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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)q1

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−11

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|δp

|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−11

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

pp′

|v−u|1−p′

p≤ |t−s|1−p′

pX

[u,v]∈P

d(fu, fv)p

|v−u|δp−1p′

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−11

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−δZt−h

s

d(fu, fu+h)pdu1

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 dudv1

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)p1

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]2d(fu, fv)

|v−u|1/p+γp

dudv, [s, t]⊂[0, T ].

Applying the Garsia-Rodemich-Rumsey inequality with Ψ(·) = (·)pand p(·) = (·)1/p+γgives

d(fs, ft)≤8Zt−s

0Fs,t

u21

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)p1

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>0kfs+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)pdu1

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−11/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×Lip1×˜

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 ≤CkV1−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