/q∈(0,1]1/q\in (0,1] and integrability p∈[q,∞]p\in [ q,\infty ], where the case p∈{q,∞}p\in \{ q,\infty \} corresponds to the known cases. Interestingly, the result is obtained as consequence of known q-variation rough path estimates

Full text

arXiv:1609.03132v2 [math.PR] 25 Apr 2017
ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE
PETER K. FRIZ AND DAVID J. PR ¨
OMEL
Abstract. It is known, since the seminal work [T. Lyons, Differential equations driven by
rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a
controlled differential equation is locally Lipschitz continuous in q-variation resp. 1/q-H¨older
type metrics on the space of rough paths, for any regularity 1/q ∈(0,1].
We extend this to a new class of Besov-Nikolskii-type metrics, with arbitrary regularity
1/q ∈(0,1] and integrability p∈[q, ∞], where the case p∈ {q, ∞} corresponds to the known
cases. Interestingly, the result is obtained as consequence of known q-variation rough path
estimates.
Key words: controlled differential equation, Besov embedding, Besov space, Itˆo-Lyons map,
p-variation, Riesz type variation, rough path.
MSC 2010 Classification: Primary: 34A34, 60H10; Secondary: 26A45, 30H25, 46N20.
1. Introduction
We are interested in controlled differential equations of the type
(1.1) dYt=V(Yt) dXt, t ∈[0, T ],
where X= (Xt) is a suitable (n-dimensional) driving signal, Y= (Yt) is the (m-dimensional)
output signal and V= (V1,...,Vn) are vector fields of suitable regularity. A fundamental
question concerns the continuity of the solution map X7→ Y, strongly dependent on the used
metric.
A decisive answer is given by rough path theory, which identifies a cascade of good metrics,
determined by some regularity parameter δ≡1/q ∈(0,1], and essentially given by q-variation
resp. δ-H¨older type metrics. As long as the driving signal Xpossesses sufficient regularity, say
Xis a continuous path of finite q-variation for q∈[1,2) (in symbols X∈Cq-var([0, T ]; Rn)),
Lyons [Lyo94] showed that the solution map X7→ Yassociated to equation (1.1) is a locally
Lipschitz continuous map with respect to the q-variation topology. However, this strong
regularity assumption on Xexcludes many prominent examples from probability theory as
sample paths of stochastic processes like (fractional) Brownian motion, martingales or many
Gaussian processes.
In order to restore the continuity of the solution map associated to a controlled differential
equation for continuous paths Xof finite q-variation for arbitrary large q < ∞, it is not
sufficient anymore to consider a path “only” taking in the Euclidean space Rn, cf. [Lyo91,
LCL07]. Instead, Xmust be viewed as ⌊q⌋-level rough path, which in particular means X
takes values in a step-⌊q⌋free nilpotent group G⌊q⌋(Rn): Let us recall that for Z∈C1-var(Rn)
Date: April 26, 2017.
1

2 FRIZ AND PR ¨
OMEL
its ⌊q⌋-step signature is given by
S⌊q⌋(Z)s,t :=1,Zs<u<t
dZu,...,Zs<u1<···<u⌊q⌋<t
dZu1⊗ · · · ⊗ dZu⌊q⌋.
The corresponding space of all these lifted paths is
G⌊q⌋(Rn) := {S⌊q⌋(Z)0,T :Z∈C1-var([0, T ]; Rn)} ⊂
⌊q⌋
M
k=0 Rn⊗k,
which we equip with the Carnot-Caratheodory metric dcc , see Subsection 3.2 for more details.
While in the case of q∈[1,2) this reduces to a classical path X: [0, T ]→Rn, in the case
of q > 2 this means, intuitively, Xis a path enhanced with the information corresponding
to the “iterated integrals” up to order ⌊q⌋. In the context of rough path theory, the solution
map X7→ Y, taking now a ⌊q⌋-level rough path X(in symbols X∈Cq-var([0, T ], G⌊q⌋(Rn)))
as input, is often called Itˆo-Lyons map.
In most applications, the output is regarded as path, Y∈Cq-var([0, T ]; Rm), although -
depending on the route one takes - it can be seen as rough path [Lyo98, LQ02, FV10] or
controlled rough path [Gub04, FH14]. It is a fundamental property of rough path theory
that solving differential equations - that is, applying the Itˆo-Lyons map - entails no loss of
regularity: if the driving signal enjoys δ-H¨older (resp. q-variation) regularity, then so does
the output signal.
Let us explain the basic idea which underlies this work. To this end – only estimates matter
– take Xsmooth and rewrite (1.1) in the classical form ˙
Y=V(Y)˙
X. Take Lp-norms on both
sides to arrive at
(1.2) kYkW1,p;[0,T ]≤ kVk∞kXkW1,p;[0,T ]
in terms of the semi-norm kXkW1,p;[0,T ]:= (RT
0|˙
Xt|pdt)1/p. Here, of course, we have regularity
δ= 1 (⇔q= 1), and the extreme cases p∈ {1,∞} (= {q, ∞}) amount exactly to the variation
resp. H¨older estimates
kYk1-var;[0,T]≤ kVk∞kXk1-var;[0,T ],
kYk1-H¨ol;[0,T ]≤ kVk∞kXk1-H¨ol;[0,T ]
(1.3)
since indeed kXk1-var;[0,T ]≈ kXkW1,1[0,T]resp. kXk1-H¨ol;[0,T ]=kXkW1,∞;[0,T ]. Conversely, one
may view (1.2) as interpolation of the estimates (1.3), by regarding W1,p, for any p∈[1,∞],
as interpolation space of W1,1and W1,∞. This discussion suggests moreover that the solution
map X7→ Yis also continuous in W1,p , even locally Lipschitz in the sense
(1.4) kY1−Y2kW1,p;[0,T ].kX1−X2kW1,p;[0,T ],
as indeed may be seen by some fairly elementary analysis. (Mind, however, that the solution
map X7→ Yis highly non-linear so that there is little hope to appeal to some “general theory
of interpolation”.)
The estimates (1.3) and (1.4), in case p= 1 and p=∞, are well-known (e.g. [Lyo98, LQ02,
FV10]) to extend to arbitrarily low regularity δ≡1/q ∈(0,1], provided that, essentially,
k · k1-var;[0,T ]is replaced by k · kq-var;[0,T ](with the correct rough path interpretation on the
right-hand sides above).

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 3
The question arises if the well-studied q-variation and δ-H¨older formulation of rough path
theory are not the extreme cases of a more flexible formulation of the theory, that comes - in
the spirit of Besov (Nikolskii) spaces - with an additional integrability parameter p∈[q, ∞].
(Here, having qas lower bound on pis quite natural in view of known Besov embeddings: in
the Besov-scale (Bδ,p
r), with additional fine-tuning parameter r, one has, always with δ= 1/q,
Cq-var ≈Nδ,q ≡Bδ,q
∞, in the form of tight (but strict) inclusions Nδ+ε,q ⊂Cq-var ⊂Nδ,q ; see
Remark 2.1.)
The first contribution of this paper is to given an affirmative answer to the above question,
in the generality of arbitrarily low regularity δ > 0. With focus on the interesting case of
regularity δ < 1, we have, loosely stated,
Theorem 1.1. Let δ≡1/q ∈(0,1] and p∈[q, ∞]. Then, for Lipγvector fields Vwith γ > q,
the Itˆo-Lyons map (as defined below in (3.5)) is locally Lipschitz continuous from a Besov-
Nikolskii-type (rough) path space with regularity/integrability (δ, p)into a Besov-Nikolskii-type
path space of identical regularity/integrability (δ, p).
Somewhat surprisingly, it is possible to prove this via delicate application of classical q-
variation estimates1in rough path theory; that is, morally, from the case p=q. On the other
hand, a precise definition of the involved spaces - to make this reasoning possible - is a subtle
matter. First, care is necessary for rough paths take values in a non-linear space, the step-⌊q⌋
free nilpotent group G⌊q⌋(Rn) equipped with the Carnot-Caratheodory metric dcc, which is
a no standard setting for classical Besov resp. Nikolskii spaces (Bδ,p
rresp. Nδ,p := Bδ,p
∞).
Another and quite serious difficulty is the lack of super-additivity of Nikolskii norms. Recall
that the ”control”
ω(s, t) := kXkp
p-var;[s,t]
has the most desirable property of super-additivity, i.e. ω(s, t) + ω(t, u)≤ω(s, u), a simple
fact that is used throughout Lyons’ theory. For instance, as a typical consequence
kXkp
p-var;[0,T ]= sup
P⊂[0, T ]X
[u,v]∈P
kXkp
p-var;[u,v],
where the supremum is taking over all partition of the interval [0, T ]. Several other (rough)
path space norms also have this property, as exploited e.g. in [FV06]. However, this convenient
property fails for the Besov spaces of consideration (unless δ= 1) and indeed, in general with
strict inequality,
kXkp
Nδ,p;[0,T ]≤sup
P⊂[0, T ]X
[u,v]∈P
kXkp
Nδ,p;[u,v]=: kXkp
ˆ
Nδ,p;[0,T ].
This leads us to use the Besov-Nikolskii-type space ˆ
Nδ,p, defined as those Xfor which the
right-hand side above is finite, as the correct space (in rough path or path space incarnation)
to which we refer in Theorem 1.1, at least in the new regimes δ < 1, p ∈(q, ∞).
A better understanding of these spaces is compulsory, and this is the second contribution
of this paper. For instance, it is reassuring that one has tight inclusions of the form Nδ+ε,p ⊂
ˆ
Nδ,p ⊂Nδ,p (Corollary 2.12). In fact, an exact characterization is possible in terms of Riesz
type variation spaces, in reference to Riesz [Rie10], who considered such spaces (although
with regularity parameter δ= 1). We have
1The use of control functions is pure notational convenience.

4 FRIZ AND P R ¨
OMEL
Theorem 1.2. Consider δ= 1/q < 1and p∈(q, ∞). Then the Besov-Nikolskii-type space
ˆ
Nδ,p coincide with the Riesz type variation spaces Vδ,p and ˜
Vδ,p defined respectively via finite-
ness of
kXkp
Vδ,p := sup
P⊂[0, T ]X
[u,v]∈P
dcc(Xv, Xu)p
|v−u|δp−1,
kXkp
˜
Vδ,p := sup
P⊂[0, T ]X
[u,v]∈P
kXkp
1
δ
-var;[u,v]
|v−u|δp−1,
for a rough path Xand the Carnot-Caratheodory distance dcc. More general, this is also true
for arbitrary metric spaces instead of G⌊q⌋(Rn).
Moreover, all associated inhomogenous rough path distances are locally Lipschitz equivalent.
Let us also note that the above introduced Riesz type variation spaces agree (trivially) with
the q-variation space in the extreme case of p=q≡1/δ. (In the Besov scale, this usually
fails. For instance, we have the strict inclusion W1,1⊂C1-var; not every rectifiable path is
absolutely continuous.)
We conclude this introduction with some pointers to previous works. The case of regularity
δ > 1/2, essentially a Young regime, was considered in [Z¨ah98, Z¨ah01]. Our result can also
be regarded as extension of [PT16], which effectively dealt with regularity δ= 1/q > 1/3 and
accordingly integrability p≥q= 3. We note that path spaces with “mixed” H¨older-variation
regularity, similar in spirit to the Riesz type spaces (with tilde) also appear as tangent spaces
to H¨older rough path spaces [FV10, p.209], see also [Aid16]. Moreover, regularity of Cameron-
Martin spaces associated to Gaussian processes with “H¨older dominated ρ-variation of the
covariance” (a key condition in Gaussian rough path theory, cf. [FH14, Ch. 10], [FGGR16])
can be expressed with the help of “mixed” H¨older-variation regularity, see e.g. [FH14, p.151].
Organization of the paper: In Section 2 we define and give various characterizations of
our spaces, starting for the reader’s convenience with the (much) simpler situation δ= 1.
In particular, Theorem 1.2. is an effective summary of Theorem 2.11 and Lemmas 3.4, 3.6
and 3.7. Section 3 is devoted to establish the local Lipschitz continuity of the Itˆo-Lyons
map in suitable rough path metrics and Theorem 1.1. can be found in Theorem 3.3 and
Corollaries 3.5 and 3.8.
Acknowledgment: P.K.F. is partially supported by the European Research Council through
CoG-683164 and DFG research unit FOR2402. D.J.P. gratefully acknowledges financial sup-
port of the Swiss National Foundation under Grant No. 200021 163014. Both authors are
grateful for the excellent hospitality of the Hausdorff Research Institute for Mathematics,
where the work was initiated.
2. Riesz type variation
In this section we introduce a class of function spaces which unifies the notions of H¨older
and q-variation regularity. For this purpose we generalize an old version of variation due to
F. Riesz and provide two alternative but equivalent characterizations of the so-called Riesz
type variation and additionally various embedding results. As explained in the Introduction,
the later application in the rough path framework requires us to set up all the function spaces
for paths taking values in a metric spaces.

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 5
Let us briefly fix some basic notation: Pis called partition of an interval [s, t]⊂[0, T ] if
P={[ti, ti+1] : s=t0< t1<··· < tn=t, n ∈N}. In this case we write P ⊂ [s, t] indicating
that Pis a partition of the interval [s, t]. Furthermore, for such a partition Pand a function
χ:{(u, v) : s≤u < v ≤t} → Rwe use the abbreviation
X
[u,v]∈P
χ(u, v) :=
n−1
X
i=0
χ(ti, ti+1).
If not otherwise specified, (E , d) denote a metric space, T∈(0,∞) is finite real number and
C([0, T ]; E) stands for the set of all continuous functions f: [0, T ]→E.
Two frequently used topologies to measure the regularity of functions are the H¨older con-
tinuity and the q-variation:
The H¨older continuity of a function f∈C([0, T ]; E) is measured by
kfkδ-H¨ol;[s,t]:= sup
u,v∈[s,t], u<v
d(fu, fv)
|v−u|δ, δ ∈(0,1],
and Cδ-H¨ol([0, T ]; E) stands for the set of all functions f∈C([0, T ]; E) such that kfkδ-H¨ol :=
kfkδ-H¨ol;[0,T ]<∞. The case δ= 1, that is the H¨older continuity of order 1, is usually refer to
as Lipschitz continuity.
The q-variation of a function f∈C([0, T ]; E) is defined by
(2.1) kfkq-var;[s,t]:= sup
P⊂[s,t]X
[u,v]∈P
d(fu, fv)q1
q
, q ∈[1,∞),
where the supremum is taken over all partitions Pof the interval [s, t]. The set of all functions
f∈C([0, T ]; E) with kfkq-var := kfkq-var;[0,T ]<∞is denoted by Cq-var([0, T ]; E). The notion
of q-variation can be traced back to N. Wiener [Wie24]. The special case of 1-variation is also
called bounded variation. A comprehensive list of generalizations of q-variation and further
references can be found in [DN99].
Remark 2.1. Classical function spaces as fractional Sobolev or more general Besov spaces
do not provide a unifying framework simultaneously covering the space of H¨older continuous
functions and the space of continuous functions with finite q-variation. For example, let us
replace for a moment (E, d)by the Euclidean space (R,|·|)and denote the homogeneous Besov
spaces by Bδ,p
r([0, T ]; R). While the H¨older space Cδ-H¨ol([0, T ]; R)is a special case of Besov
spaces, namely the homogeneous Besov space Bδ,∞
∞([0, T ]; R), for δ∈(0,1), the q-variation
space Cq-var([0, T ]; R)is not covered by the wide class of Besov spaces. Indeed, classical
embedding theorems, [You36] and [LY38], yield the following continuous embeddings:
Bα,p
∞([0, T ]; R)⊂Cp-var([0, T ]; R)⊂B1/p,p
∞([0, T ]; R),
for p∈(1,∞)and α∈(1/p, 1), see also [Sim90] and [FV06]. In particular, it is known that
the second embedding is not an equality. An example can be found in [Ter67]. The relation
between the space of functions with finite q-variation and Besov spaces was investigated in
the literature for a long time, see for example [MS61],[Pee76],[BLS06] and [Ros09]. For a
comprehensive introduction to function spaces we refer to [Tri10].
To set up a class of function spaces covering precisely and simultaneously the H¨older
spaces and the q-variation spaces, we introduce a generalized version of a variation due to

6 FRIZ AND P R ¨
OMEL
F. Riesz [Rie10]. For δ∈(0,1] and p∈[1/δ, ∞) the Riesz type variation of a function
f∈C([0, T ]; E) is given by
(2.2) kfkVδ,p;[s,t]:= sup
P⊂[s,t]X
[u,v]∈P
d(fu, fv)p
|v−u|δp−11
p
,
for a subinterval [s, t]⊂[0, T ] and for p=∞we set
(2.3) kfkVδ,∞;[s,t]:= sup
u,v∈[s,t], u<v
d(fu, fv)
|v−u|δ.
The set Vδ,p([0, T ]; E) denotes all continuous functions f∈C([0, T ]; E) such that kfkVδ,p :=
kfkVδ,p;[0,T ]<∞. The case of δ= 1 was originally defined by F. Riesz [Rie10] and a similar
generalization as given in (2.2) was already mentioned in [Pee76, p. 114, (14’)].
Proposition 2.2. Let (E, d)be a metric space and T∈(0,∞). For δ∈(0,1] and p∈[1/δ, ∞]
one has the following relations
Cδ-H¨ol([0, T ]; E) = Vδ,∞([0, T ]; E)⊂Vδ,p([0, T ]; E)⊂Vδ,1/δ([0, T ]; E) = C1/δ-var([0, T ]; E).
More precisely, the 1/δ-variation of a function f∈Vδ,p([0, T ]; E)satisfies the bound
kfk1/δ-var;[s,t]≤ kfkVδ,p;[s,t]|t−s|δ−1
p
for every subinterval [s, t]⊂[0, T ].
Before we come to the proof, we need the following remark about super-additive functions.
Remark 2.3. Setting ∆T:= {(s, t) : 0 ≤s≤t≤T}a function ω: ∆T→[0,∞)is called
super-additive if
ω(s, t) + ω(t, u)≤ω(s, u)for 0≤s≤t≤u≤T.
Furthermore, if ωand ˜ωare super-additive and α, β > 0with α+β≥1, then ωα˜ωβis
super-additive. The proof works as [FV10, Exercise 1.8 and 1.9].
Proof of Proposition 2.2. The identifies
Cδ-H¨ol([0, T ]; E) = Vδ,∞([0, T ]; E) and Vδ,1/δ([0, T ]; E) = C1/δ-var ([0, T ]; E)
are ensured by the definitions of the involved function spaces.
The first embedding can be seen by
kfkp
Vδ,p = sup
P⊂[0, T ]X
[u,v]∈P d(fu, fv)
|v−u|δp
|v−u| ≤ Tkfkp
Cδ;[0,T ], f ∈Cδ([0, T ]; E).
The second embedding is trivial for δ= 1/p. For δ > 1/p we first observe that
(2.4) d(fs, ft)≤d(fs, ft)p
|t−s|δp−11
p
|t−s|δ−1
p≤ kfkVδ,p;[s,t]|t−s|δ−1
p,[s, t]⊂[0, T ],
for f∈Vδ,p([0, T ]; E) and thus
d(fs, ft)1
δ≤ kfk
1
δ
Vδ,p;[s,t]|t−s|1−1
δp =: ω(s, t).

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 7
Since kfkp
Vδ,p;[s,t]and |t−s|are super-additive as functions in (s, t)∈∆Tand (δp)−1+ 1 −
(δp)−1≥1, ωis a super-additive by Remark 2.3. Hence, using the super-additivity of ω, we
arrive at the claimed estimate
kfk1/δ-var;[s,t]≤ kfkVδ,p;[s,t]|t−s|δ−1
p.

The next lemma justifies the definition of the Riesz type variation in the case of p=∞,
cf. (2.3), and collects some embedding results of these sets of functions.
Lemma 2.4. Let (E, d)be a metric space, T∈(0,∞)and [s, t]⊂[0, T ]. Suppose δ∈(0,1)
and p∈[1/δ, ∞].
(1) If δ > 1/p, then Vδ,p([0, T ]; E)⊂C(δ−1/p)-H¨ol ([0, T ]; E)with the estimate
d(fs, ft)≤ kfkVδ,p;[s,t]|t−s|δ−1
p, f ∈Vδ,p([0, T ]; E).
(2) If δ, δ′∈(0,1) and p, p′∈[1/δ, ∞]with δ′< δ and p′< p, then one has
Vδ,p([0, T ]; E)⊂Vδ′,p ([0, T ]; E)and Vδ,p([0, T ]; E)⊂Vδ,p′([0, T ]; E)
with the estimates for f∈Vδ,p([0, T ]; E)
kfkVδ′,p;[s,t]≤(t−s)δ−δ′kfkVδ,p;[s,t]and kfkVδ,p′;[s,t]≤(t−s)1
p′−1
pkfkVδ,p;[s,t].
(3) For every f∈Vδ,∞([0, T ]; E)one has
lim
p→∞ kfkVδ,p;[s,t]=kfkVδ,∞;[s,t].
Proof. (1) The first assertion follows directly by the estimate (2.4).
(2) Let Pbe a partition of the interval [s, t]⊂[0, T ]. For f∈Vδ,p([0, T ]; E) the estimates
X
[u,v]∈P
d(fu, fv)p
|v−u|δ′p−1≤ |t−s|(δ−δ′)pX
[u,v]∈P
d(fu, fv)p
|v−u|δp−1
and (using H¨older’s inequality)
X
[u,v]∈P
d(fu, fv)p′
|v−u|δp′−1=X
[u,v]∈P d(fu, fv)
|v−u|δ−1
pp′
|v−u|1−p′
p≤ |t−s|1−p′
pX
[u,v]∈P
d(fu, fv)p
|v−u|δp−1p′
p
lead to (2) by taking the supremum over all partition of [s, t].
(3) Due to Lemma 2.4 (1), we have
kfkVδ,∞;[s,t]≤lim inf
p→∞ kfkVδ,p;[s,t], f ∈Vδ,∞([0, T ]; E).
Furthermore, for p > q ≥1 we get
kfkVδ,p;[s,t]=sup
P⊂[s,t]X
[u,v]∈P
d(fu, fv)q
|v−u|δq−1
d(fu, fv)p−q
|v−u|δ(p−q)1
p
≤ kfk
q
p
Vδ,q;[s,t]kfk1−q
p
Vδ,∞;[s,t]
and thus
lim sup
p→∞ kfkVδ,p;[s,t]≤ kfkVδ,∞;[s,t].


8 FRIZ AND P R ¨
OMEL
In the following we introduce two different but equivalent characterizations of the Riesz
type variation (2.2). The first one is based on the classical notion of q-variation due to Wiener
and thus is particularly convenient for applications in rough path theory. The second one
relies on certain Besov spaces, namely Nikolskii spaces, which allows to related the Riesz type
variation spaces to classical function spaces as fractional Sobolev spaces. See Lemma 2.6 and
Theorem 2.11 for the equivalence.
In order to give a characterization of Riesz type variation of a function f∈C([0, T ]; E) in
terms of q-variation, we introduce a mixed H¨older-variation regularity by
(2.5) kfk˜
Vδ,p;[s,t]:= sup
P⊂[s,t]X
[u,v]∈P
kfkp
1
δ-var;[u,v]
|v−u|δp−11
p
, δ ∈(0,1], p ∈[1/δ, ∞),
for a subinterval [s, t]⊂[0, T ] and in the case of p=∞we define
kfk˜
Vδ,∞;[s,t]:= sup
P⊂[s,t]
sup
[u,v]∈P
kfk1
δ-var;[u,v]
|v−u|δ.
Moreover, we denote by ˜
Vδ,p([0, T ]; E) the set of all functions f∈C([0, T ]; E) such that
kfk˜
Vδ,q := kfk˜
Vδ,q;[0,T ]<∞.
An alternative way to measure Riesz type variation of a function f∈C([0, T ]; E) is related
to homogeneous Nikolskii spaces. Hence, we briefly recall the notation of homogeneous Nikol-
skii spaces, which correspond to the homogeneous Besov spaces Bδ,p
∞([0, T ]; E). For δ∈(0,1]
and p∈[1,∞) we define
kfkNδ,p;[s,t]:= sup
|t−s|≥h>0
h−δZt−h
s
d(fu, fu+h)pdu1
p
for a subinterval [s, t]⊂[0, T ] and for p=∞we further set
kfkNδ,∞;[s,t]:= sup
|t−s|≥h>0
h−δsup
u∈[s,t−h]
d(fu, fu+h).
The set of all functions f∈C([0, T ]; E) such that kfkNδ,p := kfkNδ,p ;[0,T ]<∞is denoted by
Nδ,p([0, T ]; E).
Using the definition of Nikolskii regularity, we introduce a refined Nikolskii type regularity
by
(2.6) kfkˆ
Nδ,p;[s,t]:= sup
P⊂[s,t]X
[u,v]∈P
kfkp
Nδ,p;[u,v]1
p
, δ ∈(0,1], p ∈[1,∞),
for f∈C([0, T ]; E) and a subinterval [s, t]⊂[0, T ]. For p=∞we set
kfkˆ
Nδ,∞;[s,t]:= sup
P⊂[s,t]
sup
[u,v]∈P
kfkNδ,∞;[u,v].
Furthermore, ˆ
Nδ,p([0, T ]; E) stands for the set of all functions f∈C([0, T ]; E) such that
kfkˆ
Nδ,q := kfkˆ
Nδ,q;[0,T ]<∞.
Remark 2.5. While k · kp
ˆ
Nδ,p;[s,t]is a super-additive function in (s, t)∈∆Tby its definition,
this is not true for the Nikolskii regularity k · kp
Nδ,p;[s,t]itself if δ∈(0,1). The later can be
seen particularly by Remark 2.13.

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 9
In the next two subsections we show that the just introduced two ways of measuring path
regularity are indeed equivalent to the Riesz type variation. We start by considering the
special case of regularity δ= 1, that is the space V1,p, in Subsection 2.1. The equivalence for
general Riesz type variation spaces is content of Subsection 2.2
2.1. Characterization of the space V1,p.The special case δ= 1 or in other words the set
V1,p([0, T ]; Rn) coincides with the original definition due to F. Riesz [Rie10] and is already
fairly well-understood. For the sake of completeness we present here the full picture assuming
E=Rnsince it is will be general enough for the later applications concerning the solution
map associated to a controlled differential equation, see Subsection 3.1.
It is well-known that the Riesz type variation space V1,p([0, T ],Rn) corresponds to the
classical Sobolev space W1,p([0, T ]; Rn), see e.g. [FV10, Proposition 1.45]. Let us recall the
definition of the Sobolev space W1,p([0, T ]; Rn) (cf. [FV10, Definition 1.41]). For p∈[1,∞]
and T∈(0,∞) a function f∈C([0, T ]; Rn) is in W1,p([0, T ]; Rn) if and only if fis of the
form
ft=f0+Zt
0
f′
sds, t ∈[0, T ],
for some f′∈Lp([0, T ]; Rn). Moreover, we define kfkW1,p := kf′kLpfor f∈W1,p([0, T ]; Rn).
Including the three known characterizations of V1,p ([0, T ],Rn), we end up with the following
five different ways to measure the Riesz type variation.
Lemma 2.6. Let T∈(0,∞),p∈(1,∞)and Rnbe equipped with the Euclidean norm |·|.
The space V1,p([0, T ]; Rn)has the following different characterizations
V1,p([0, T ]; Rn) = ˜
V1,p([0, T ]; Rn) = ˆ
N1,p([0, T ]; Rn) = N1,p ([0, T ]; Rn) = W1,p ([0, T ]; Rn)
with
kfkV1,p =kfk˜
V1,p =kfkˆ
N1,p =kfkW1,p =kfkN1,p for f∈C([0, T ]; Rn).
Proof. For f∈C([0, T ]; Rn) and p∈(1,∞) the identifies
kfkV1,p =kfkW1,p =kfkN1,p
can be found in [FV10, Proposition 1.45] and [Leo09, Theorem 10.55].
Next we observe that
kfkW1,p =kfkp
V1,p ≤ kfkp
˜
V1,p ≤sup
P⊂[0, T ]X
[u,v]∈P
kfkp
W1,p;[u,v]|v−u|p−1
|v−u|p−1≤ kfkp
W1,p ,
where we used [FV10, Theorem 1.44] (see also [FV06, Theorem 1]) for the second estimate
and the super-additivity of kfkp
W1,p;[u,v]as a function in (u, v)∈∆Tin the last one.
As last step note that
kfkp
ˆ
N1,p = sup
P⊂[0, T ]X
[u,v]∈P
kfkp
W1,p;[u,v]
due to [Leo09, Theorem 10.55], which implies
kfkp
W1,p =kfkp
ˆ
N1,p ≤ kfkp
W1,p
using once more the super-additivity of kfkp
W1,p;[u,v]as a function in (u, v)∈∆T.

10 FRIZ AND P R ¨
OMEL
2.2. Characterizations of Riesz type variation. While Sobolev spaces and Nikolskii
spaces coincide with the Riesz type variation spaces for regularity δ= 1, this is not true
anymore for the fractional regularity δ∈(0,1). However, the characterizations of Riesz type
variation via q-variation due to Wiener (2.5) and via classical Nikolskii spaces (2.6) still work
as we will see in this subsection.
We start by recalling the definition of fractional Sobolev spaces. For δ∈(0,1) and p∈
[1,∞) the fractional Sobolev (also called Sobolev-Slobodeckij ) regularity of a function f∈
C([0, T ]; E) is given by
kfkWδ,p;[s,t]:= ZZ[s,t]2
d(fu, fv)p
|v−u|1+δp dudv1
p
for a subinterval [s, t]⊂[0, T ] and we abbreviate k · kWδ,p := k · kWδ,p;[0,T ]. The set of all
functions f∈C([0, T ]; E) such that kfkWδ,p <∞is denoted by Wδ,p([0, T ]; E).
As an auxiliary result we first need an explicit embedding of Nikolskii regular functions
Nδ′,p([0, T ]; E) into the set of functions with fractional Sobolev regularity Wδ,p([0, T ]; E).
Lemma 2.7. Suppose that (E, d)is a metric space and T∈(0,∞). Let p∈[1,∞)and
δ, δ′∈(0,1) be such that δ′> δ. For f∈Nδ′,p([0, T ]; E)it holds
kfkWδ,p;[s,t]≤2
(δ′−δ)p1
p
kfkNδ′,p;[s,t](t−s)δ′−δ
for any s, t ∈[0, T ]with s < t. In particular, Nδ′,p([0, T ]; E)⊂Wδ,p([0, T ]; E).
Proof. The fractional Sobolev regularity can be rewritten as
kfkp
Wδ,p;[s,t]=ZZ[s,t]2
d(fu, fv)p
|v−u|1+δp dudv= 2 Zt−s
0Zt−h
s
d(fu, fu+h)p
|h|1+δp dudh
for s, t ∈[0, T ] with s < t and for every f∈Wδ,p([0, T ]; E). Since f∈Nδ′,p([0, T ]; E), one
has
Zt−h
s
d(fu, fu+h)pdu≤ kfkp
Nδ′,p;[s,t]hδ′p.
Therefore, we conclude for δ′> δ > 0 that
kfkp
Wδ,p;[s,t]≤2Zt−s
0
kfkp
Nδ′,p;[s,t]hδ′p
|h|1+δp dh≤2
(δ′−δ)pkfkp
Nδ′,p;[s,t](t−s)(δ′−δ)p,
for every interval [s, t]⊂[0, T ], and thus Nδ′,p([0, T ]; E)⊂Wδ,p([0, T ]; E). 
The next proposition presents that functions of refined Nikolskii type regularity are also of
finite q-variation and H¨older continuous. It can be seen as a refinement of [FV06, Theorem 2].
For the sake of notational brevity, we use in the following Aϑ.Bϑ, for a generic parameter
ϑ, meaning that Aϑ≤CBϑfor some constant C > 0 independent of ϑ.
Proposition 2.8. Suppose that (E, d)is a metric space and T∈(0,∞). Let δ∈(0,1) and
p∈(1,∞)be such that α:= δ−1/p > 0, and set q:= 1
δ.
(1) If f∈Nδ,p([0, T ]; E), then f∈Cα-H¨ol ([0, T ]; E)and
d(fs, ft).kfkNδ,p;[s,t](t−s)δ−1
p,[s, t]⊂[0, T ].

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 11
(2) The q-variation of any f∈ˆ
Nδ,p([0, T ]; E)can be estimated by
kfkq-var;[s,t].kfkˆ
Nδ,p;[s,t](t−s)α,[s, t]⊂[0, T ],
and one has ˆ
Nδ,p([0, T ]; E)⊂Cα-H¨ol([0, T ]; E)and ˆ
Nδ,p([0, T ]; E)⊂Cq-var([0, T ]; E).
Proof. (1) Choose γ < δ such that γ−1/p > 0. Because f∈Nδ,p([0, T ]; E), Lemma 2.7
yields f∈Wγ,p([0, T ]; E) and we have
kfkp
Wγ,p ;[s,t]=Fs,t := ZZ[s,t]2d(fu, fv)
|v−u|1/p+γp
dudv, [s, t]⊂[0, T ].
Applying the Garsia-Rodemich-Rumsey inequality with Ψ(·) = (·)pand p(·) = (·)1/p+γgives
d(fs, ft)≤8Zt−s
0Fs,t
u21
p
dp(u) = 8
(γ−1/p)kfkWγ,p;[s,t](t−s)γ−1
p,
using γ−1
p>0, see for instance [FV10, Theorem A.1] for a version of the Garsia-Rodemich-
Rumsey lemma suitable for functions with values in a metric space. Furthermore, Lemma 2.7
yields
d(fs, ft)≤8
(γ−1/p)2
(δ−γ)p1
p
kfkNδ,p;[s,t](t−s)δ−γ(t−s)γ−1
p
.kfkNδ,p;[s,t](t−s)δ−1
p,
(2.7)
which gives f∈C(δ−1/p)-H¨ol([0, T ]; E).
(2) Assuming f∈ˆ
Nδ,p([0, T ]; E) the estimate (2.7) leads to
d(fs, ft).kfkˆ
Nδ,p;[s,t](t−s)δ−1
p,[s, t]⊂[0, T ].
Recalling α=δ−1/p > 0 and q=1
δ, we note that
ω(s, t) := kfkq
ˆ
Nδ,p;[s,t](t−s)αq ,0≤s≤t≤T ,
is super-additive. Indeed, since kfkp
ˆ
Nδ,p;[s,t]and |t−s|are super-additive as functions in
(s, t)∈∆Tand q/p + 1 −1/δp ≥1, Remark 2.3 ensures the super-additivity of ω.
Hence, by the super-additivity of ωwe deduce that
kfkq
q-var;[s,t]≤Cω(s, t) = Ckfkq
ˆ
Nδ,p;[s,t]|t−s|αq ,
for some constant C > 0 depending only on δand p.
In particular, we have proven that ˆ
Nδ,p([0, T ]; E)⊂Cα-H¨ol([0, T ]; E) and ˆ
Nδ,p([0, T ]; E)⊂
Cq-var([0, T ]; E). 
Remark 2.9. Proposition 2.8 (2) does not hold for k · k ˆ
Nδ,p;[s,t]replaced by k · kNδ,p;[s,t], see
Remark 2.13 below.
Remark 2.10. Alternatively to the given proofs of Lemma 2.7 and Proposition 2.8, one could
use the abstract Kuratowski embedding to extend the known Besov embeddings from Banach
spaces to general metric spaces and then proceed further as presented above. For example
note, if (E, k · k)is a Banach space, then classical Besov embeddings lead to
kft−fsk ≤ sup
|t−s|≥h>0kfs+h−fsk
|h|δ−1/p |t−s|δ−1/p ≤CkfkNδ,p;[s,t]|t−s|δ−1/p,

12 FRIZ AND P R ¨
OMEL
for every f∈Nδ,p([0, T ]; E),δ∈(0,1),p∈(1,∞)such that δ > 1/p, and some constant
C > 0, cf. [Sim90, Theorem 10]. However, we preferred here to give direct proofs.
On the other hand, the embedding Nδ,p([0, T ]; E)⊂Wδ,p([0, T ]; E)does not hold true, which
prevents to deduce Proposition 2.8 as a corollary of [FV06, Theorem 2]. Hence, the elaborated
embedding of Lemma 2.7 is essential to obtain Proposition 2.8.
The next theorem is the main result of the first part: the characterization of Riesz type
variation via k · k˜
Vδ,p and k · k ˆ
Nδ,p .
Theorem 2.11. Let T∈(0,∞)and (E, d)be a metric space. Suppose that δ∈(0,1) and
p∈(1,∞)such that δ > 1/p. Then, k · kVδ,p ,k · k˜
Vδ,p and k · k ˆ
Nδ,p are equivalent, that is
kfkVδ,p .kfk˜
Vδ,p .kfkˆ
Nδ,p .kfkVδ,p
for every function f∈C([0, T ]; E), and thus
Vδ,p([0, T ]; E) = ˜
Vδ,p([0, T ]; E) = ˆ
Nδ,p([0, T ]; E).
Proof. For a function f∈C([0, T ]; E) and an interval [s, t]⊂[0, T ] recall that
kfkNδ,p;[s,t]=sup
|t−s|≥h>0
h−δp Zt−h
s
d(fu, fu+h)pdu1
p
.
Let us fix h∈(0, t −s] and take a partition P(h) := {[ti, ti+1] : s=t0<··· < tM=t−h}
such that
|tM−tM−1| ≤ hand |ti+1 −ti|=hfor i= 0,...,M −2, M ∈N.
Since supu∈[ti,ti+1]d(fu, fu+h)p≤ kfkp
1/δ-var;[ti,ti+2]for i= 0,...,M −1 with tM+1 := t−h, we
observe that
Zt−h
s
|d(fu, fu+h)|pdu=
M−1
X
i=0 Zti+1
ti
d(fu, fu+h)pdu≤
M−1
X
i=0
sup
u∈[ti,ti+1]
d(fu, fu+h)p(ti+1 −ti)
≤1
2(2h)δp
M−1
X
i=0
kfkp
1/δ-var;[ti,ti+2]
(2h)δp−1.hδpkfkp
˜
Vδ,p;[s,t],
which implies kfkp
Nδ,p;[s,t].kfkp
˜
Vδ,p;[s,t]. Therefore, the super-additivity of kfkp
˜
Vδ,p;[s,t]as
function in (s, t)∈∆Treveals
kfkˆ
Nδ,p .kfk˜
Vδ,p .
For the converse inequality Proposition 2.8 gives
kfk1
δ-var;[u,v].kfkˆ
Nδ,p;[u,v]|v−u|δ−1
p,0≤u < v ≤T ,
for δ∈(0,1) and p∈(1,∞) such that δ > 1/p, which leads to
khkp
˜
Vδ,p = sup
P⊂[0, T ]X
[u,v]∈P
khkp
1
δ-var;[u,v]
|v−u|δp−1≤sup
P⊂[0, T ]X
[u,v]∈P
khkp
ˆ
Nδ,p;[u,v]|v−u|δp−1
|v−u|δp−1≤ khkp
ˆ
Nδ,p ,
where we applied the super-additivity of kfkp
ˆ
Nδ;p;[s,t]as a function in (s, t)∈∆T.
It remains to show
kfkVδ,p .kfk˜
Vδ,p .kfkVδ,p, f ∈C([0, T ], E ).

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 13
The first inequality follows immediately from the definitions and the observation
d(fu, fv)p≤ kfkp
1/δ-var;[u,v],[u, v]⊂[0, T ].
The second inequality can be deduced from Proposition 2.2, which gives the estimate
kfkp
1/δ-var;[u,v]≤ kfkp
Vδ,p;[u,v]|v−u|δp−1,
and the super-additivity of kfkp
Vδ;p;[s,t]as a function in (s, t)∈∆T.
As a next step we briefly want to understand how the set Vδ,p ([0, T ]; E) of functions with
finite Riesz type variation are related to other types of measuring the regularity of functions.
The characterization of Riesz type variation in terms of Nikolskii regularity allows to deduce
the following result connecting the set Vδ,p([0, T ]; E) with the notion of classical fractional
Sobolev and Nikolskii regularity.
Corollary 2.12. Let T∈(0,∞)and (E, d)be metric space. If δ∈(0,1) and p∈(1,∞)such
that δ > 1/p, then one has the inclusions
(2.8) Wδ,p([0, T ]; E)⊂Vδ,p([0, T ]; E)⊂Nδ,p ([0, T ]; E)
and
Nδ+ǫ,p([0, T ]; E)⊂ˆ
Nδ,p([0, T ]; E)⊂Nδ,p([0, T ]; E)
for ǫ∈(0,1−δ).
Proof. For the first embedding let f∈Wδ,p([0, T ]; E). Applying Theorem 2.11 and [Sim90,
Theorem 11], which can be extended to general metric spaces by Kuratowski’s embedding
theorem, we get
kfkp
Vδ,p .kfkp
ˆ
Nδ,p = sup
P⊂[0, T ]X
[s,t]∈P
kfkp
Nδ,p;[s,t].sup
P⊂[0, T ]X
[s,t]∈P
kfkp
Wδ,p;[s,t]≤ kfkp
Wδ,p.
For the second embedding let f∈Nδ,p([0, T ]; E) and we apply again Theorem 2.11 to
obtain
kfkp
Nδ,p ≤ kfkp
ˆ
Nδ,p .kfkp
Vδ,p .
The first embedding for the refined Nikolskii type space ˆ
Nδ+ǫ,p is a consequence of Theo-
rem 2.11 and the embedding
Nδ+ǫ,p([0, T ]; E)⊂Wδ,p([0, T ]; E)⊂Vδ,p([0, T ]; E),
where we used Lemma 2.7 and (2.8).
The second embedding for the refined Nikolskii type space ˆ
Nδ,p follows directly from its
definition. 
Remark 2.13. Both embeddings are proper embeddings, which means in both cases the equal-
ity does not hold.
Indeed, an example of a set of functions which are included in V1/2+H,2([0, T ]; R)but not
in W1/2+H,2([0, T ]; R)consists of the Cameron-Martin space of a fractional Brownian motion
with Hurst index H∈(0,1/2), see [FV06] and [FH14, Section 11] and the references therein.
To see that the second embedding is not an equality, we recall that the sample paths of a
Brownian motion belong the Nikolskii space N1/2,p ([0, T ]; R)for p∈(2,∞), which was proven
by [Roy93] (cf. [Ros09, Proposition 1]). However, it is also well-known that sample paths
of a Brownian motion are not contained in C2-var([0, T ]; R). In other words, they cannot

14 FRIZ AND P R ¨
OMEL
be contained in V1/2,p([0, T ]; R)for p∈(2,∞)since this is a subset of C2-var([0, T ]; R)by
Proposition 2.2.
2.3. Separability considerations. In order to embed the Riesz type variation spaces into
separable Banach spaces, we need to restricted the general metric space Eand focus here on
the case E=Rnequipped with the Euclidean norm |·|. As usual |·| induces the metric
d(x, y) := |y−x|for x, y ∈Rnand thus k · kVδ,p,k · k ˜
Vδ,p and k · k ˆ
Nδ,p become semi-norms,
which can be easily modified to proper norms by adding for instance the Euclidean norm of
the functions evaluated at zero, cf. (2.9). An immediate consequence of Theorem 2.11 is the
following equivalence.
Corollary 2.14. Let T∈(0,∞)and Rnbe equipped with the Euclidean norm |·|. If δ∈(0,1)
and p∈(1,∞)are such that δ > 1/p, then the semi-norms k · kVδ,p,k · k ˜
Vδ,p and k · k ˆ
Nδ,p are
equivalent.
In order to turn Cδ-H¨ol([0, T ]; Rn) and Cp-var([0, T ]; Rn) into Banach spaces, one usually
introduces the norms
(2.9) |f(0)|+kfkδ-H¨ol and |g(0)|+kgkp-var
for f∈Cδ-H¨ol([0, T ]; Rn) and g∈Cp-var([0, T ]; Rn), respectively. These Banach spaces are
not separable, see [FV10, Theorem 5.25].
To restore the separability, one can consider the closure of smooth paths. Let C∞([0, T ]; Rn)
be the space of smooth functions f∈C([0, T ]; Rn). For δ∈(0,1) and p∈(1,∞) we define
C0,δ-H¨ol([0, T ]; Rn) := C∞([0, T ]; Rn)k·kδ-H¨ol and C0,p-var([0, T ]; Rn) := C∞([0, T ]; Rn)k·kp-var.
These two Banach spaces are separable and one has the obvious embeddings
C0,δ-H¨ol([0, T ]; Rn)⊂Cδ-H¨ol ([0, T ]; Rn) and C0,p-var([0, T ]; Rn)⊂Cp-var([0, T ]; Rn).
The Riesz type variation space Vδ,p([0, T ]; Rn) can be embedded into C0,α-H¨ol([0, T ]; Rn)
and C0,p-var([0, T ]; Rn).
Lemma 2.15. Let T∈(0,∞)and Rnbe equipped with the Euclidean norm | · |. If δ∈(0,1)
and p∈(1,∞)are such that δ > 1/p, then one has the embeddings
Vδ,p([0, T ]; Rn)⊂C0,p-var([0, T ]; Rn)and Vδ,p([0, T ]; Rn)⊂C0,α-H¨ol([0, T ]; Rn)
for α∈(0, δ −1/p).
Proof. For f∈Vδ,p([0, T ]; Rn) and δ > 1/p we apply Lemma 2.4 to obtain
lim
ε→0sup
P⊂[0, T ],|P|<ε X
[s,t]∈P
|ft−fs|p.lim
ε→0sup
P⊂[0, T ],|P|<ε X
[s,t]∈P
kfkp
Vδ,p;[s,t]|t−s|δp−1
≤ kfkp
Vδ,p lim
ε→0εδp−1= 0,
where |P| denotes the mesh size of the partition P, and thus f∈C0,p-var([0, T ]; Rn) due to
Wiener’s characterization of C0,p-var([0, T ]; Rn), see [FV10, Theorem 5.31].
Using Wiener’s characterization of C0,α-H¨ol([0, T ]; Rn) for α∈(0, δ −1/p), we get the second
embedding because of
lim
ε→0sup
[s,t]⊂[0,T ],|t−s|<ε
|ft−fs|
|t−s|α≤lim
ε→0sup
[s,t]⊂[0,T ],|t−s|<ε |ft−fs|p
|t−s|δp−11/p
εδ−1/p−α= 0
for f∈Vδ,p([0, T ]; Rn). 

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 15
3. Continuity of the Itˆ
o-Lyons map
The dynamics of a controlled differential equation driven by a path X: [0, T ]→Rnof finite
q-variation is formally given by
(3.1) dYt=V(Yt) dXt, Y0=y0, t ∈[0, T ],
where y0∈Rmis the initial condition, V:Rm→ L(Rn,Rm) is a smooth enough vector field
and T∈(0,∞). Here L(Rn,Rm) denotes the space of linear operators from Rnto Rm. If the
driving signal X∈Cp-var([0, T ]; Rn) for p∈[1,2), Lyons [Lyo94] first established the existence
and uniqueness of a solution Yto the equation (3.1). Moreover, he proved that the Itˆo-Lyons
map is a locally Lipschitz continuous map with respect to the q-variation topology. In order to
restore the continuity for more irregular paths X, say X∈Cq-var([0, T ]; Rn) for an arbitrary
large q < ∞, Lyons introduced the notion of rough paths in his seminal paper [Lyo98], see
Subsection 3.2. Based on Lyons’ estimate, one can deduce the local Lipschitz continuity of
the Itˆo-Lyons map with respect to a H¨older topology, see for example [Fri05].
The aim of this section is to particularly unify these two results by establishing the local
Lipschitz continuity of the Itˆo-Lyons map on Riesz type variation spaces. For this purpose
we combine Lyons’ estimates with our characterization of Riesz type variation in terms to
q-variation to deduce the locally Lipschitz continuity of the Itˆo-Lyons map with respect to
an inhomogeneous Riesz type distance. See Proposition 3.1 for the continuity result in the
regime of bounded variation paths. For the result in the general rough path setting we refer
to Theorem 3.3 and the Corollaries 3.5 and 3.8.
To quantify the regularity of the vector field Vin the controlled differential equation (3.1),
we introduce for α > 0 the space Lipα:= Lipα(Rm;L(Rn,Rm)) in the sense of E. Stein,
cf. [FV10, Definition 10.2]. For α > 0 and ⌊α⌋:= max{n∈N:n≤α}the space Lipα
consists of all maps V:Rm→ L(Rn,Rm) such that Vis ⌊α⌋-times continuously differentiable
with (α− ⌊α⌋)-H¨older continuous partial derivatives of order ⌊α⌋(or with continuous partial
derivatives of order αin the case α=⌊α⌋). On the space Lipαwe introduce the usual norm
k · kLipαand further denote the supremum norm by k · k∞. For the supremum norm on
C([0, T ]; Rn) we write k · k∞;[0,T ]:= sup0≤t≤T| · |.
3.1. Continuity w.r.t. ˜
V1,p.In this subsection we derive the local Lipschitz continuity of
the solution map on the Riesz type variation spaces V1,p([0, T ]; Rn). To that end the equiva-
lent characterization of V1,p ([0, T ]; Rn) given by ˜
V1,p([0, T ]; Rn) turns out to be particularly
convenient. The solution map Φ is defined by
(3.2) Φ: Rm×Lip1ט
V1,p([0, T ]; Rn)→˜
V1,p([0, T ]; Rm) via Φ(y0, V, X ) := Y,
where Ydenotes the solution to the integral equation
(3.3) Yt=y0+Zt
0
V(Ys) dXs, t ∈[0, T ].
First notice that the integral appearing in equation (3.3) can be defined as a classical Riemann-
Stieltjes integral with respect to bounded variation functions because of the embedding
˜
V1,p([0, T ]; Rn)⊂C1-var([0, T ]; Rn) for all p∈(1,∞) due to Proposition 2.2 and Lemma 2.6.
Proposition 3.1. For X∈˜
V1,p([0, T ]; Rn)with p∈(1,∞),V∈Lip1and every initial
condition y0∈Rm, the controlled differential equation (3.3) has a unique solution Y∈
˜
V1,p([0, T ]; Rn)and the solution map Φas defined in (3.2) is locally Lipschitz continuous.

16 FRIZ AND P R ¨
OMEL
More precisely, for yi
0∈Rm,Xi∈˜
V1,p([0, T ]; Rn),Vi∈Lip1such that
kXik˜
V1,p ≤band kVikLip1≤l, i = 1,2,
for some b, l > 0and corresponding solution Yi, there exist a constant C=C(b, l, p)≥1such
that
kY1−Y2k˜
V1,p ≤CkV1−V2k∞+|y1
0−y2
0|+kX1−X2k˜
V1,p .
Proof. Since Xi∈˜
V1,p([0, T ]; Rn), Xiis in particular of bounded variation and thus the
integral equation (3.3) is well-defined and admits a unique solution Yi∈C1-var([0, T ]; Rn) for
each i= 1,2. Moreover, for every subinterval [s, t]⊂[0, T ] the local Lipschitz continuity of
the solution map Φ in 1-variation, c.f. [FV10, Theorem 3.18] and [FV10, Remark 3.19], yields
kY1−Y2k1-var;[s,t]≤2 exp(3bl)|y1
s−y2
s|lc(s, t) + kV1−V2k∞c(s, t) + lkX1−X2k1-var;[s,t]
where c(s, t) can be chosen such that
kX1k1-var;[s,t]+kX2k1-var;[s,t]≤c(s, t).k(s, t)(t−s)1−1/p
with k(s, t) := kX1k˜
V1,p;[s,t]+kX2k˜
V1,p;[s,t]. Dividing both sides by |t−s|1−1/p and taking
them to the power pleads to
kY1−Y2kp
1-var;[s,t]
|t−s|p−1
.exp(3blp)|y1
s−y2
s|plpk(s, t)p+kV1−V2kp
∞k(s, t)p+lpkX1−X2kp
1-var;[s,t]
|t−s|p−1.
From this inequality we deduce, by summing over a partition of [0, T ] and taking then the
supremum over all partitions, that
kY1−Y2k˜
V1,p .exp(3vl)kV1−V2k∞b+kY1−Y2k∞;[0,T ]bl +lkX1−X2k˜
V1,p ,
where we used the super-additivity of k(s, t)pand k(0, T )≤2b. Finally, kY1−Y2k∞;[0,T ]can
be estimated by [FV10, Theorem 3.15] to complete the proof. 
Remark 3.2. An immediate consequence of Lemma 2.6 is that the local Lipschitz continuity
as stated in Proposition 3.1 of the solution map Φgiven by (3.2) also holds with respect to
the (equivalent) Sobolev or Nikolskii metric.
3.2. Continuity w.r.t. general Riesz type variation. In order to give a meaning to the
controlled differential equation (3.1) for driving signals Xwhich are not of bounded variation,
we introduce here the basic framework of rough path theory. For more comprehensive mono-
graphs about rough path theory we refer to [LQ02, FV10, FH14], and for the convenience of
the reader the following definitions are mainly borrowed from [FV10].
As already explained in the Introduction, a rough path takes values in the metric space
(GN(Rn), dcc) and not “only” in the Euclidean space (Rn,k · k). Let us recall the basic
ingredients:

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 17
For N∈Nand a path Z∈C1-var(Rn) its N-step signature is given by
SN(Z)s,t :=1,Zs<u<t
dZu,...,Zs<u1<···<uN<t
dZu1⊗ · · · ⊗ dZuN
∈TN(Rn) :=
N
M
k=0 Rn⊗k,
where Rn⊗kdenotes the k-tensor space of Rnand R⊗0:= R. We note that TN(Rn) is an
algebra (“level-Ntruncated tensor algebra”) under the tensor product ⊗. The corresponding
space of all these lifted paths is the step-Nfree nilpotent group (w.r.t. ⊗)
GN(Rn) := {SN(Z)0,T :Z∈C1-var ([0, T ]; Rn)} ⊂ TN(Rn).
For every g∈GN(Rn) the so-called “Carnot-Caratheodory norm”
kgkcc := inf ZT
0
kdγsk:γ∈C1-var([0, T ]; Rn) and SN(γ)0,T =g,
where RT
0kdγskis the length of γbased on the Euclidean distance, is finite and the infimum
is attained, see [FV10, Theorem 7.32]. This leads to the Carnot-Caratheodory metric dcc via
dcc(g, h) := kg−1⊗hkcc, g, h ∈GN(Rn),
where g−1is the inverse of gin the sense g−1⊗g= 1, see [FV10, Proposition 7.36 and
Definition 7.41]. Hence, (GN(Rn), dcc) is a metric space.
The space of all weakly geometric rough paths of finite q-variation is then given by
Ωq:= Cq-var([0, T ]; G⌊q⌋(Rn)) := X∈C([0, T ]; G⌊q⌋(Rn)) : kXkq-var <∞,
where k · kq-var is the q-variation with respect to the metric space (G⌊q⌋(Rn), dcc) as defined
in (2.1) and ⌊q⌋:= max{n∈N:n≤q}. Note that k · kq-var on Ωqis commonly called the
homogeneous (rough path) norm since it is homogeneous with respect to the dilation map on
T⌊q⌋(Rn), cf. [FV10, Definition 7.13].
Coming back to the controlled differential equation (3.1), we first need to introduce a
solution concept suitable for this equation given the driving signal is now a weakly geometric
rough path. Let V:Rm→ L(Rn,Rm) be a sufficiently smooth vector field and y0∈Rmbe
some initial condition. For a weakly geometric rough path X∈Cq-var([0, T ]; G⌊q⌋(Rn)) we
call Y∈C([0, T ]; Rm) a solution to the controlled differential equation (also called rough
differential equation)
(3.4) dYt=V(Yt) dXt, Y0=y0, t ∈[0, T ],
if there exist a sequence (Xn)⊂C1-var([0, T ]; Rn) such that
lim
n→∞ sup
0≤s<t≤T
dcc(S⌊q⌋(Xn)s,t,Xs,t) = 0,sup
nkS⌊q⌋(Xn)kq-var;[0,T]<∞,
and the corresponding solutions Ynto equation (3.3) converge uniformly on [0, T ] to Yas n
tends to ∞, cf. [FV10, Definition 10.17].

18 FRIZ AND P R ¨
OMEL
On the space Ωq=Cq-var([0, T ]; GN(Rn)) the classical way to restore to the continuity of
the solution map associated to a controlled differential equation (3.4) (also called Itˆo-Lyons
map) is to introduce the inhomogeneous variation distance
ρq-var(X1,X2) := max
k=1,...,N ρ(k)
q-var;[0,T ](X1,X2),
for X1,X2∈Cq-var([0, T ]; GN(Rn)) and q∈[1,∞), with
ρ(k)
q-var;[s,t](X1,X2) := sup
P⊂[s,t]X
[u,v]∈P
|πk(X1
u,v −X2
u,v)|q
kk
q
,[s, t]⊂[0, T ],
for k= 1,...,N, where πk:TN(Rn)→Rn⊗kis the projection to the k-th tensor level
and each tensor level Rn⊗kis equipped with the Euclidean structure. Here we recall that
Xu,v := X−1
u⊗Xv. Note that the distance ρq-var is not homogeneous anymore with respect
to the dilation map on TN(Rn) as indicated by its name.
In the seminal paper [Lyo98], Lyons showed that the solution map Φ given by
Φ: Rm×Lipγ×C1/δ-var([0, T ]; G⌊1/δ⌋(Rn)) →C1/δ-var([0, T ]; Rm) via Φ(y0, V , X) := Y,
where Ydenotes the solution to equation (3.4) given the input (y0, V, X), is local Lipschitz
continuity with respect to the inhomogeneous variation distance ρ1/δ-var for any finite regu-
larity 1/δ > 1.
In the spirit of our characterization (2.5) of the Riesz type variation we introduce for
δ∈(0,1) and p∈[1/δ, ∞) the inhomogeneous mixed H¨older-variation distance by
ρ˜
Vδ,p (X1,X2) := max
k=1,...,N ρ(k)
˜
Vδ,p;[0,T ](X1,X2),
for X1,X2∈˜
Vδ,p([0, T ]; GN(Rn)) and for k= 1,...,N, we set
ρ(k)
˜
Vδ,p;[s,t](X1,X2) := sup
P⊂[s,t]X
[u,v]∈P
ρ(k)
1/δ-var;[u,v](X1,X2)p
k
|u−v|δp−1k
p
,[s, t]⊂[0, T ].
Furthermore, we define the Riesz type geometric rough path space by
Ωδ,p := Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) = ˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) = ˆ
Nδ,p([0, T ]; G⌊1/δ⌋(Rn)),
for δ∈(0,1), p ∈(1/δ, ∞). The identifies hold due to Theorem 2.11 since G⌊1/δ⌋(Rn) is a
metric space equipped with the Carnot-Caratheodory metric dcc .
Relying on the equivalent characterization of Riesz type variation in terms of q-variation
(cf. Theorem 2.11) and on the inhomogeneous mixed H¨older-variation distances for Riesz
type geometric rough paths, the local Lipschitz continuity of the Itˆo-Lyons map Φ given by
(3.5) Φ : Rm×Lipγט
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) →˜
Vδ,p([0, T ]; Rm) via Φ(y0, V , X) := Y,
where Ydenotes the solution to equation (3.4) given the input (y0, V, X), can be obtained
with respect to the inhomogeneous mixed H¨older-variation distance.
Theorem 3.3. Let δ∈(0,1) and γ, p ∈(1,∞)be such that δ > 1/p and γ > 1/δ. For
a Riesz type geometric rough path X∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)), for V∈Lipγand for every
initial condition y0∈Rm, the controlled differential equation (3.4) has a unique solution
Y∈˜
Vδ,p([0, T ]; Rm).

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 19
Furthermore, the Itˆo-Lyons map Φas defined in (3.5) is locally Lipschitz continuous, that
is, for yi
0∈Rm,Xi∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) and Vi∈Lipγsatisfying
kXik˜
Vδ,p ≤band kVikLipγ≤l, i = 1,2,
for some b, l > 0, with corresponding solution Yi, there exist a constant C=C(b, l, γ, δ, p)≥1
such that
kY1−Y2k˜
Vδ,p ≤CkV1−V2kLipγ−1+|y1
0−y2
0|+ρ˜
Vδ,p (X1,X2).
Before we come to the proof, we recall that a continuous function ω: ∆T→[0,∞) is called
control function if ω(s, s) = 0 for s∈[0, T ] and ωis super-additive, cf. Remark 2.3.
Proof. Due to Proposition 2.8 and Theorem 2.11, the assumption X∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn))
implies X∈C1/δ-var([0, T ]; G⌊1/δ⌋(Rn)) as δ > 1/p. Therefore, the controlled differential
equation (3.4) has a unique solution Y∈C1/δ-var([0, T ]; Rm) given the regularity of the
vector field V∈Lipγwith γ > 1/δ, see [FV10, Theorem 10.26].
It remains to show the Riesz type variation of Yand the local Lipschitz continuity of the
Itˆo-Lyons map Φ as defined in (3.5). For this purpose we choose a suitable control function
ω(see (3.9) for the specific definition) and introduce the distance
ρ1/δ-ω(X1,X2) := max
k=1,...,⌊1/δ⌋ρ(k)
1/δ-ω;[0,T ](X1,X2)
for X1,X2∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) and
ρ(k)
1/δ-ω;[0,T ](X1,X2) := sup
0≤s<t≤T
|πk(X1
s,t −X2
s,t)|
ω(s, t)kδ , k = 1,...,⌊1/δ⌋.
As one can see in the proof of [FV10, Theorem 10.26], one has the following two estimates
for the control function ωand a constant C=C(γ, δ)>0:
kY1−Y2k∞;[0,T ]
≤C|y1
0−y2
0|+1
lkV1−V2kLipγ−1+ρ1/δ-ω(X1,X2)exp(Cl1/δ ω(0, T ))(3.6)
and, for all u < v in [0, T ],
|Y1
u,v −Y2
u,v|
≤Cl|Y1
u−Y2
u|+kV1−V2kLipγ−1+lρ1/δ-ω(X1,X2)ω(u, v)δexp(C l1/δω(0, T )).(3.7)
From inequality (3.7) we deduce that
kY1−Y2k
1
δ
1/δ-var;[s,t]= sup
P⊂[s,t]X
[u,v]∈P
|Y1
u,v −Y2
u,v|1/δ
.lkY1−Y2k∞;[s,t]+kV1−V2kLipγ−1+lρ1/δ-ω(X1,X2)1/δ
ω(s, t) exp(Cl1/δ ω(0, T ))1/δ ,

20 FRIZ AND P R ¨
OMEL
which further leads to
kY1−Y2kp
˜
Vδ,p = sup
P⊂[0, T ]X
[s,t]∈P
kY1−Y2kp
1/δ-var;[s,t]
|t−s|δp−1
.lky1−y2k∞;[0,T ]+kV1−V2kLipγ−1+lρ1/δ-ω(X1,X2)p
×exp(Cl1/δ ω(0, T ))psup
P⊂[0, T ]X
[s,t]∈P
ω(s, t)δp
|t−s|δp−1p
.
Plugging in estimate (3.6) in the last inequality gives
kY1−Y2k˜
Vδ,p .l|y1
0−y2
0|+kV1−V2kLipγ−1+lρ1/δ-ω(X1,X2)
×exp(Cl1/δ ω(0, T ))sup
P⊂[0, T ]X
[s,t]∈P
ω(s, t)δp
|t−s|δp−1.
(3.8)
In order to complete the proof, we consider the control function
(3.9) ω(s, t) := kX1k
1
δ
1/δ-var;[s,t]+kX2k
1
δ
1/δ-var;[s,t]+
⌊1/δ⌋
X
k=1
ω(k)
X1,X2(s, t),(s, t)∈∆T,
where
ω(k)
X1,X2(s, t) := ρ(k)
1/δ-var;[s,t](X1,X2)
ρ(k)
˜
Vδ,p;[0,T ](X1,X2)1
δk
with the convention 0/0 := 0, and investigate some properties of ω. First notice that ωfulfills
all the requirements to apply [FV10, Theorem 10.26]. Moreover, it is easy to see that
(3.10) ρ1/δ-ω(X1,X2).ρ˜
Vδ,p (X1,X2)
as one has for k= 1,...,⌊1/δ⌋and 0 ≤s < t ≤Tthe following estimate
|πk(X1
s,t −X2
s,t)| ≤ ρ(k)
1/δ-ω;[s,t](X1,X2)
ρ(k)
˜
Vδ,p;[0,T ](X1,X2)ρ(k)
˜
Vδ,p;[0,T ](X1,X2)≤ω(s, t)δk ρ˜
Vδ,p (X1,X2).
The last observation on ωwe need is
(3.11) sup
P⊂[0, T ]X
[s,t]∈P
ω(s, t)δp
|t−s|δp−1.kX1kp
˜
Vδ,p +kX2kp
˜
Vδ,p + 1.
Indeed, by Proposition 2.8 we have for every partition Pof [0, T ]
X
[s,t]∈P
kXikp
1/δ-var;[s,t]
|t−s|δp−1.X
[s,t]∈P
kXikp
˜
Vδ,p;[s,t]|t−s|δp−1
|t−s|δp−1.kXikp
˜
Vδ,p , i = 1,2,

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 21
and further using
ρ(k)
1/δ-var;[s,t](X1,X2)≤ρ(k)
1/δ-var;[s,t](X1,X2)p
k
|t−s|δp−1k
p
|t−s|(δ−1/p)k
≤ρ(k)
˜
Vδ,p;[s,t](X1,X2)|t−s|(δ−1/p)k,for k= 1,...,⌊1/δ⌋,
we arrive at
X
[s,t]∈P
ω(k)
X1,X2(s, t)δp
|t−s|δp−1=ρ(k)
˜
Vδ,p;[0,T ](X1,X2)−p
kX
[s,t]∈P
ρ(k)
1/δ-var;[s,t](X1,X2)p
k
|t−s|δp−1
≤ρ(k)
˜
Vδ,p;[0,T ](X1,X2)−p
kX
[s,t]∈P
ρ(k)
˜
Vδ,p;[s,t](X1,X2)p
k≤1.
By combining these estimates we deduce (3.11).
Therefore, estimate (3.8) together with (3.10) and (3.11) reveals
kY1−Y2k˜
Vδ,p .l|y1
0−y2
0|+kV1−V2kLipγ−1+lρ ˜
Vδ,p(X1,X2)
×exp(Cl1/δ (2b+ 1))(2b+ 1),
which completes the proof. 
3.3. Inhomogeneous Riesz type distance. In the context of rough path theory it is very
convenient to work with the characterization of Riesz type variation in terms of classical q-
variation and to introduce the corresponding inhomogeneous mixed H¨older-variation distance
ρ˜
Vδ,p , as we have seen in the previous subsection. However, also the other other characteriza-
tions of Riesz type variation allow for obtaining the local Lipschitz continuity of the Itˆo-Lyons
map Φ as defined in (3.5).
Keeping in mind the Riesz type variation (2.2), we define for δ∈(0,1) and p∈[1/δ, ∞)
inhomogeneous Riesz type distance by
ρVδ,p (X1,X2) := max
k=1,...,⌊1/δ⌋ρ(k)
Vδ,p;[0,T ](X1,X2),
for X1,X2∈Vδ,p([0, T ]; G⌊1/δ⌋(Rn)), where we set
ρ(k)
Vδ,p;[s,t](X1,X2) := sup
P⊂[s,t]X
[u,v]∈P
|πk(X1
u,v −X2
u,v)|p
k
|u−v|δp−1k
p
,[s, t]⊂[0, T ],
for k= 1,...,⌊1/δ⌋. Indeed, the inhomogeneous Riesz type distance and inhomogeneous
mixed H¨older-variation distance are equivalent.
Lemma 3.4. If δ∈(0,1) and p∈(1,∞)with δ > 1/p, then
ρVδ,p (X1,X2).ρ˜
Vδ,p (X1,X2).ρVδ,p(X1,X2)
for X1,X2∈Vδ,p([0, T ]; G⌊1/δ⌋(Rn)).
Proof. The first inequality directly follows from
|πk(X1
u,v −X2
u,v)|p
k≤ρ(k)
1/δ-var;[u,v](X1,X2)p
k,[u, v]⊂[0, T ],
for k= 1,...,⌊1/δ⌋.

22 FRIZ AND P R ¨
OMEL
For the second inequality we notice
|πk(X1
u,v −X2
u,v)|1
δk ≤|πk(X1
u,v −X2
u,v)|p
k
|u−v|δp−11
δp
|u−v|1−1
δp
≤ρ(k)
Vδ,p;[u,v](X1,X2)1
δk |u−v|1−1
δp ,[u, v]⊂[0, T ],
for k= 1,...,⌊1/δ⌋. Due to Remark 2.3 the function
∆T∋(u, v)7→ ρ(k)
Vδ,p;[u,v](X1,X2)1
δk |u−v|1−1
δp
is super-additive and thus we get
ρ(k)
1/δ-var;[u,v](X1,X2)p
k≤ρ(k)
Vδ,p;[u,v](X1,X2)|u−v|δp−1.
Therefore, using the super-additive of ρ(k)
Vδ,p;[u,v](X1,X1)p
kas functions in (u, v)∈∆T, we
obtain
ρ(k)
˜
Vδ,p;[u,v](X1,X2)p
k≤ρ(k)
Vδ,p;[u,v](X1,X2)p
k
for k= 1,...,⌊1/δ⌋, which implies the second inequality. 
Applying the equivalence of the inhomogeneous distances ρVδ,p and ρ˜
V δ,p (Lemma 3.4) and
the characterization of Riesz type spaces (Theorem 2.11), the local Lipschitz continuity of the
Itˆo-Lyons map (3.5) with respect to ρVδ,p is an immediate consequence of Theorem 3.3:
Corollary 3.5. Let δ∈(0,1) and γ, p ∈(1,∞)be such that δ > 1/p and γ > 1/δ.
The Itˆo-Lyons map
Φ: Rm×Lipγ×Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) →Vδ,p([0, T ]; Rm)via Φ(y0, V, X) := Y,
where Ydenotes the solution to controlled differential equation (3.4) given the input (y0, V , X),
is locally Lipschitz continuous with respect to inhomogeneous Riesz type distance ρVδ,p .
3.4. Inhomogeneous Nikolskii type distance. To complete the picture, we provide in
this subsection an inhomogeneous distance in terms of Nikolskii regularity (cf. (2.6)), which
is locally Lipschitz equivalent to the inhomogeneous Riesz type distance and which ensures
the local Lipschitz continuity of the Itˆo-Lyons map Φ as defined in (3.5).
To that end we introduce the inhomogeneous Nikolskii type distance as follows: For
X1,X2∈ˆ
Nδ,p([0, T ]; G⌊1/δ⌋(Rn)) we set
ρ(k)
Nδ,p;[u,v](X1,X2) := sup
|v−u|≥h>0
h−δk Zv−h
u
|πk(X1
r,r+h−X2
r,r+h)|p
kdrk
p
,[u, v]⊂[0, T ],
and
ρ(k)
ˆ
Nδ,p;[s,t](X1,X2) := sup
P⊂[s,t]X
[u,v]∈P
ρ(k)
Nδ,p;[u,v](X1,X2)p
kk
p
,[s, t]⊂[0, T ],
for k= 1,...,⌊1/δ⌋. The inhomogeneous Nikolskii type distance ρˆ
Nδ,p is defined by
ρˆ
Nδ,p (X1,X2) := max
k=1,...,N ρ(k)
ˆ
Nδ,p;[0,T ](X1,X2).
In the next two lemmas (Lemma 3.6 and 3.7) we establish that the two ways of introduc-
ing an inhomogeneous distance on C([0, T ]; G⌊1/δ⌋(Rn)), given by the inhomogeneous mixed

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 23
H¨older-variation distance and the inhomogeneous Nikolskii type distance, are locally equiva-
lent.
Lemma 3.6. If δ∈(0,1) and p∈(1,∞)with δ > 1/p, then
ρˆ
Nδ,p (X1,X2).ρ˜
Vδ,p (X1,X2)
for X1,X2∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)).
Proof. Let X1,X2∈˜
Vδ,p([0, T ]; G⌊1/δ⌋(Rn)) and k= 1,...,N. For [s, t]⊂[0, T ] and h∈
(0, t −s] fixed we consider the dissection P(h) := {[ti, ti+1] : s=t0<··· < tM=t−h}such
that
|tM−tM−1| ≤ hand |ti+1 −ti|=hfor i= 0,...,M −2, M ∈N.
Since supu∈[ti,ti+1]|πk(X1
u,u+h−X2
u,u+h)| ≤ ρ1/δ-var;[ti,ti+2](X1,X2) for i= 1,...,M −1 with
tM+1 := t−h, we deduce that
Zt−h
s
|πk(X1
u,u+h−X2
u,u+h)|p
kdu≤
M−1
X
i=1
sup
u∈[ti,ti+1]
|πk(X1
u,u+h−X2
u,u+h)|p
kh
≤1
2(2h)δp
M−1
X
i=1
ρ1/δ-var;[ti,ti+2](X1,X2)p
k
|ti−ti+2|δp−1
.hδpρ(k)
˜
Vδ,p;[s,t](X1,X2)p
k.
Therefore, by the super-additivity of ρ(k)
ˆ
Nδ,p;[s,t](X1,X2)p
kand ρ(k)
˜
Vδ,p;[s,t](X1,X2)p
kas functions
in (s, t)∈∆Twe obtain
ρ(k)
ˆ
Nδ,p;[0,T ](X1,X2).ρ(k)
˜
Vδ,p;[0,T ](X1,X2)
for every k= 1,...,⌊1/δ⌋, which implies that ρˆ
Nδ,p (X1,X2).ρ˜
Vδ,p (X1,X2). 
Lemma 3.7. Let δ∈(0,1) and p∈(1,∞)with δ > 1/p. For Riesz type geometric rough
paths X1,X2∈ˆ
Nδ,p([0, T ]; G⌊1/δ⌋(Rn)) there exists a constant
C:= Cδ, p, kX1kˆ
Nδ,p ,kX2kˆ
Nδ,p ≥1,
depending only on δ,pand the upper bound of kX1kˆ
Nδ,p and kX2kˆ
Nδ,p , such that
ρ˜
Vδ,p (X1,X2)≤Cρ ˆ
Nδ,p(X1,X2).
Proof. Let X1,X2∈ˆ
Nδ,p([0, T ]; G⌊1/δ⌋(Rn)) be Riesz type geometric rough paths. In order
to prove the inequality in Lemma 3.7, it is sufficient to show that there exists a constant
C=Cδ, p, kX1kˆ
Nδ,p ,kX2kˆ
Nδ,p ≥1 such that
(3.12) |πj(X1
s,t −X2
s,t)|1
δj ≤Cj
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
j1
δp
|t−s|1−1
δp =: ω(j)(s, t)
for all s, t ∈[0, T ] with s < t and for every j= 1,...,⌊1/δ⌋.

24 FRIZ AND P R ¨
OMEL
Indeed, for each j= 1,...,⌊1/δ⌋the function ω(j): ∆T→[0,∞) is the super-additive:
ω(j)(s, t)+ω(j)(t, u)
≤Cj
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
j+ρ(i)
ˆ
Nδ,p;[u,t](X1,X2)p
j1
δp |t−s|+|u−t|1−1
δp
≤Cj
X
i=1
ρ(i)
ˆ
Nδ,p;[s,u](X1,X2)p
j1
δp
|u−s|1−1
δp =ω(j)(s, u),
for 0 ≤s≤t≤u≤T, where we used H¨older’s inequality and p/j ≥1. This implies
ρ(j)
1/δ-var;[s,t](X1,X2) = sup
P⊂[s,t]X
[u,v]∈P
|πj(X1
u,v −X2
u,v)|1
δj jδ
≤Cj
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj
p
|t−s|j
p(δp−1),
where the super-additivity of ω(j)is applied in last line. Hence, we get further
ρ(j)
˜
Vδ,p;[0,T ](X1,X2) = sup
P⊂[0, T ]X
[u,v]∈P
ρ(j)
1/δ-var;[u,v](X1,X2)p
j
|u−v|δp−1j
p
≤Csup
P⊂[0, T ]X
[u,v]∈P
j
X
i=1
ρ(i)
ˆ
Nδ,p;[u,v](X1,X2)p
jj
p
≤C
j
X
i=1
ρ(i)
ˆ
Nδ,p;[0,T ](X1,X2),
which immediately gives by taking the maximum over j= 1,...,⌊1/δ⌋that
ρ˜
Vδ,p (X1,X2)≤Cρ ˆ
Nδ,p(X1,X2).
In order to prove inequality (3.12) for each j= 1,...,⌊1/δ⌋, we argue via induction. For
j= 1 inequalities (3.12) is an easy consequence of Theorem 2.11. We now assume that (3.12)
is true for the levels j= 1,...,k −1 and establish the inequality for level k. Let us fix
s, t ∈[0, T ] with s < t and define
Zs
u:= πk(X1
s,s+u−X2
s,s+u), u ∈[0, t −s].
For u, h ∈[0, t −s] with u+h∈[0, t −s] and using
X1
s,s+u+h−X1
s,s+u=X1
s,s+u⊗(X1
s+u,s+u+h−1), π0(X1
s+u,s+u+h−1) = 0
and
πj(X1
s+u,s+u+h−1) = πj(X1
s+u,s+u+h) for j > 0,
we obtain
Zs
u+h−Zs
u=πk(X1
s,s+u+h−X1
s,s+u)−πk(X2
s,s+u+h−X2
s,s+u)
=
k
X
j=1
πk−j(X1
s,s+u)⊗πj(X1
s+u,s+u+h)−
k
X
j=1
πk−j(X2
s,s+u)⊗πj(X2
s+u,s+u+h)
=
k
X
j=1
πk−j(X1
s,s+u)⊗πj(X1
s+u,s+u+h−X2
s+u,s+u+h)

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 25
+
k
X
j=1
πk−j(X1
s,s+u−X2
s,s+u)⊗πj(X2
s+u,s+u+h).
Keeping in mind π0(X1
s,s+u−X2
s,s+u) = 0, we arrive at
Zs
u+h−Zs
u=
k−1
X
j=1
πk−j(X1
s,s+u)⊗πj(X1
s+u,s+u+h−X2
s+u,s+u+h)
+
k−1
X
j=1
πk−j(X1
s,s+u−X2
s,s+u)⊗πj(X2
s+u,s+u+h)
+πk(X1
s+u,s+u+h−X2
s+u,s+u+h).
Hence, we get the following estimate
sup
|t−s|≥h>0
h−δZt−h
s
|Zs
u+h−Zs
u|pdu1
p
.∆1+ ∆2+ ∆3
where we set
∆1:=
k−1
X
j=1
sup
|t−s|≥h>0
h−δZt−h
s
kX1
s,s+ukp(k−j)|πj(X1
s+u,s+u+h−X2
s+u,s+u+h)|pdu1
p
,
∆2:=
k−1
X
j=1
sup
|t−s|≥h>0
h−δZt−h
s
|πk−j(X1
s,s+u−X2
s,s+u)|pkX2
s+u,s+u+hkpj du1
p
,
∆3:= sup
|t−s|≥h>0
h−δZt−h
s
|πk(X1
s+u,s+u+h−X2
s+u,s+u+h)|pdu1
p
.
Due to Proposition 2.8 and δ > 1/p, we have
kX1
s,s+ukp(k−j)≤ kX1kp(k−j)
1/δ-var;[s,t].kX1kp(k−j)
ˆ
Nδ,p;[s,t]|t−s|(δ−1
p)p(k−j).
Moreover, the induction hypothesis gives
|πj(X1
s+u,s+u+h−X2
s+u,s+u+h)|(1−1
j)p.j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj−1
|t−s|(j−1)(δ−1
p)p.
Therefore, ∆1can be estimated by
∆1.
k−1
X
j=1
kX1kk−j
ˆ
Nδ,p;[s,t]|t−s|(k−j)(δ−1
p)j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj−1
p
|t−s|(j−1)(δ−1
p)
×sup
|t−s|≥h>0
h−δZt−h
s
|πj(X1
s+u,s+u+h−X2
s+u,s+u+h)|p
jdu1
p

26 FRIZ AND P R ¨
OMEL
≤
k−1
X
j=1
kX1kk−j
ˆ
Nδ,p;[s,t]ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)1
jj
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj−1
p
|t−s|(k−1)(δ−1
p)
≤
k−1
X
j=1
kX1kk−j
ˆ
Nδ,p;[s,t]j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj
p
|t−s|(δ−1
p)(k−1).
For ∆2we first observe again due to Proposition 2.8 that
kX2
s+u,s+u+hkpj .kX2
s+u,s+u+hkpkX2kp(j−1)
ˆ
Nδ,p;[s,t]|t−s|(δ−1
p)(j−1)p
and by the induction hypothesis that
|πk−j(X1
s,s+u−X2
s,s+u)|p.k−j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
k−jk−j
|t−s|(δ−1
p)(k−j)p.
Combining the last two estimates, we get
∆2.
k−1
X
j=1
kX2kj−1
ˆ
Nδ,p;[s,t]|t−s|(δ−1
p)(j−1)
×sup
|t−s|≥h>0
h−δZt−h
s
|πk−j(X1
s,s+u−X2
s,s+u)|pkX2
s+u,s+u+hkpdu1
p
.
k−1
X
j=1
kX2kj
ˆ
Nδ,p;[s,t]|t−s|(δ−1
p)(k−1)k−j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
k−jk−j
p
.
For ∆3we briefly need to introduce the inhomogeneous H¨older distance for level kby
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2) := sup
u,v∈[s,t]; u6=v
|πk(X1
s+u,s+v−X2
s+u,s+v)|
|u−v|(δ−1
p)k.
This time we simply estimate
∆3≤ρ(k)
ˆ
Nδ,p;[s,t](X1,X2)1
kρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)1−1
k|t−s|(δ−1
p)(k−1).
Applying Proposition 2.8 to Zs
·we get
|πk(X1
s,t −X2
s,t)|=|Zs
0−Zs
t−s|.sup
|t−s|≥h>0
h−δZt−h
s
|Zs
u+h−Zs
u|pdu1
p
|t−s|δ−1
p.
Putting the estimates for ∆1, ∆2and ∆3, we deduce further
|πk(X1
s,t −X2
s,t)| ≤ ˜
C|t−s|δ−1
pk−1
X
j=1 j
X
i=1
ρ(i)
ˆ
Nδ,p;[s,t](X1,X2)p
jj
p
|t−s|(δ−1
p)(k−1)
+ρ(k)
ˆ
Nδ,p;[s,t](X1,X2)1
kρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)1−1
k|t−s|(δ−1
p)(k−1)

ROUGH PATH METRICS ON A BESOV–NIKOLSKII TYPE SCALE 27
.˜
C|t−s|(δ−1
p)kk
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)
+k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
k1
p
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)1−1
k
.˜
C|t−s|(δ−1
p)k k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
kk
p
+k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
k1
p
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)1−1
k,
for some constant ˜
C=˜
C(δ, p, kX1kˆ
Nδ,p ,kX2kˆ
Nδ,p )≥1, which can be rewritten as
|πk(X1
s,t −X2
s,t)|
|t−s|(δ−1
p)k.˜
C k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
kk
p
+k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
k1
p
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)1−1
k.
In other words, we showed that
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)
˜ω(k)(s, t).˜
C1 + ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)
˜ω(k)(s, t)1−1
k,
with
˜ω(k)(s, t) := k
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
kk
p
.
Hence, there exists a constant C=C(δ, p, kX1kˆ
Nδ,p ,kX2kˆ
Nδ,p )≥1 such that
ρ(k)
(δ−1/p)-H¨ol;[s,t](X1,X2)
˜ω(k)(s, t)≤C.
In particular, we have
|πk(X1
s,t −X2
s,t)| ≤ Ck
X
j=1
ρ(j)
ˆ
Nδ,p;[s,t](X1,X2)p
kk
p
|t−s|(δ−1
p)k,
which implies (3.12) for level kand the proof is complete. 
Combining the local equivalence of the inhomogeneous distances ρ˜
Vδ,p and ρˆ
Nδ,p (Lemma 3.6
and 3.7) with the local Lipschitz continuity of the Itˆo-Lyons map (3.5) with respect to ρ˜
Vδ,p
(Theorem 3.3), we deduce same continuity result with respect to ρˆ
Nδ,p :
Corollary 3.8. Let δ∈(0,1) and γ, p ∈(1,∞)be such that δ > 1/p and γ > 1/δ.
The Itˆo-Lyons map
Φ: Rm×Lipγ׈
Nδ,p([0, T ]; G⌊1/δ⌋(Rn)) →ˆ
Nδ,p([0, T ]; Rm)via Φ(y0, V , X) := Y,

28 FRIZ AND P R ¨
OMEL
where Ydenotes the solution to controlled differential equation (3.4) given the input (y0, V , X),
is locally Lipschitz continuous with respect to inhomogeneous Nikolskii type distance ρˆ
Nδ,p .
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Peter K. Friz, Technische Universit¨
at Berlin and Weierstrass Institute Berlin, Germany
David J. Pr¨
omel, Eidgen¨
ossische Technische Hochschule Z¨
urich, Switzerland