Approximations of the brownian rough path with applications to stochastic analysis

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www.elsevier.com/locate/anihpb

Approximations of the Brownian rough path with applications to stochastic analysis

Peter Friz

^a

, Nicolas Victoir

^b,^∗

a251 Mercer St., New York, NY 10012, USA bMagdalen College, Oxford OX1 4AU, UK

Received 25 August 2003; received in revised form 15 March 2004; accepted 11 May 2004 Available online 11 September 2004

Abstract

A geometricp-rough path can be seen to be a genuine path of finitep-variation with values in a Lie group equipped with a natural distance. The group and its distance lift(R^d,+,0)and its Euclidean distance.

This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Lévy Area).

As a first application, extending an idea due to Millet and Sanz-Solé, we characterize the support of the Enhanced Brown- ian Motion (without relying on correlation inequalities). Secondly, we prove Schilder’s theorem for this Enhanced Brownian Motion. As all results apply in Hölder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang [Stochastic Process. Appl. 102 (2) (2002) 265–283]. Lyons’ fine estimates in terms of control functions [Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] allow us to show that the Itô map is still continuous in the topologies we introduced. This provides new and simplified proofs of the Stroock–Varadhan support theorem and the Freidlin–Wentzell theory. It also provides a short proof of modulus of continuity for diffusion processes along old results by Baldi.

Résumé

Unp-rough path est un chemin dep-variation finie à valeurs dans un groupe de Lie muni d’une distance sous-riemannienne.

Le groupe et sa distance géneralisent(R^d,+,0)et la distance euclidienne.

Cette approche nous permet d’obtenir un module de continuité tres précis pour le rough path brownien (le mouvement brownien et son aire de Lévy). Pour ce dernier, nous prouvons un théorème du support (adaptant une idée de Millet et Sans- Solé) et un théorème de Schilder. Comme tous les résultats sont prouvés en utilisant des topologies de type Hölder ou plus fines, cela géneralise le papier de Ledoux, Qian, Zhang [Stochastic Process. Appl. 102 (2) (2002) 265–283]. Les résultats de T. Lyons [Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] permettent de prouver rapidement que l’application d’Itô est continue pour les topologies que nous avons introduites. Cela nous donne de nouvelles preuves du théorème du support de Stroock–Varadhan

* Corresponding author.

E-mail addresses: [email protected] (P. Friz), [email protected] (N. Victoir).

doi:10.1016/j.anihpb.2004.05.003

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et de la théorie de Freidlin–Wentzell. Nous obtenons au passage une preuve simple du module de continuité pour les processus de diffusions, obtenu précédemment par Baldi.

MSC: 60F10; 60H10

Keywords: Rough paths; Support theorem; Large deviation; Modulus of continuity of diffusions

1. Introduction

Starting with [20], Terry Lyons developed a general theory of integration and differential equations of the form

dy_t=f (y_t) dx_t. (1)

To include the important example of stochastic differential equations,x is allowed to be “rough” in some sense.

Standard Hölder regularity of Brownian motion, for instance, implies finitep-variation only forp >2. Another is- sue was to explain (deterministically) the difference between stochastic differential equations based on Stratonovich versus Itô integration. Last but not least, motivated from examples like Fractional Brownian motion, driving signals much rougher than Brownian motion should be included.

All this has been accomplished in a beautiful way and the reader can nowadays find the general theory exposed in [21,23,24].

Loosely speaking, for generalp1, one needs to “enhance” the driving signalx, with values in some Banach spaceV, to X∈V ⊕V^⊗²· · · ⊕V^⊗[^p^] such that the resulting object X satisfies certain algebraic¹and analytic conditions. Forxof finite variation, this enhancement will simply consist of all the iterated integrals ofx,

X^k_s,t:=

s<u₁<···<u_k<t

dx_u₁⊗ · · · ⊗dx_u_k, k=1, . . . ,[p].

These are the Smooth Rough Paths. Consider a time horizon of[0,1](valid for the rest of the paper) and introduce thep-variation metric, defined as

d(X, Y )= max

k=1,...,[p]

sup

D

l

|X^k_t_l−1_,t

l−Y_t^k_l−1_,t

l|^{p/ k} k/p

,

where sup_Druns over all finite divisions of[0,1]. Here| · |denotes (compatible) tensor norms inV^⊗^k. Closure of Smooth Rough Paths with respect to this metric yields the class of Geometric Rough Paths, denoted byGΩ_p(V ).

The solution map, also called Itô map, to (1) is then a continuous map from GΩp(V )→GΩp(W ), provided f:W→L(V , W )satisfies mild regularity conditions. This is Lyons’ celebrated Universal Limit Theorem. In par- ticular, smooth approximationsX(n)which converge inp-variation toX∈GΩ_p(V )will cause the corresponding solutions Y (n)to converge to Y in p-variation. Hence, one deals with some kind of generalized Stratonovich theory.

However, the so important case ofp∈(2,3), on which this paper will focus, allows for more. For the sake of concreteness, we will setV =R^dfrom here on. Following [24, p. 149] and also [23] the driving signal only needs to be a Multiplicative functional of finitep-variation. By definition, this is a continuous map

(s, t )→(X_s,t¹ , X_s,t² )∈R^d⊕(R^d)^⊗²=:T²,

where 0st1, satisfying the algebraic Chen condition

X_s,u=X_s,t⊗X_t,u ⇔ X_s,u¹ =X_s,t¹ +X¹_t,u, X²_s,u=X²_s,t+X_t,u² +X¹_s,t⊗X¹_t,u, (2)

1 For algebraic convenienceXis often enhanced toR⊕V⊕V^⊗2· · · ⊕V^⊗[p]with scalar component constant 1.

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wheneverstu, and the analytic conditiond_p-var(X,0) <∞i.e.

sup

(0t₀<...<t_n1)

l

|X^k_t

l−1,tl|^{p/ k}<∞, k=1,2. (3)

(Often k=1,2 are referred to as first resp. second level.) The class of such rough paths is denoted Ωp(R^d).

Condition (2) is known as Chen relation and expresses simple additive properties wheneverX²is obtained as some iterated integral including the cases of Stratonovich resp. Itô Enhanced Brownian Motion. Whenever a first order calculus underlies this integration (which is the case for Stratonovich integration), one has

Symm(X²)=1

2X¹⊗X¹. (4)

Geometricp-Rough Paths satisfy this condition, but paths satisfying this condition and which satisfy the condition d_p-var(X,0)forms a set slightly bigger thanGΩ_p(V )[13]. Clearly,

{Smooth Rough Paths} ⊂GΩp(R^d)⊂Ωp(R^d).

One can indeed choose in which space to work with and the Lyons theory will provide meaning, existence and uniqueness to the purely deterministic rough differential equation

dY=V (y0+Y_0t¹) dX,

whereV =(V₁, . . . , V_d)are, in general non-commuting, vector fields with mild regularity conditions. As before, the Itô mapX →Y is, continuous underp-variation topology. This rough differential equation indeed generalizes ordinary and stochastic (Stratonovich and Itô) differential equations. For instance, it is known (and also follows from the results in this paper) that a.s. the Stratonovich Enhanced Brownian Motion (EBM) B∈GΩp(R^d)for all p∈(2,3). ChoosingX=B the projection of the rough pathY to its first level will solve the associated Stratonovich stochastic differential equation. That is

y0+Y_0t¹ solves dy=

i

Vi(y)◦dβⁱ.

For the rest of the paper,pdenotes a fixed real in(2,3). The contributions of this paper may be summarized as follows:

(a) We look at geometricp-rough paths from a new angle. Observe thatG:= {X∈T²: (4) holds}is the free nilpotent Lie group of step 2 [23,31], a simply connected Lie group which lifts(R^d,+,0). Chen’s condition is equivalent to the fact that x_t=X_0,t is aG-valued path such thatX_s,t=x⁻_s¹⊗x_t=x_s,t.We put a homogenous, sub-additive norm on(G,⊗). Geometricp-rough paths are then easily seen to be the closure of “smooth”G-valued paths under thep-variation metric. Standard proofs for Kolmogorov’s criterion or the Garsia, Rumsey, Rodemich inequality adapt with no changes from(R^d,+)-valued to(G,⊗)-valued processes. With this observation, regularity results for the EBM B,as Hölder continuity and Lévy modulus of continuity, follow after simple moment estimates.

Sometimes, it will be convenient to work in the associated Lie algebra of the groupG. For instance, the EBM viewed through this chart is nothing else than the well studied Gaveau diffusion [14]. Its importance in the context of limit theorems was already highlighted in Malliavin’s book, [26].

(b) We introduce a number of different topologies onGΩp(R^d),effectively reducing this space to geometric rough paths for which the associated norms are finite. For instance, we are able to deal with Hölder and “modulus type” norms. Exploiting fine estimates in Lyons’ Limit Theorem we have continuity of the Itô map in all these topologies. In former applications of rough path theory to stochastic analysis result were always obtained inp- variation topology, leaving open a gap between (usually well known) results in Hölder and stronger topologies.

(c) Lyons’ Universal Limit Theorem implies a Lévy modulus of continuity for diffusions, along the results by Baldi [2].

(d) We establish convergence of several different approximations to the EBM. The EBM is usually defined as the limit of a sequence of smooth rough paths, which is shown to be Cauchy [24]. Here, we define directly the

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EBM, and its regularity allows us to prove convergence of some sequences of smooth rough paths to the EBM. The first idea, common to works in [22,26,17,12], is that approximations are obtained by conditioning with respect to dyadic filtrations. After establishing uniform regularity of approximations, easily obtained by Doob’s inequality, we can use an interpolation argument to show convergence in interesting topologies of some sequences of smooth rough paths to the EBM.

(e) IfW is the Wiener measure on the Wiener space equipped with the, say,α-Hölder topology,α <1/2 , the supportWis the closure of smooth path under the α-Hölder metric. One proof, as noticed in [28], reduces to some convergence results of adapted linear approximationωⁿ, which we take to be the dyadic approximation ofωtime shifted of−2⁻ⁿ(to make it adapted). Thenωⁿ converges toωin theα-Hölder topology, which shows that the support ofWis contained in the closure of smooth path under theα-Hölder metric. Reciprocally, given a smooth pathg,g+ω−ωⁿconverges tog, which shows the inverse inclusion, using the Cameron–Martin theorem.

We extend this proof to the “Wiener measure” on the free nilpotent group of step 2, i.e. the law of the Enhanced Brownian motion. The main difficulty is to introduce a translation operatorT, to make sense ofg+ω−ωⁿ for group valued paths. By means of continuity of the Itô map, from (b), this immediately implies the support theorem for diffusions. In the context of rough paths this improves work by Ledoux, Qian, Zhang (support theorem in p-variation topology, [25]) and by the first cited author (Hölder topology, [12]).

As for the history of the support theorem, it was originally obtained by Stroock, Varadhan [37] in sup-topology, then by Ben Arous, Gruadinaru, Ledoux [7] and Millet, Sanz-Solé [28] in Hölder norm of exponent less than 1/2.

Probably the more abstract proof of this result can be found in [1]. Extension to modulus space has been obtain in [16] and to Orlicz–Besov space in [27]. We limit ourselves to “modulus norm”, in the spirit of [16], and we will not recover fully the results in [16,27]. On the other hand, we have a description of the support of the Stratonovich enhanced diffusion the goes beyond the last quoted results.

(f) Schilder’s theorem for EBM is obtained. As before, continuity of the Itô map will give the Freidlin–Wentzell large deviation result in the topologies mentioned in (b). Strassen’s law is obtained as corollary. Again, we improve [25] and recover well known large deviations results in Hölder and modulus norm [5]. Once again though, we do not deal with Orlicz–Besov metrics (see [10]).

Constants in this paper may vary from line to line.

2. Rough paths

2.1. Free nilpotent Lie group of step 2

We fix the dimensiond (d2 to avoid trivialities) and we denote by L(R^d)=R^d⊕so(d), where so(d) R^d(d⁻^1)/2is the space of real antisymmetricd×dmatrices. With the bracket

[,]:L(R^d)×L(R^d)→L(R^d), (a¹, a²), (b¹, b²)

→ [a, b] =(0, a¹⊗b¹−b¹⊗a¹),

L(R^d)becomes a (step 2 nilpotent) Lie algebra. The group multiplication in the associated simply connected Lie GroupG(R^d)=exp(L(R^d)),[31,39,41], is given by the Baker–Campbell–Hausdorff formula.²

⊗:G(R^d)×G(R^d)→G(R^d), exp(a)⊗exp(b)→exp

a+b+1 2[a, b]

.

2 Due to step 2 nilpotency, only the first bracket appears.

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Its neutral element is exp(0), and the inverse of exp(a)is exp(−a). We can identifyG(R^d)with the nonlinear submanifold ofR^d⊕R^d^×^dgiven by

g=(g¹, g²)∈R^d⊕(R^d)^⊗²: symmetric part ofg²equals1 2g¹⊗g¹

with usual (truncated) tensor multiplication, that is, g⊗h=(g¹+h¹, g²+g¹⊗h¹+h²).

(G(R^d),⊗,exp(0))is the free nilpotent group of step 2 overR^d, [23,31]. Note thatG(R^d)is invariant under the dilation operatorδ_t, fort∈R,δ_t being defined by

G(R^d)→G(R^d), exp(a¹, a²)→exp(t a¹, t²a²).

We define on the group g = inf

x¹,...,xⁿ∈R^d

ni=1exp(xⁱ)=g

n i=1

|x_i|_R^d, (5)

where|.|_R^d is the Euclidean norm onR^d. For all g,g is finite by Chow theorem [15,29], although it can be seen quite directly here with the use of the Baker–Campbell–Hausdorff formula..is a sub-additive, symmetric homogeneous norm [11] onG(R^d), that is

(i) gif and only ifg=exp(0),

(ii) for allg∈G(R^d)andt∈R,δ_tg = |t|g, (iii) for allg, h∈G(R^d),g⊗hg + h, (iv) for allg,g = g⁻¹.

From this sub-additive, symmetric homogeneous norm, we construct a left invariant distance onG(R^d)(which is a Carnot–Caratheodory distance [15,29]) by defining

d(g, h)= h⁻¹⊗g.

If| · |_R^d_⊗R^d denotes a norm onR^d⊗R^d, then exp(a¹, a²)= |a¹|_R^d +

|a²|_R^d_⊗R^d

defines another homogeneous norm onG(R^d)(that is a norm satisfying (i) and (ii)), and as all homogeneous norms are equivalent [18], one can find some positive constantsc₁, c₂such that for allg∈G(R^d)

c₁|||g|||gc₂|||g|||. (6)

This implies the following:

Corollary 2. For some constantC, h⁻¹⊗g⊗hC

g +

hg

, (7)

and for anyk2, d

_k

i=1

gi, k

i=1

hi

C

k i=1

d(gi, hi)+

d(gi, hi) ^k

j=i+1

hj

. (8)

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Proof. Ifg=e^b, h=e^a,h⁻¹⊗g⊗h=e^b⊗e^−[^a,b^], hence

|||h⁻¹⊗g⊗h||||||g||| + |||e^−[^a,b^]||||||g||| +

|||g||| · |||h|||.

The inequality is then proved using inequality (6). The second inequality is a consequence of the first one. We show it fork=2, the general case follows in exactly the same way, by induction.

d(g₁⊗g₂, h₁⊗h₂)= h⁻₂¹⊗h⁻₁¹⊗g₁⊗g₂

= h⁻₂¹⊗h⁻₁¹⊗g1⊗h2⊗h⁻₂¹⊗g2 h⁻₂¹⊗h⁻₁¹⊗g1⊗h2 + h⁻₂¹⊗g2

C

d(g₁, h₁)+

d(g₁, h₁)h₂

+d(g₂, h₂). 2

(G(R^d),⊗,exp(0))equipped with a homogeneous norm is a simple generalization of(R^d,+,0)equipped with a norm.

We letC₀([0,1], G(R^d))to be the space of continuous function from[0,1]toG(R^d)such that their value at time 0 is exp(0). With a slight abuse, we will call such elementsG(R^d)-valued paths. If x∈C₀([0,1], G(R^d))and s < t, we will denote by xs,t the element x⁻_s¹⊗xt.

Remark 3. Let|.|_R^d_⊗R^d be a compatible tensor norm,³invariant under matrix-transposition. Then an explicit norm satisfying (i)–(iv) is given by

max

|a¹|_R^d,

a²+1

2a¹⊗a¹ R^d⊗R^d

,

where(a¹, a²)∈L(R^d).

2.2. p-variation

Let(G,⊗, e)be a group equipped with a homogeneous norm..Here, we think ofGbeing either(R^d,+,0) or(G(R^d),⊗,exp(0)).A pathx:[0,1] →Gis said to have finitep-variation if

sup

(0t₀<···<t_n1)

i

x_t_i_,t_i₊₁^p<∞.

Note that a pathxis continuous and of finitep-variation if and only if (see [24]) x_s,t^pω(s, t ) for allst

for some control functionω.By definition, this means

(i) ω:{(s, t ),0st1} →R⁺is continuous near the diagonal.

(ii) ωis super-additive, i.e.

∀s < t < u, ω(s, t )+ω(t, u)ω(t, u). (9)

(iii) ω(t, t )=0 for allt∈ [0,1].

We will say in such case thatxhas finitep-variation controlled byω. We will construct control functions in the following way,

3 That is|a⊗b|_R^d_⊗R^d |a|_R^d|b|_R^d.

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Proposition 4. Consider a continuous mapf:R⁺→R⁺,increasing, convex,f (0)=0. Then(s, t ) →f (t−s)is a control function.

Example 5.f (t −s)=c(t−s) for some constantc >0. This is equivalent to 1/p-Hölder continuity for the controlled path.

We define ap-variation distance between twoG-valued pathsxandy:

d_p-var(x, y)= sup

(0t₀<···<t_n1)

i

d(x_t_i_,t_i₊₁, y_t_i_,t_i₊₁)^p.

2.3. Definition of a geometricp-rough path

We will denote byπ₁resp.π₂the natural injection fromG(R^d)ontoR^dresp. onto so(d). If x is aG(R^d)-valued path of finitep-variation, then clearly,π₁(x):t→π₁(x_t)is aR^d-valued path of finitep-variation. We will say that x lies aboveπ₁(x). Conversely, assumexis aR^d-valued path of finitep-variation. Ifp <2, then

S(x):t→exp

x_t+1 2 t 0

(x_u⊗dx_u−dx_u⊗x_u)

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Ifp∈(2,3),then there exists aG(R^d)-valued path x lying abovex [40], but uniqueness is not true anymore [23,40].

Definition 6. For p∈ [2,3),the set of geometric p-rough path is the closure of{S(x), x of finite 1-variation} under thep-variation metricdp-var. Such class is denoted byGΩp(R^d).

We see in particular that any G(R^d)-valued path of finite q-variation, for q > p, is a p-rough path [13].

Once again, the set ofG(R^d)-valued path of finitep-variation, that we denoteGΩ_p+(R^d)is strictly bigger than GΩp(R^d).

Note that if x is aG(R^d)-valued path, then(s, t )→x⁻_s¹⊗xt is a geometric multiplicative functional, in the sense of [23]. Reciprocally, if y_s,t is a multiplicative functional, then x_t =y_0,t is aG(R^d)path starting at exp(0) and y_s,t=x⁻_s¹⊗x_t.Geometric multiplicative functionals andG(R^d)-valued path are the same objects. The main theorem in rough path theory is the continuity of the Itô map.

2.4. The Itô map

Theorem 7. Let x∈GΩ_p+(R^d),ε >0 andVbe a linear map fromR^dinto the Lip[p+ε,Rⁿ]vector fields.⁴There exists a solution y∈GΩ_p+(Rⁿ)to the rough differential equation

dyt=V π1(yt)

dxt, y₀¹=a, (11)

i.e. there exists an extension of x to z∈GΩ_p+(R^d⊕Rⁿ)such that z projects onto zt=(x_t,yt),

4 A functionf which is(k+ε)-Lipschitz onR^d,fork∈Nis aktimes differentiable function whosekth derivative isε-Hölder, using the classical definition of Hölder functions with parameter in[0,1).See [23,34].

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and z satisfies z=

h(π₁(z)) dz, with

h:R^d⊕Rⁿ→Hom(R^d⊕Rⁿ,R^d⊕Rⁿ), (x, y)→

(dX, dY )→(dX, f (y)dY ) .

If thep-variation of x is controlled byω, then thep-variation of z (and hence y) is controlled byCω, whereC depends onp, ε, the(p+ε)-Lipschitz norm ofVand the supremum ofωon the consider interval. Moreover, for alls < tsuch thatω(s, t ) <1,

zs,t^pKp,ε,fω(s, t ),

whereK_p,ε,_V is a constant which only depends onp,ε, and the(p+ε)-Lipschitz norm ofV.

If x andx are two elements of˜ GΩ_p+(R^d)such thatx⁻_s,t¹⊗ ˜xs,t^pεω(s, t ), then the corresponding solution of Eq. (11) z andz satisfy˜ z⁻_s,t¹⊗ ˜zs,t^pη(ε)ω(s, t )whereηis a continuous function such thatη(0)=0 (i.e. the map x→z is continuous, and hence the Itô map x→y is continuous).

Proof. A simple translation of the first and second level estimates in [23] or [24] to our norm||| · |||or · . 2 A simple corollary of it, observed in [12], is the continuity of the Itô map in “Hölder type” norm. The same simple argument gives the continuity of the Itô map in “modulus topologies”. First, we let

Ξ_p=

ϕ:[0,1] →R⁺, withϕ(0)=0 andϕ^pis strictly increasing and convex .

Such set is obviously not empty,t→t^1/p being one example of an element of Ξ_p. Let us look at some more complicated one.

Example 8. Letα >0. Then, the functionx→(x(−lnx)^a)^p/2is strictly increasing and convex in a neighborhood of 0 (it can be checked by differentiating it twice). Letχ_a,p=infx>0d²(x(−lnx)^a)^p/2

dx² <0.Then define φ_a,p(x)=

√

x(−lnx)^a ifx∈ [0, χ_a,p],

φ^p_a,p(χ_a,p)+(φ_a,p^p )(χ_a,p)(x−χ_a,p)1/p

ifx∈ [χ_a,p,1].

In other words,φ_a,p^p (x)is the smallest convex function dominating(x(−lnx)^a)^p/2. Remark that fora >1, lim

x→0

φ1,p(x) φ_a,p(x)=0.

For a functionϕ∈Ξ_p,we define a distance between twoG(R^d)-valued paths x andx˜ d_ϕ(x,x)˜ = sup

0s<t1

x⁻_s,t¹⊗ ˜xs,t ϕ(t−s) . For a singleG(R^d)-valued path x, we let

xϕ= sup

0s<t1

x_s,t ϕ(t−s).

We also letd_∞(x,x)˜ =sup₀_s<t₁x⁻_s,t¹⊗ ˜x_s,tandx_∞=sup₀_s<t₁x_s,t. It is straightforward to check thatdϕandd_∞are distances on the spaceC₀(G(R^d)).

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Corollary 9. Let x,x˜∈GΩ_p+(R^d),ε >0 andV be a linear map fromR^d into the Lip[p+ε,Rⁿ]vector fields, and z,˜z∈GΩ_p+(R^d⊕Rⁿ)the corresponding solution of Eq. (11). There exists a continuous functionδsatisfying δ(0)=0, such that

dϕ(x,x)˜ ε⇒dϕ(z,z)˜ δ(ε).

Remark 10. The requirement thatϕ∈Ξ_p to defined_ϕ is only for convenience (so that(s, t )→ϕ^p(t−s) is a control). Indeed, ifϕ˜ is another increasing function such that ϕ˜ is equivalent to ϕ at 0, then the topologies on G(R^d)-valued paths space induced bydϕandd_ϕ_˜are identical.

2.5. The translation operator on rough path space

We define the translation operator, first introduced in a more general situation in [23]. Letq be real such that 1/q+1/p >1.

The following definition is motivated by replacingxbyx+f in (10).

Definition 11. Let x∈GΩ_p+(R^d)andf be aR^d-valued path of finiteq-variation. We letx_t=π₁(x_t). Then we defineT_f(x)by

π₁ T_f(x)

s,t=f_s,t+x_s,t and

π₂

T_f(x)_s,t

=π₂(x_s,t)+1 2π₂

t s

f_s,u⊗df_u

+1 2 t s

f_s,u⊗dx_u +1

2 t

s

xs,u⊗dfu− t s

dfu⊗xs,u− t s

dxu⊗fs,u

,

where the integrals are well defined Young integrals.

Remark that it is easily checked thatTf(x)_s,t=Tf(x)⁻_0,s¹⊗Tf(x)_0,t.

Theorem 12. Let x be aG(R^d)-valued path of finitep-variation controlled byε^pω(s, t ), andf be aR^d-valued path of finiteq-variation controlled by(s, t ). Assume moreover that(s, t )^1/qCω(s, t )^1/p. Then for alls < t andε <1,

d

T_f(x)_s,t, S(f )_s,t

C√

εω(s, t )^1/p.

Proof. For alls < t, we have S(f )⁻_s,t¹⊗T_f(x)_s,t=exp

x_s,t−1

2[f_s,t, x_s,t] +π₂(x_s,t)

⊗exp

1 2 t

s

f_s,u⊗dx_u+1 2 t

s

x_s,u⊗df_u

⊗exp

−1 2 t

s

dfs,u⊗xu−1 2

t s

dxs,u⊗fu

.

Hence, by inequality (6) and Young inequality [42] (which says that for alls < t,|_t

s f_s,u⊗dx_u|Cε(s, t )^1/q× ω(s, t )^1/p and similar inequalities for the other Young integrals), we get that

S(f )⁻_s,t¹⊗Tf(x)_s,tεω^1/p(s, t )+

ε²ω^2/p(s, t )+Cε(s, t )^1/qω(s, t )^1/pC√

εω(s, t )^1/p. 2

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3. The enhanced Brownian motion

In what follows we will lift Brownian motion as(R^d,+)-valued to a(G(R^d),⊗)-valued process. Early work by Gaveau [14] and in particular the presentation in [26] use related algebraic ideas. See also [30].

3.1. Two classical properties of the Brownian motion

Let(C₀(R^d),F, (Ft)_t,P)be the Wiener space. The evaluation operatorB is then underPa Brownian motion starting at 0.

Applying Garsia, Rodemich and Rumsey inequality, it is not too difficult to see [36,38] that for alls < t, Bs,t 4

√α

t−s

0

log(1+^4F_u2)

u du,

whereF is aL¹-random variable and a constantα >0, sufficiently small.

We denote byW^1,2the Cameron–Martin space

h:[0,1] →R^d, h(t )= t 0

h(t ) dtwithh∈L²([0,1])

.

Cameron–Martin theorem, e.g. [19,32], states that iff ∈W^1,2is(Ft)_t-adapted, then the law of(B_t)₀_t₁(i.e. the Wiener probabilityP) and the law of(B_t+f (t ))₀_t₁(that we will denoteP^f) are equivalent.

We will now extendB to a G(R^d)-valued path of finite p-variation, for all p >2 (and hence a geometric p-rough path), and show that our “enhanced” Brownian motion has a similar modulus of continuity and extend Cameron–Martin theorem to our enhanced Brownian motion.

3.2. Their extensions to the enhanced Brownian motion 3.2.1. Definition of the enhanced Brownian motion

The enhanced Brownian motion was first defined in [33]. See also [24].

Forn∈NandxaR^d-valued path, we denote byxⁿthe path which agrees withxat the points₂^kn, k=0, . . . ,2ⁿ, and which is linear in the intervals[₂^kn,^k₂⁺n¹], k=0, . . . ,2ⁿ−1. Asxⁿhas finite 1-variation, we can define xⁿ= S(xⁿ),that is the naturalG(R^d)-valued path lying abovexⁿ.

We denote byΓ the almost surely defined map B→

t→ lim

n→∞S(Bⁿ)_t ,

and we let(B_t)₀_t₁be theG(R^d)-valued pathΓ (B). Note that [24]

Bt=a.s.exp

Bt+1 2 t

s

(Bu⊗ ◦dBu− ◦dBu⊗Bu)

,

where we have used Stratonovich integration. We will call(B_t)_t₀the enhanced Brownian motion.

Remark 13. Almost surely,Γ ◦π₁(B)=B andπ₁◦Γ (B)=B.

Remark 14. The inverse of exp, denoted by log, provides a global chart forG(R^d). Gaveau’s diffusion, [26], is exactly our EBM seen through this chart.

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3.2.2. Modulus of continuity for the enhanced Brownian motion

We now cite Garsia, Rodemich and Rumsey inequality, but with the functionf below beingG(R^d)-valued (while usually,f takes values in a normed vector space). Using the properties · onG(R^d),the proofs of the following theorem is the same proof than the classical Garsia, Rodemich and Rumsey theorem.

Theorem 15. LetΨ and p be continuous strictly increasing functions on [0,∞) with p(0)=Ψ (0)=0 and Ψ (x)→ ∞asx→ ∞. Givenf ∈C₀([0,1], G(R^d)), if

1 0

Ψ

f (s)⁻¹⊗f (t ) p(|t−s|)

ds dtF, (12)

then for 0s < t1, f (s)⁻¹⊗f (t )8

t−s

0

Ψ⁻¹ 4F

u²

dp(u).

Applying the Garsia, Rodemich and Rumsey inequality, we obtain a modulus of continuity for the enhanced Brownian motion

Theorem 16. Define

ζ (x)= 1 2√

2 x 0

log(1+1/u²)

u du.

Then there exists a random variableZ1a.s. and inL¹, and a constantC >0 such that for alls < t, B_s,tCZ^1/4ζ

t−s

√Z

. (13)

Proof. As in the proof of Lévy’s modulus of continuity in [36,38] we use the Garsia, Rodemich and Rumsey inequality withf (t )=B_t,p(x)=√

xandΨ (x)=exp(αx²)−1. We obtain Bs,tC

t−s

0

log(1+^4F_u2)

u du,

whereF is the (now random) left-hand side of (12). SinceB_s,t^law=√

t−sB_0,1the expectation ofF is estimated by

E exp

αB0,1²

. (14)

We claim that this last expression is finite for a small enoughα >0. Remark thatπ1(B), resp. π2(B)are some elements of the first (resp. second) Wiener–Itô Chaos. By general integrability properties of the Wiener–Itô chaos [32, p. 207], there existsα >˜ 0

E exp

˜

απ1(B0,1)²

R^d

<∞, E

exp

˜

απ₂(B0,1)

R^d⊗R^d

<∞.

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By inequality (6), the finiteness of the expectation of exp(αB_0,1²)is easily obtain.⁵ It remains just to define Z=max{4F,1}(we wantZ1 for technical convenience later on), and do a change of variable to obtain inequality (13). 2

Remark 17. Asζ (x)∼x→0C√

−xlnx,

δlim→0 sup

0s<t1

|t−s|δ

B_s,t

√−δlnδ C. (15)

On the other hand we can trivially get a deterministic lower bound by noting|B_s,t|_R^d B_s,tand using Lévy’s result. All this is known (with an equality) for hypoelliptic diffusions on Nilpotent group in [2] and elliptic diffusions in [3].

Lemma 18. There exists a constantCsuch that for allx, y∈ [0,1], ζ (xy)Cζ (x)ζ (y).

Proof. Forasmall enough, there exists constantsK₁andK₂such that ifx∈(0, a] K₁√

−xlnxζ (x)K₂√

−xlnx.

Hence, from the inequality∀x, y∈(0, a],−ln(xy)⁻_ln^{2 ln}2a^aln(x)ln(y), we obtain that for allx, y∈ [0, a],ζ (xy) Cζ (x)ζ (y)for a constantC.

For a fixedb, it is easily seen that 0<infx∈(0,1/b)ζ (xb)

ζ (x) <sup_x_∈_(0,1/b)^{ζ (xb)}_{ζ (x)} <∞. Hence, ifx, y∈ [0,1], ζ (xy)=ζ

axay a²

sup

z∈(0,a²)

ζ (z/a²)

ζ (z) ζ (axay)

C sup

z∈(0,a²)

ζ (z/a²)

ζ (z) ζ (ay)ζ (ay)

C

sup_z_∈_(0,a2) ζ (z/a²)

ζ (z)

(inf_z_∈_(0,1/a)^{ζ (za)}_{ζ (z)})²ζ (x)ζ (y). 2 Proposition 19.

B_s,tMζ (t−s), (16)

whereMis a random variable for which there exists a constantλ >0 such thatE(exp(λM²)) <∞. Proof. From the previous lemma, we can set

M=CZ^1/4ζ 1

√Z

=2C 1 0

ln

1+ Z

v⁴

dv.

Hence, by Jensen inequality,

5 One could prove this directly as we know the density of B0,1[22].

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E

exp(λM²)

E

1 0

exp

4λC²ln

1+ Z v⁴

dv

E

1 0

1+ Z

v⁴ 4λC²

dv

E

1 0

2Z v⁴

4λC²

dv

<∞

forλsmall enough. 2

The last estimate looks like a control butζ^pis not convex on the entire interval[0,1]. We defineφ_pto bepth- root of the smallest convex function dominatingx→ζ (x)^p(remark thatφ_pis very similar toφ_1,pof Example 8, as √^{ζ (x)}

−xlnx→x→01).

Corollary 20. Thep-variation of B is controlled by(s, t )→C^pZ^p/4φ^p_p(^t√⁻^s

Z)and also by(s, t )→M^pφ_p^p(t−s).

3.2.3. A Cameron–Martin theorem on the group

This section, despite being short and quite trivial, will be crucial in the proof of the support theorem. For f ∈W^1,2, the translation of B byf isT_f(B), and it is well defined asf has finite 1-variation and B has almost surely finitep-variation, 2< p <3.

Theorem 21. Letf∈W^1,2be an(Ft)-adapted path. Then the law of B is equivalent to the law ofT_f(B).

Proof. By the Definition 11 and properties of Young and Stratonovich integral,T_f(B)=Γ (f +B). Hence, the law of B isP◦Γ⁻¹while the law ofT_f(B)isP^f ◦Γ⁻¹. Hence, by the Cameron–Martin theorem, these two laws are equivalent. 2

4. Modulus of continuity for solution of SDEs

Theorem 22. Lety_t be the solution of the Stratonovich differential equation dy_t=f₀(y_t) dt+f (y_t)◦dB_t,

wheref₀ is(1+ε)-Lipschitz andf is(2+ε)-Lipschitz. For h∈W^1,2, we denote byF (h) the solution of the ordinary differential equation⁶

dF (h)_t=f₀ F (h)_t

dt+f F (h)_t

dh_t.

We also denote by F the extension of the Itô mapF to the set ofG(R^d)-valued path of finitep-variation. There exists a constantCsuch that

lim

h→0 sup

0s<t1

|t−s|δ

F(B)_s,t

√−δlnδ C a.s.

In particular, lim

h→0 sup

0s<t1

|t−s|δ

|y√t−ys|_R^d

−δlnδ C a.s.

6 Forh∈W^1,2but not piecewiseC¹this ODE still makes sense as rough differential equation with driving signal of finitep=1 variation, see [21].