A Brief Introduction to Rough Paths

A brief introduction to rough paths

Giovanni Zanco

September 2, 2016

These notes are based on a short course I taught at the University of Pisa in April, 2016, and originated from a course taught by Jan Maas at IST Austria in autumn 2015, for which I have been the teaching assistant. They provide a brief introduction to the theory of rough paths of Hölder regularity between 1/3 and 1/2, with some hints at the theory for rough paths of arbitrary regularity. The exposition and the results presented are heavily based on the book [Friz and Hairer, 2014], which I suggest as a reference for a ﬁrst study of the subject. Some of the material presented here is also adapted from [Baudoin, 2013], [Friz and Victoir, 2010] and from personal handwritten notes by Jan Maas. Further references are given throughout the text. The proofs given here follow very closely the cited references. These notes have been developed as a handout, with the goal of organising and extending the material presented during the mentioned courses, thus giving a very concise overview of the basic results and of the lines along which the theory of rough paths developed. They are not supposed to be exhaustive, and many important topics are omitted, together with many details. For all sections but the last one basic knowledge in real analysis and stochastics is required, but nothing more. The last section contains more advanced topics, but it is kept at a very informal level, with few rigorous proofs. I warmly thank Jan Maas, for his course provided me with a general and comprehensive outline upon which these short notes developed. The current shape of this material has also beneﬁted from the comments given and the question raised during the course in Pisa; I thank all the participants for their interest and help, and in particular I acknowledge Franco Flandoli for his useful suggestions about the exposition of some topics discussed herein.

Contents

1 Introduction 2

2 Elements of Young integration 3

3 Rough Paths 6

1 4 Some comments on the general theory 11

5 Rough integration 13

6 Rough differential equations 23

7 Stochastic processes as rough paths 25

8 Stochastic differential equations 30

9 Applications to a stochastic partial differential equation 31

1 Introduction

The central objects of these lectures will be differential equations of the form

dYt = f (Yt) dXt (1) where X : [0,T ] → E is a driving signal, Y : [0,T ] → V is the output (unknown) and f : V → L(E; V ) is a smooth function, E and V are Banach spaces and L(E; V ) denotes the space of continuous linear maps from E to V . A common choice is E = Rd and V = Rn, so that f : Rn → Rn×d. We expect X and Y to be continuous functions. The usual way to interpret equation (1) together with some initial datum y0 is in its integral form Z t Yt = y0 + f (Ys) dXs (2) 0 and a standard scheme to solve (2) is (i) to give meaning to the integral;

(ii) to apply some fixed point result. To deal with item (i) we need an integration theory that is satisfactory in the sense that it allows to work with signals and unknowns of suitable regularity, depending on the problem we are interested in. To deal with item (ii) we need to space of solutions to (2) to have some nice metric structure. If f is smooth and X is differentiable, then the classical theory applies: equation (2) is interpreted as Z t Yt = y0 + f (Ys) X˙ s ds 0 and the solution Y can be found as the fixed point of the map M defined on continuous functions by Z t M(Y )t := y0 + f (Ys) X˙ s ds . 0

We will be interested here in situations in which Y is a α-Hölder function (and hence t 7→ f (Yt) is α-Hölder as well) and X is a β-Hölder function. If α + β > 1 we can interpret the integral

2 appearing in (2) as a Young integral (see section 2) and find the solution as the unique fixed point of the map M above in the space Cα. If α + β ≤ 1 the question is more tricky; rough paths theory provides a convenient answer. Of course there are well known probabilistic results that allow to define the so-called stochastic integrals and to solve stochastic differential equations like (2). However they typically do not provide pathwise solutions (if X is a Brownian motion one cannot fix a Brownian path X(ω) and solve (2) for that particular realisation of X), rather solutions in a probabilistic sense; indeed all stochastic integrals require some probabilistic property of X and Y to be well defined (semimartingale structure, adaptedness, etc.) rather than some regularity property of the typical paths of X and Y . We will see that many classical results about stochastic differential equations are recovered in the theory of rough paths. What does the study of rough paths then add to the classical theories of stochastic analysis? Among many interesting answers to these question, we will focus mainly on the following one. A celebrated result by T. Lyons (see [Lyons, 1991]) states the following:

Theorem 1.1. There exists no separable Banach space B ⊂ C ([0, 1]; R) with the properties:

(i) sample paths of Brownian motion belong to B almost surely;

R · ˙ (ii) The map (g, h) 7→ 0 g(t)h(t) dt extends from smooth functions to a continuous map on B × B taking values in C ([0, 1]; R).

For example the solution map B 7→ Y of the Stratonovich differential equation

dYt = f (Yt) ◦ dBt is measurable but not continuous, in general, with respect to any reasonable topology (and indeed not all smooth approximations to B give convergence to the Stratonovich solution for the above equation). Rough paths provide a framework in which continuity of the solution map can, to a certain extent, be restored.

Here we will consider driving signals with Hölder regularity α ∈ (1/3, 1/2]. The original theory prefers to work with functions of ﬁnite p-variation rather than with Hölder functions, and allows to consider signals of arbitrarily low regularity, but requires heavy algebraic methods that would need too long to be introduced. Restricting to α ∈ (1/3, 1/2] allows to avoid the study of signatures, to avoid many algebraic difﬁculties and is anyway interesting enough to see many features of the theory and tackle some interesting problems. Some hints of the general theory will however be given hereinafter; the interested reader can refer to [Friz and Victoir, 2010] and [Lyons et al., 2007].

2 Elements of Young integration

We recall the following deﬁnitions that will be frequently used in these notes.

3 Definition 2.1 (Hölder continuous functions). Let α > 0. A function X defined on an interval [0,T ] ⊂ R and taking values in a Banach space E is α-Hölder continuous (often α-Hölder for brevity) if the quantity |Xt − Xs|E kXkα := sup α t6=s |t − s| is finite. The space of all α-Hölder continuous functions from [0,T ] into E is denoted by Cα ([0,T ]; E).

When no confusion can arise on the domain of the functions at hand we will simply write Cα(E), or even Cα if the co-domain is clear as well. Hölder continuous functions are continuous, and any α-Hölder function with α > 1 is constant. The quantity k·kα is a semi-norm (it does not separates constant functions); however the quantity

kXkCα := |X0|E + kXkα is a norm that makes Cα(E) a Banach space (in general not separable). This norm is equivalent β α to k · k∞ + k · kα. If α < β then obviously C ⊂ C . We say that a E-valued function X belongs to Ck,α(E) if it is k times differentiable with k,α its k-th derivative being α-Hölder. The space C (E) endowed with the norm kXkCk,α = (k) kXkCk + kX kα is also a Banach space.

By partition of an interval I, in the sequel, we will mean a finite family Π of (essentially) disjoint sub-intervals [s, t] of I such that ∪[s,t]∈Π[s, t] = I. Therefore choosing a partition is equivalent to choosing a finite number of points t0 = 0 < t1 < ··· < tN = T and dividing I into the sub-intervals [ti, ti+1]. The one-point overlap between adjacent intervals will cause no trouble. The mesh of a partition Π is defined as |Π| := max[s,t]∈Π |t − s|. Definition 2.2 (Finite p-variation functions). Let p > 0. A function X defined on an interval [0,T ] ⊂ R and taking values in a Banach space E has finite p-variation if

1   p X p kXkp−var := sup |Xt − Xs|E < ∞ Π [s,t]∈Π where the supremum is taken over all partitions Π of [0,T ]. The space of all continuous functions from [0,T ] into E with ﬁnite p-variation is denoted by Cp−var ([0,T ]; E).

As above we will write Cp−var(E) or even Cp−var when spaces are clear from the context. Any function with ﬁnite p-variation for some p < 1 is constant, and functions of ﬁnite 1-variation are known as bounded variation (BV) functions. The quantity k · kp−var is a semi-norm on Cp−var, but kXkCp−var := |X0|E + kXkp−var p−var is a norm (equivalent to k·k∞ +k·kp−var) that turns C into a Banach space (not separable, in general). If p < q then Cp−var ⊂ Cq−var.

4 Any α-Hölder function is easily seen to have ﬁnite 1/α-variation. Conversely, any continuous function X with ﬁnite p-variation can be written as X = Y ◦ τ where Y is 1/p-Hölder and τ : [0,T ] → [0, 1] is continuous and increasing.

Smooth functions are not dense neither in Cα for any α < 1 nor in Cp−var for any p > 1. The closure of the set of smooth functions∗ under the Cα norm is denoted by C0,α and the closure of the set of smooth functions under the Cp−var norm is denoted by C0,p−var. Of course we have that C0,α ⊂ Cα and C0,p−var ⊂ Cp−var, but we remark that the inclusion is strict. However it can be easily proved that the difference is tiny, in the sense that for every α < β ≤ 1 the inclusion Cβ ⊂ C0,α holds, and for every 1 ≤ p < q the inclusion Cp−var ⊂ C0,q−var holds. Moreover we have that the closure of C1−var in Cp−var is again C0,p−var.

We will now briefly recall some ideas about the construction of Young integral, considering the real-valued case for simplicity. Let X ∈ Cβ ([0, 1]; R) and Y ∈ Cα ([0, 1]; R). A first way to define the integral of Y against X consists in studying a Riemann sum approximation along the dyadic partition of [0, 1]. Set

I0 = Y0 (X1 − X0) , I1 = Y0 X 1 − X0 + Y 1 X1 − X 1 , 2 2 2 . . 2n−1 X In = Y j X j+1 − X j . 2n 2n 2n j=0

We thus have that

2n−1 X |In+1 − In| = Y j+1/2 − Y j X j+1 − X j+1/2 n n 2n 2 2 2n j=0 2n−1 X −(j+1)α −(j+1)β ≤ kY kα 2 kXkβ 2 j=0 n −(n+1)(α+β) = kXkβkY kα2 2 and therefore if α + β > 1 the sequence (In) is Cauchy and we can deﬁne the Young integral as

Z 1 Ys dXs := lim In . 0 n→∞

An alternative deﬁnition using ﬁnite p-variation spaces is based on the following Young estimate.

∗by smooth we usually mean functions in C∞; however here using piecewise C1 functions would yield the same result.

5 Proposition 2.3. Let X,Y ∈ C1−var ([0, 1] : R) and choose p, q ≥ 1 such that 1/p + 1/q > 1. Then Z 1 1 Y dX − Y (X − X ) ≤ kXk kY k , (3) s s 1 1 0 1−1/p−1/q p−var q−var 0 1 − 2 where the integral above is well deﬁned as a Riemann-Stieltjes integral.

Now for X ∈ C0,p−var and Y ∈ C0,q−var with 1/p + 1/q > 1, there exist two sequences (n) (n) 1−var (n) (n) X n, Y n in C such that X → X in p-variation norm and Y → Y in q- variation norm. The above estimate the yields convergence of the sequence R Y (n) dX(n) and we can deﬁne the Young integral of Y against X as

Z 1 Z 1 (n) (n) Ys dXs := lim Ys dXs . 0 n→∞ 0

The Young integral is easily seen to be independent of the sequences X(n) and Y (n) and to satisfy again inequality (3). The extension to general X ∈ Cp−var and Y ∈ Cq−var is obtained through the inclusions stated above. The ﬁrst construction shows explicitly the role of the condition α + β > 1; it can actually be shown that there exist sequences X(n), Y (n) of smooth functions such that both X(n) → 0 and (n) 1 R (n) (n) Y → 0 in C 2 but Y dX → ∞.The second construction instead suggests the general principle that an estimate like (3) comparing the integral with the global increment over the domain of integration might be of help in deﬁning an integral.

3 Rough Paths

To simplify formulas in the sequel we introduce a convenient shorthand.

Notation 3.1. Given a function X on [0,T ] we will denote by Xs,t its increment between s and t, i.e. Xs,t := Xt − Xs .

To avoid confusion, the value of a function F of two variables will be then denoted by F(s,t), (s, t) ∈ [0,T ]2.

To deﬁne a rough path we start from the following observation: suppose that X belongs to α d n n C [0,T ]; R , α ∈ (1/3, 1/2], and that Y takes values in R and solves (1) with f : R → Rn×d smooth. Then we expect that, at least on small scales, the output be “similar” to the noise, i.e.

Ys,t = f (Ys) Xs,t + R(s,t) (4) where R is some remainder that we expect to control (in some sense to be clariﬁed later on), and thus

ˆ f(Y )s,t = f (Yt) − f(Ys) = Df(Ys)Ys,t + Re(s,t) = Df(Ys)f(Ys)Xs,t + R(s,t) (5)

6 ˆ for suitable remainder terms Re and R. Then, neglecting the remainder and setting Zs = d×d n ∼ n×d×d ∼ d n×d Df(Ys)f(Ys) ∈ L R ; R = R = L R ; R , we should have Z t Z t Z t f(Ys) dXs = f(Y0) + f(Ys) − f(Y0) dXs ≈ f(Y0)X0,t + Z0 Xs,t ⊗ dXs ; 0 0 0 (6) R we reduced the problem to calculating X dX. Since α ≤ 1/2 this is of course again an ill- R t posed task; rough paths theory proposes to postulate the value of 0 Xs,t dXs and to consider as a path not only X but a couple (X, X ) where the second component plays the role of the iterated integral of X against itself. This allows then to deﬁne a rough path integral R Y dX for Y in a suitable class of functions.

From the point of view of the solution map introduced above, consider the stochastic differential equation in R2 0 0 1 dY = f (Y ) dB , f(x) = x + t t t 1 0 0 for B a standard Brownian motion in R2, that is the system of equations

( 1 1 dYt = dW 2 1 2 dYt = Wt dWt . The solution map is then Z · 1 1 2 W 7→ W , Ws dWs . 0 This is not continuous, but if we add by deﬁnition to B its iterated integrals as a second component B , continuity is straightforwardly restored.

The new object X that we will introduce has in principle nothing to do with integration; in fact such an integration does not exist and, in a sense, we want to define it using X . Anyway we would like to recover the classical case when X is smooth so that its integral against itself is well defined. Therefore suppose for a moment that X is smooth and define the function of two variables X by Z t X (s,t) := Xs,r ⊗ dXr ; s then it follows that Z t Z u Z t X (s,t) − X (s,u) − X (u,t) = Xs,r ⊗ dXr − Xs,r ⊗ dXr − Xu,r ⊗ dXr s s u Z t = Xu,s dXr u = Xs,u ⊗ Xu,t . This suggests the definition below, for which we introduce another convenient notation.

7 2 α 2 Notation 3.2. Let F : [0,T ] → E. We write F ∈ C2 [0,T ] ; E if |F | (s,t) E kF kα = sup α < ∞ . s6=t |t − s|

Note that we use the same symbol k · kα with different meanings depending on whether it refers to functions of one or two variables. If X ∈ Cα ([0,T ]; E) then it is immediate to check α that (s, t) 7→ Xs,t belongs to C ([0,T ]; E).

Deﬁnition 3.3. Let α ∈ (1/3, 1/2]. The space C α ([0,T ]; E) of (E-valued) α-Hölder rough paths X α 2α 2 is the space of pairs (X, ) ∈ C ([0,T ]; E) ⊕ C2 [0,T ] ; E ⊗ E such that the identity

X (s,t) − X (s,u) − X (u,t) = Xs,u ⊗ Xu,t (7) holds true for every s, u, t ∈ [0,T ].

Identity (7) is known as Chen’s relation. It implies that X (t,t) = 0 for every t ∈ [0,T ]. As before we will write C α(E) or C α if no confusion can arise.

α α 2α The space C is a closed set in C ⊕ C2 but it is not a linear space due to the nonlinear constraint given by Chen’s relation. On C α we deﬁne the rough path norm p X, X α := kXkα + kX kα (8) 9 9 which is not a norm in the common sense (the space is not linear), but is homogeneous with respect to the natural scaling (X, X ) 7→ (λX, λ2X ). We also introduce the rough path metric

|X − Y | |X − Y | X Y s,t s,t E (s,t) (s,t) E ρα (X, ), (Y, ) := |X0 − Y0|E + sup α + sup 2α . s6=t |t − s| s6=t |t − s|

α Theorem 3.4. (C , ρα) is a complete metric space. The proof of this fact is not difficult and can be done as an exercise. It is also easy to show that C β ⊂ C α for any 1/3 < α < β ≤ 1/2. Some remarks on the function X are required. It is true that, if α ∈ (1/3, 1/2). given X ∈ Cα there exists an associated rough path (X, X ) ∈ C α (this result is known as Lyons-Victoir theorem, [Lyons and Victoir, 2007]), but the choice of X is not unique. Indeed if X could be determined uniquely by X we would be adding no information to our function, so that we could not expect to obtain better results than those already available from classical theories. If α = 1/2 the Lyons-Victoir theorem applies only when E is finite-dimensional. Non-uniqueness of X is easily shown: if we change X (s,t) to Xf(s,t) := X (s,t) + gt − gs for any continuous function g then relation (7) is again satisfied; therefore Chen’s relation is surely not enough to define X given X. In particular X is determined up to increments of a 2α-Hölder function g, and in general there is no canonical choice for g. However if (X, X ) and (X, Xf) are both rough paths and we set δ(s,t) := X (s,t) − Xf(s,t) and gt := δ(0,t), we immediately see that δ(s,t) − δ(s,u) − δ(u,t) = 0 and δ(u,t) = δ(0,t) − δ(0,u) = gt − gu. Therefore X is uniquely determined by Chen’s relation and by the one-variable function X (0,t). This means that knowledge

8 of the paths t 7→ Xt and t 7→ X (0,t) completely identiﬁes the rough path (X, X ). In this sense rough paths are really paths and not two-dimensional objects. Therefore we will write X s,t for X (s,t) in the sequel.

Example 3.5. Let (X, X ) ∈ C α(Rd), α ∈ (1/3, 1/2], and h : [0,T ] → Rd. If we translate the path X by the function h, is there a canonical way to obtain a new rough path associated to X + h? Set h Th(X, X ) = (X + h, X ) where X h is deﬁned as Z t Z t Z t X h X s,t = s,t + hs,r ⊗ dXr + Xs,r ⊗ dhr + hs,r ⊗ dhr . s s s

1,2 If h ∈ W the three integrals above are well deﬁned (in classical sense) and the operator Th is continuous from C α into itself.

Example 3.6. A typical example of non-standard rough path behaviour is given in R2 by

2 Z t (n) 1 cos 2πn t X (n) (n) (n) Xt = 2 , s,t = Xs,r ⊗ dXr . n sin 2πn t s

(n) α (n) (n) α For α < 1/2 it easily seen that X → 0 in C but X , X → (0, X ) in C (i.e. with respect to ρα), where X is given by

0 1 X = π(t − s) . s,t −1 0

The iterated integral “remembers” the oscillations of the paths, that therefore appear in the limit object.

It is worth noting that X in the example above is anti-symmetric, and in general we do not expect X s,t ∈ E ⊗ E to be symmetric. Nevertheless the study of the symmetric part of X is important in the context of rough paths. Suppose for a while that X is a smooth function taking Rd X R t X values in and set s,t = s Xs,r ⊗ dXr. Then the symmetric part of is given by

Z t Z t 1 X i,j X j,i 1 i j j i 1 i j 1 i j s,t + s,t = Xs,r dXr + Xs,r dXr = d Xs,rXs,r = Xs,tXs,t 2 2 s 2 s 2 hence 1 Sym(X ) = X ⊗ X . (9) s,t 2 s,t s,t This equation is false for a general rough path (X, X ) but motivates the following deﬁnition. α X α Deﬁnition 3.7. The set of geometric rough paths Cg consists of those (X, ) ∈ C that satisfy equation (9).

9 Relation (9) holds for smooth paths considered as rough paths with their canonical lift X R t s,t = s Xs,r ⊗ dXr. The closure of the set of these canonical lifts of smooth paths in α 0,α α C is a set denoted by Cg that can be shown to be strictly smaller than Cg . Similarly to what β 0,α happens for Hölder spaces, it can be easily shown that Cg ⊂ Cg for 1/3 < α < β ≤ 1/2.

We end this section with a useful approximation criterion.

(n) (n) β Proposition 3.8. Let β ∈ (1/3, 1/2] and consider a sequence X , X in C such that (n) X (n) (n) X (n) X supn kX kβ < ∞ and supn k k2β < ∞. If X0,t → X0,t and 0,t → 0,t pointwise, β (n) (n) then (X, X ) belongs to C and X , X → (X, X ) with respect to ρα for every α ∈ (1/3, β).

(n) X (n) Proof. Using Chen’s relation it is easy to show that pointwise convergence of Xs,t , s,t holds for every s, t. Using the uniform Hölder bounds, this allows to show uniform convergence as follows. Let C be a constant such that

(n) β (n) 2β |Xs,t| = lim |X | ≤ C|t − s| , |X s,t| = lim |X | ≤ C|t − s| , n→∞ s,t n→∞ S,t which exists by assumption, and choose ε > 0 and t1, . . . , tk ∈ [0,T ] such that maxi |ti − β ti−1| < ε/C. Then

(n) (n) (n) (n) ε (n) Xs,t − X ≤ X ¯ − X + |Xs,s¯| + X ¯ + X + X ≤ 4 + X ¯ − X s,t s,¯ t s,¯ t¯ t,t s,s¯ t,t¯ C s,¯ t s,¯ t¯

s¯ t¯ t s t X − X(n) where and are the nearest i’s to and respectively. Since s,¯ t¯ s,¯ t¯ can be made small for n large enough uniformly over all couples ti, tj, we see that the convergence is actually X (n) uniform. The convergence of s,t is handled similarly (but using Chen’s relation to write the increments). We can now use the uniform convergence: ∃ εn → 0 such that for every n

(n) X X (n) Xs,t − Xs,t ∨ s,t − s,t < εn .

(n) β X X (n) 2β 1−θ θ Since Xs,t − Xs,t ≤ 2C|t − s| and s,t − s,t ≤ 2C|t − s| , using a ∧ b ≤ a b (valid for a, b > 0 and 0 < θ < 1) we get

(n) 1−α/β α X X (n) 1−α/β 2α Xs,t − Xs,t . εn |t − s| , s,t − s,t . εn |t − s| for every 1/3 < α < β, where the expression “A . B” has to be interpreted as “A ≤ B neglecting constants”. This implies convergence with respect to the metric ρα by deﬁnition. ^q

10 4 Some comments on the general theory

Consider a differential equation in Rn

d Z t X Z t yt = y0 + f(yr) dxr = y0 + fj(yr) dxj(r) 0 j+1 0

Rd Rn d Rn Rn×d where x takes values in , y in and f = (fj)j=1 : → . To each fj one associates in a canonical way the differential operator given on smooth functions by

n X ∂ fjG(z) = hfj(z), ∇G(z)i , i.e. fj = fj,l(·) . ∂zl l=1 Then for any G : Rd → R, by a change of variables, we get

d X Z t G(yt) = G(y0) + fjG(yr) dxj(r) . j=1 0 Changing again variables in the integral and iterating yields

∞ X X Z G(yt) = G(y0) + (fj1 ··· fjk G)(y0) dxj1 (t1) ··· xjk (tk) (10) k=1 |j|=k 0≤t1<···

bpc 1 Z Z j X ⊗j ⊗j d(x, y) = dx − dy . p j=1 j −var A fundamental result by Lyons ([Lyons, 1998]) then shows that if x is a p-rough path its signature S (x) is completely determined by its ﬁrst bpc terms, i.e.

bpc X Z S (x) = 1 + dx⊗k . (11) k k=1 ∆

11 The idea of representing a path as a formal power series and to translate operations on paths into operations on series dates back to Chen ([Chen, 1958]). In this framework, Chen’s relation is a consequence of properties of integrals and formal series. To choose α ∈ (1/3, 1/2] corresponds to consider only the ﬁrst iterated integral in (11) (recall the correspondence between α-Hölder functions and continuous functions of ﬁnite 1/α-variation stated in section 2), and the two ap- proaches become equivalent. In order to consider signals of lower Hölder regularity one would need to consider more terms in the signature, and some algebraic structure is needed to develop the theory. We will give now a brief description of such algebraic interpretation in the main case we are considering, that is α ∈ (1/3, 1/2]; this should give a hint of the general theory.

Looking at terms of different degree in (11) as different components, it is natural to see an α d 2 α-rough path (X, X ) ∈ C [0,T ]; R , α ∈ (1/3, 1/2], as a function on [0,T ] taking values in T (2) = T (2) Rd := R ⊕ Rd ⊕ Rd ⊗ Rd

(2) that associates to every couple (s, t) the element (1,Xs,t, X s,t) ∈ T . (2) Deﬁne a product on T as ˆ ˆ ˆ (a, b, c) (ˆa, b, cˆ) := (aa,ˆ ab +ab, ˆ acˆ +ac ˆ + b ⊗ b) .

Then (1, 0, 0) is the unit element and the product acts as a concatenation of paths, in the sense that X X X (1,Xs,t, s,t) = (1,Xs,u, s,u) (1,Xu,t, u,t) for any s ≤ u ≤ t ∈ [0,T ]. T (2) Rd Rd Rd The set 1 := 1 ⊕ ⊕ ⊗ is a group with respect to the operation (every element has an inverse) and actually it is a Lie group, hence a rough path (X, X ) can be thought of as representing the increments of the group-valued path

t 7→ (1,X0,t, X 0,t) =: X0,t ,

−1 and Xs,t is then given by X0,s X0,t. Using the identiﬁcations 1 ↔ (1, 0, 0), b ↔ (0, b, 0), c ↔ (0, 0, c) we deﬁne the exponential map 1 exp(b + c) := 1 + b + c + b b . 2

T (2) Rd Rd Rd ∼ Rd Rd×d T (2) ∼ Rd Rd×d T (2) Then 0 := 0 ⊕ ⊕ ⊗ = ⊕ and 1 = exp ⊕ . 0 is a ˆ ˆ ˆ (2) Lie algebra with respect to the bracket [b + c, b +c ˆ] = b b − b b; let g be its sub-algebra generated by elements of the form (0, b, 0). Since g(2) is closed under [·, ·], exp g(2) is a Lie T (2) X α α sub-group of 1 . A rough path (X, ) ∈ C is geometric (i.e. it belongs to Cg ) if and only (2) if the map t 7→ X0,t takes values in exp g . exp g(2) is a nice space and also has a metric structure; its algebraic properties allow to deﬁne geometric rough paths for signals of arbitrarily low regularity considering more iterated integrals in the signature S (X), while its metric structure allows to obtain analytic properties of

12 the paths. For example it can be shown that geometric β-rough paths are uniform limits of lifts of smooth paths and the convergence holds with respect to ρα for any α < β. Non-geometric rough paths can be studied as well, for example embedding them in spaces of geometric rough paths (as in [Hairer and Kelly, 2015]) or adding some algebraic structure (as in [Gubinelli, 2010]).

5 Rough integration

We will ﬁrst show how to deﬁne the integral Z f(Y ) dX under some assumptions that allow to exploit the intuitive ideas outlined at the beginning of section 3. We will then rephrase the construction of the integral, stating and proving the main results in a more abstract fashion, to obtain the most general case.

Similarly to the case described by equations (4), (5) and (6), we assume now that E and V are separable Banach spaces; we will need here to endow the tensor product E ⊗ E with a norm such that we have a continuous embedding of L(E; L(E; V )) into L(E ⊗ E; V ), and we will do so henceforth†. X α 1 1 2 Assume that (X, ) is in C ([0,T ]; E) for some α ∈ ( /3, /2] and that f ∈ Cb (V ; L(E; V )). We now assume that the function Y : [0,T ] → V is such that

Y Ys,t = f(Ys)Xs,t + R(s,t) ∀ s, t ∈ [0,T ] (12)

Y 2α 2 where R ∈ C2 [0,T ] ; V . Then, by Taylor’s formula ˆ f(Y )s,t = Df(Ys)f(Ys)Xs,t + R(s,t)

ˆ 2α for some R ∈ C2 ([0,T ]; L(E; V )). Set

Zs = Df(Ys)f(Ys) ∈ L(E; L(E,V )) ⊂ L(E ⊗ E; V ) ; now equation (6) suggests that, at least for very small t, we should expect the identity

Z t f(Ys) dXs = f(Y0)X0,t + Z0X 0,t (13) 0 to hold up to terms of order t3α or higher, whatever the meaning of the left-hand side is. This hints at considering Riemann-type sums of terms like the one on the right-hand side of (13) along partitions of [0,T ], hoping that these Riemann sums will converge - as the mesh of the partition goes to 0 - to something that we will then call integral of f(Y ) against X.

†The projective tensor norm is the most common example of a norm that satisﬁes this requirement.

13 To carry this intuition out, consider a partition Π of [0,T ] consisting of m ≥ 1 intervals (and therefore the set Π? of the partition point of Π consist of m − 1 points). If m ≥ 2 there exists a ? − + − + point t0 ∈ Π such that [t , t0] ∈ Π and [t0, t ] ∈ Π for some t and t and

+ − 2 t − t ≤ T. m − 1

− + ? ? Indeed if this were not the case, denoting by tj and tj the nearest Π -neighbours of tj ∈ Π , P + − 2 P + − we would have ? t − t > (m − 1) T , while surely ? t − t ≤ 2T . tj ∈Π j j m−1 tj ∈Π j j Set Z X f(Ys) dXs := [f(Ys)Xs,t + ZsX s,t] Π [s,t]∈Π

? where Z is deﬁned as above. If we remove the point t0 from Π , we obtain a new partition of [0,T ] that we denote by Π \{t0}. The Lipschitz property of f and Df and the assumption on Y , together with some simple cancellations then yield

Z Z

f (Ys) dXs − f(Ys) dXs Π Π\{t0}

− − − X − + X + = f (Yt ) Xt ,t0 + Zt t ,t0 + f (Yt0 ) Xt0,t + Zt0 t0,t

− f (Yt− ) Xt−,t+ − Zt− X t−,t+ − + − X − X − + X + = f(Y )t ,t0 Xt0,t + Zt t ,t0 − t ,t + Zt0 t0,t

= Z − X − + Rˆ − X + − Z − X + + X − ⊗ X + + Z X + t t ,t0 (t ,t0) t0,t t t0,t t ,t0 t0,t t0 t0,t

= Z − X + + Rˆ − X + t ,t0 t0,t (t ,t0) t0,t

= (Df (Y ) f (Y ) − Df (Y − ) f (Y − )) X + + Rˆ − X + t0 t0 t t t0,t (t ,t0) t0,t

h X + 2α 2 − α Y − 2α ≤ k k2α t − t0 kfk∞ D f ∞ kfk∞ t0 − t + R 2α t0 − t

2 − α − 2αi + kDfk∞ kfk − ∞ kXkα t0 − t + kRk2α t0 − t ˆ − 2α + α + kRk2α t0 − t kXkα t − t0

2 3α ≤ C T m − 1 for a certain constant C depending on f, X, and X .

14 Iterating this procedure up to removing all partition point we get the estimate Z m ! X 1 3α 3α f(Ys) dXs − f(Y0)X0,T − Z0X 0,T ≤ C 2 T 3α Π l=2 (l − 1)

≤ Cζ(3α)T 3α where C is some constant and ζ denotes Riemann’s zeta function‡. To show convergence of the Riemann sums as the mesh of the partition tends to 0 and to see that the limit does not depend on the chosen partition Π it is enough to show that Z Z

sup f(Ys) dXs − f(Ys) dXs → 0 as ε → 0 , (14) |Π1|∨|Π2|<ε Π1 Π2 where the supremum is taken over all partitions of [0,T ]. To this end fix two partitions Π1 and ? ? ? Π2 and notice that if Θ is their common refinement (defined by Θ = Π1 ∪ Π2) we have Z Z

f(Ys) dXs − f(Ys) dXs Π1 Π2 Z Z Z Z

≤ f(Ys) dXs − f(Ys) dXs + f(Ys) dXs − f(Ys) dXs Π1 Θ Θ Π2 by the triangle inequality. Therefore we can suppose that Π2 is a reﬁnement of Π1 without loss of generality. This implies that

Z Z Z ! X f(Ys) dXs − f(Ys) dXs = f(Ys)Xs,t + ZsX s,t − f(Ys) dXs Π1 Π2 Π2∩[s,t] [s,t]∈Π1 X ≤ C |t − s|3α [s,t]∈Π

3α−1 ≤ CT |Π1| which proves (14) since α > 1/3.

We have thus shown that we can define the rough path integral of f(Y ) against X over [0,T ] as Z T X f(Ys) dXs := lim [f(Ys)Xs,t + Df(Ys)f(Ys)X s,t] |Π|→0 0 [s,t]∈Π provided f is smooth and Y satisfies the assumptions above, that essentially say that oscillations of Y are of the same scale of oscillations of X. This justifies the following definition, on which the general construction of the rough path integral is based.

‡actually the fact that exactly Riemann’s zeta function pops out of calculation is of no importance here; we only need a bound on the ﬁnite sum that is uniform in m.

15 Deﬁnition 5.1. Let X ∈ Cα ([0,T ]; E), α ∈ (1/3, 1/2]. We say that Y ∈ Cα ([0,T ]; V ) is α Y 2α 2 controlled by X if there exist functions Z ∈ C ([0,T ]; L(E; V )) and R ∈ C2 [0,T ] ; V such that for every s, t Y Ys,t = ZsXs,t + R(s,t) . The function Z is called the Gubinelli derivative of Y and is denoted by Y 0.

Notice that it is not required that X be a rough path for this deﬁnition to be well-posed.

α The space of V -valued paths controlled by X is denoted by CX ([0,T ]; V ) (or, as usual, by α α either CX (V ) or CX depending on the occasion) and it is endowed with the norm

0 Y kY k α := kY k α + Y α + R . CX C C 2α The semi-norm 0 0 Y Y,Y α,X := kY kα + Y α + R 2α will be useful in the sequel. A very important fact, not difﬁcult to prove, is the following.

α Theorem 5.2. C , k·k α is a Banach space. X CX

The norm k · k α measures, in a sense, the smoothness of Y with respect to X, as the CX following examples clarify.

α Example 5.3. (i) X ∈ C is controlled by itself, because Xs,t = Id ·Xs,t + 0. In particular kXk α = kXk α + 1. CX C

2α α (ii) Any Y ∈ C belongs to C , since Y = 0 · X + Y . In particular kY k α = X s,t s,t s,t CX kY kCα + kY k2α.

2 (iii) Let g ∈ Cb (E; V ). Then for a certain ξ ∈ V 1 g(X ) − g(X ) = Dg(X )X + D2g(ξ)X ⊗ X . t s s s,t 2 s,t s,t

α 0 Therefore g(x) ∈ CX with g(X) = Dg(X) and we have the bound

1 2 2 kg(X)kCα ≤ |g(X0)|V + |Dg(X0)|V + kgkC2 kXkα + D g kXkα . X 2 ∞ The essential difference between example (ii) and the others is that in example (ii) there is no bound on kY k α in terms of kXk . CX α We have seen above that the rough integral was deﬁned as a limit of Riemann sums along partitions. There the increment was given by

f(Ys)Xs,t + Df(Ys)f(Ys)X s,t.

16 Compare this to the Young integral case where

Z T X Ys dXs = lim YsXs,t |Π|→0 0 [s,t]∈Π and we have the estimate

Z t α+β Yr dXr − YsXs,t ≤ CkykCα kXkCβ |t − s| ; s in this last equation, the right hand side is o(|t − s|) if α + β > 1, and this means that YsXs,t is (locally) a good approximation for the integral. The abstract approach to rough integration that we present here is based on a lemma that shows when an integral obtained as a limit of sums of increments is reasonably well approximated by the increment over the whole interval. We will then just need to check that, in the rough path framework, we can ﬁnd an increment that satisﬁes the required properties.

Set 2 2 ∆[0,T ] = (s, t) ∈ [0,T ] : s < t

2 and consider F : ∆[0,T ] → E. If F is deﬁned by the increments of some function f, i.e. P F(s,t) = fs,t = ft − fs, then for any partition Π of [0,T ] we have F(s,t) = fT − f0. Deﬁne now the quantities

δF (s,u,t) E δF(s,u,t) := F(s,t) − F(s,u) − F(u,t) , kδF kβ := sup β ; s,u,t∈[0,T ] |t − s| s

α,β n 2 o C2 := F : ∆[0,T ] → E : F(t,t) = 0 ∀ t ∈ [0,T ] , kF kα < ∞ , kδF kβ < ∞ .

Lemma 5.4 (Sewing lemma). Let 0 < α ≤ 1 < β. There exists a unique continuous map

α,β α I : C2 ([0,T ]; V ) → C ([0,T ]; V ) such that (IF )0 = 0 and the estimate

β (IF )s,t − Fs,t ≤ C|t − s| (15) holds for every s, t ∈ [0,T ], for some constant C depending only on β.

The proof is essentially an abstract form of the construction given above for the integral R f(Y ) dX. We give it again below for completeness.

17 Proof. Without loss of generality we can integrate on [0,T ]. For a partition Π of [0,T ] set Z X F := F(s,t) . Π [s,t]∈Π R We want to show that lim|Π|→0 Π F exists; this will deﬁne IF . Step 1. We show that given any partition Π we have Z

F(0,T ) − F ≤ C kδF kβ T β . Π ? − + As before if m ≥ 2 there exists a point t0 ∈ Π such that [t , t0] ∈ Π and [t0, t ] ∈ Π for some t− and t+ and + − 2 t − t ≤ T. m − 1 Removing the point t0 from Π we obtain, with the same notations as above,

Z Z X X F − = F(s,t) − F(s,t) Π Π\{t0} [s,t]∈Π [s,t]∈Π\{t0}

− + − + = F(t ,t0) + F(t0,t ) − F(t ,t )

− + = δF(t ,t0,t ) 2 β ≤ kδF k T . β m − 1 Iterating until we get to the trivial partition we obtain Z m X 1 β β F(0,T ) − F ≤ kδF kβ (2T ) ≤ CkδF kβT . (16) (l − 1)β Π l=2 Step 2. This step is really identical to the second part of the construction of the rough integral given before. Take any two partitions Π1 and Π2; we can suppose without loss of generality that Π2 reﬁnes Π1. Then, since every [s, t] ∈ Π1 either belongs to Π2 or is divided in sub-intervals that belong to Π2, we have

Z Z Z ! X F − F = F(s,t) − F Π2 Π2 Π2∩[s,t] [s,t]∈Π1 X ≤ CkδF kβ |t − s|β [s,t]∈Π β−1 ≤ CkδF kβT |Π1| which proves that Z Z

sup f(Ys) dXs − f(Ys) dXs → 0 as ε → 0 . |Π1|∨|Π2|<ε Π1 Π2

18 Therefore the limit Z lim F =: (IF )T |Π|→0 Π exists and deﬁnes a linear function I that, thanks to the estimate above is bounded, hence continuous.

The property (IF )0 = 0 is trivial. Moreover δ (IF )(s,u,t) = 0 for any s, u, t ∈ [0,T ]. Estimate (15) follows immediately from (16). For uniqueness, if I and Ie are two continuous linear maps satisfying (15), their difference I −Ie is β-Hölder with β > 1 and therefore constant. Since I0 = Ie0 = 0, we have (I − Ie)0 = 0 and the two maps are equal. The passage from [0,T ] to any interval [s, t] ⊂ [0,T ] is straightforward. ^q

The general rough path integral of a path Y controlled by X against X is now realized 0X choosing F(s,t) = YsXs,t + Ys s,t, as showed by the following result.

Theorem 5.5 (Gubinelli). For α ∈ (1/3, 1/2], take a rough path (X, X ) ∈ C α([0,T ]; E) and a 0 α controlled path (Y,Y ) ∈ CX ([0,T ]; L(E,V )). Then (i) For any s, t ∈ [0,T ], s < t, the object

Z t X 0X Yr dXr := lim YuXu,v + Yu u,v |Π|→0 s [u,v]∈Π

exists (the limit here is taken with respect to partitions of [s, t]) and for a constant C = C(α, T ) we have the estimate

Z t 0X 3α Y X 0 Yr dXr − YsXs,t − Ys s,t ≤ C|t − s| kXkαkR k2α + k k2αkY kα . s

R t α 0 (ii) The path t 7→ Zt := 0 Yr dXr belongs to CX with Gubinelli derivative Z = Y , the map 0 R · α (Y,Y ) → ( 0 Y dX,Y ) is continuous from CX into itself and 0 0 X Y kZ,Z kα,X . kY kα + kY k∞k k2α + kXkα kY k∞ + kR k2α .

0X Proof. We want to apply the sewing lemma with F(s,t) = YsXs,t + Ys s,t, therefore we need to check that kδF kβ < ∞ for some β > 1 (F(0,0) = 0 and kF kα < ∞ are obviously satisﬁed). By Chen’s relation 0X 0X 0X δF(s,u,t) = YsXs,t − YsXs,u − YuXu,t + Ys s,t − Ys s,u − Yu u,t 0 X X 0X = −Ys,uXu,t + Ys ( s,t − s,u) − Yu u,t 0 Y 0 X 0X = − Ys Xs,u + R(s,u) Xu,t + Ys ( u,t + Xs,u ⊗ Xu,t) − Yu u,t 0 X Y = Ys,u u,t − R(s,u)Xu,t .

0 Y kδF k3α ≤ kY kαkX k2α + kR k2αkXkα =: CX,Y and we can apply the sewing lemma exactly if α > 1/3. The ﬁrst estimate then follows immediately from step 1 in the proof of the sewing lemma. For item (ii), the estimate in item (i) implies that

Z t 0 0X 3α |Zt − Zs| = Yr dXr ≤ Ys Xs,t + Ys s,t + CX,Y |t − s| s where CX,Y is the bound for kδF k above. This can be bounded by

α 0 2α 3α kY k∞kXkα|t − s| + kY k∞kX k2α|t − s| + CX,Y |t − s| yielding Y X 0 kZkα . kXkα kY k∞ + kR k2α + k k2αkY k∞, thus Z ∈ Cα. R t 0X By item (i) Zs,t = s Yr dXr = YsXs,t + Ys s,t + Re(s,t) and kRek3α < ∞; moreover 0X 0 X 2α α 0 0X |Ys s,t| ≤ kY k∞k k2α|t − s| thus Z ∈ CX and Z = Y (the term Ys s,t is the 2α- remainder). 0 0 Z The last estimate for kZ,Z kα,X = kZkα + kZ kα + kR k2α follows immediately. Finally, continuity is given by the sewing lemma. ^q

Remark 5.6. The integral just obtained depends on Y 0 and X , but these are usually hidden in 1 α β α β the notation. If /3 < β < α then C ⊂ C and CX ⊂ CX . The value of the integral does not depend on the choice of the space, but the bounds do.

1 X 1 2 X 2 α i α For (X , ), (X , ) ∈ C , take Y ∈ CXi , i = 1, 2, and consider the quantity

1 2 10 20 Y 1 Y 2 dˆα Y ,Y := Y − Y + R − R . (17) α 2α

This depends on X1 and X2 (hidden in the notation) and it is not a metric since Y 1 and Y 2 may R i live in different spaces. Now assume that there exists M ∈ such that Y α < M and C i X i X i i i0 R · i i i α ρα (0, 0), (X , ) < M, i = 1, 2, and set Z , Z = 0 Yr dXr,Y ∈ CXi . Then it can be shown that

h 0 0 i dˆ(Z1,Z2) ≤ C Y 1 − Y 2 + Y 1 − Y 2 + ρ (X1, X 1), (X2X 2) + dˆY 1,Y 2 M 0 0 0 0 α (18) where the constant CM depends on M and also on T and α, and the same bound holds for 1 2 Z − Z α.

20 R α Remark 5.7. If X and Y are smooth and X is deﬁned by Xs,t dXs, then (X, X ) ∈ C , α Y ∈ CX and the rough path integral coincides with the Riemann-Stieltjes integral. However if we change X the integral changes: let g be a 2α-Hölder function and set for s, t ∈ [0,T ] α α α Xet = Xt, Xfs,t = X s,t + gt − gs. Then (X,e Xf) ∈ C , any Y ∈ C also belongs to C and X Xe Z t Z t Z t 0 Yr dXer = Yr dXr + Yr dgr , s s s the ﬁrst two terms being rough integrals and the last one a Young integral. This is essentially a general form of the Itô-Stratonovich correction (see also section 7)

Consider now three Banach spaces E, V , U, a rough path (XX ) ∈ Cα([0,T ]; E) and two 0 α 0 α controlled paths (W, W ) ∈ CX ([0,T ]; V ) and (Y,Y ) ∈ CX ([0,T ]; L(V ; U)). Then, seeing W 0 ∈ L(E; V ) as an operator in L(E ⊗ E; E ⊗ V ), we can deﬁne an integral

Z t Yu ¯dWu (19) s 0 0 X using in the Sewing lemma the function F(u,v) = YuWu,v + YuWu u,v and repeating the proof of Gubinelli’s theorem to show that everything works. α α We can also embed in a canonical way the space CX of controlled paths in the space C of rough paths with the injection given by (Y,Y 0) 7→ (Y, Y ) where Z Y 0 0X s,t = lim Fu,v ,Fu,v = Yu ⊗ Yu,v + Yu ⊗ Yu u,v . |Π|→0 Π

ˆ ˆ0 α Then given (Z, Z ) ∈ CY we have for every s, t ∈ [0,T ]

Z t Z t Zˆr dYr = Zr ¯dYr s s

0 α ˆ 0 ˆ0 0 where (Z,Z ) ∈ CX is deﬁned by Zt = Zt, Zt = ZtYt .

It is an interesting question whether the Gubinelli derivative Y 0 is uniquely determined by Y and X. The idea one should have in mind is that the more regularity X has, the less information it gives about Y 0. Suppose that X is real-valued and that

|Xs,t| lim sup 2α = +∞ . t↓s |t − s|

0 Y Then, since Ys,t = Ys Xs,t + R(s,t), we have

RY 2α 0 Ys,t (s,t) |t − s| Ys,t = − 2α · Xs,t |t − s| Xs,t

21 and the requirement that kRY k < ∞ implies that Y 0 is the limit lim Ys,t and thus it is 2α s,t t↓s Xs,t uniquely determined. If this happens for all s in some dense set, then Y 0 is unique. For general E-valued X the right condition to ask for is that on a dense set

|hϕ, Xs,ti| ∗ lim sup 2α = +∞ ∀ ϕ ∈ E \ 0 . (20) t↓s |t − s| A rough path satisfying condition (20) is sometimes called a truly rough path. If X is truly α rough than any Y ∈ CX has a unique Gubinelli derivative (see [Friz and Shekhar, 2013]).

Consider a rough path (X, X ) of arbitrary Hölder regularity α and let S (X) be its signature, given by (11) with p = 1/α. Denote by X (k) the k-th iterated integral appearing in (11), so that X (1) = X and X (2) = X . At a very informal level, for a smooth function G and a partition Π of [s, t] we would expect that Z t h i X X 2 X (3) G(Xr) dXr ≈ G(Xu)Xu,v + DG(Xu) u,v + D G(Xu) (u,v) + ... s [u,v]∈Π

X (k) ∼ |v −u|kα k and since (u,v) , the terms with large enough should become negligible. Indeed (3) α ∈ (1/3, 1/2] X ∼ |v − u|3α k = 2 with we have (u,v) and we can stop at .

We conclude this section showing that the ﬁrst construction of the rough integral proposed earlier is actually only a special case of the formal construction that makes use of the sewing lemma. This is quite straightforward from the proofs given above; we only need to verify that X α Y α f(Y ) is controlled by X if Y is. To this end let (X, ) ∈ C (E), (Y, ) ∈ CX (V ) and 2 ϕ ∈ Cb (V ; L(V ; E)) be given. Set 0 0 (ϕ(Y ))t = ϕ (Yt) , (ϕ(Y ))t = Dϕ(Yt)Yt ; 0 0 0 2 then kϕ(Y )kα ≤ kDϕk∞kY kα and kϕ(Y ) kα ≤ kDϕk∞kY kα + kY k∞kD ϕk∞kY kα. Furthermore ϕ(Y ) 0 Y R(s,t) = ϕ(Yt) − ϕ(Ys) − Dϕ(Ys)Ys Xs,t = ϕ(Y )s,t − Dϕ(Ys)Ys,t − Dϕ(Ys)Rs,t hence ϕ(Y ) 1 2 2 Y R ≤ kD ϕk∞kY k + kDϕk∞ R . (21) 2 α 2α It easily follows that we have the bound 0 0 0 ϕ(Y ), ϕ(Y ) ≤ Ckϕk 2 Y,Y 1 + (1 + kXk ) kY,Y k α . α,X C α,X α CX 0 α with C depending on α and T . Therefore (ϕ(Y ), ϕ(Y ) ) ∈ CX , thus the integral of ϕ(Y ) against X is well deﬁned in the rough paths sense. Remark 5.8. What we just showed works as well for general functions ϕ taking values in some Banach space U and, as before, only uses the fact that X ∈ Cα. The particular case of L(V ; E)- valued function ϕ is needed to give meaning to R ϕ(Y ) dX. α R α R Moreover it can seen with similar computation that if Y,W ∈ CX ( ) then YW ∈ CX ( ) as well with Gubinelli derivative Y 0W + YW 0.

22 6 Rough differential equations

We now have a way to deﬁne integrals and we have a nice space, with a Banach space structure, that is preserved by the integral map and where we might hope to solve rough differential equations like (1) by a ﬁxed point argument.

X β 1 1 3 Theorem 6.1. Let (X, ) ∈ C ([0, 1]; E), β ∈ ( /3, /2], f ∈ Cb (V ; L(E; V )), ξ ∈ V . There 0 β exists a unique element (Y,Y ) ∈ CX ([0, 1]; V ) such that Z t Yt = ξ + f(Ys) dXs ∀t ∈ [0, 1] . 0

Remark 6.2. If f ∈ C3 with possibly unbounded derivatives then existence and uniqueness hold up to some time τ ≤ 1, i.e. solutions are only local in time. To pass from [0, 1] to an arbitrary interval [a, b] is trivial.

Proof. Choose α such that 1/3 < α < β and let T ∈ (0, 1]. Then (X, X ) ∈ C α and the map Z · α α 0 MT : CX ([0,T ]; V ) → CX ([0,T ]; V ) ,M(Y,Y ) = ξ + f(Ys) dXs, f(Y ) 0 is well defined. α We first look for a suitable ball in CX that is invariant under the action of M. First notice that α M leaves the initial condition unchanged, thus the affine subspace of CX given by

0 α 0 (Y,Y ) ∈ CX : Y0 = ξ, Y0 = f(ξ)

α is left invariant by M. Since C , k · k α is a Banach space, such afﬁne subspace is a complete X CX metric space and its balls have the form

0 α 0 0 BT,K = (Y,Y ) ∈ CX : Y0 = ξ, Y0 = f(ξ), kY,Y kα,X ≤ K , because the initial data are ﬁxed and thus it is enough to use the semi-norm k · kα,X . Thanks to item (ii) in Gubinelli’s theorem we have, using (21) 0 0 X f(Y ) MT (Y,Y ) . kf(Y )kα + kf(Y ) k∞k k2α ∗ kXkα kf(Y )kα + R α,X 2α X 2 Y . C(f) kY kα + k k2α + kXkα 1 + kY kα + R 2α h i β−α β−α 2(β−α) X . C(f) T (kXkβK + K) + T kXkβ(1 + K) + T k k2β

β−α β−α α 0 because kXkα ≤ T kXkβ, T ≤ 1 implies that T ≥ T and kY kα ≤ kY k∞kXkα + Y α kR k2αT . The last line in the sequence of inequalities above can be made smaller than K choosing T sufﬁciently small, and invariance is thus proved.

23 Contractivity is shown using the same kind of estimates and the linearity of the integral map: 0 0 given (Y,Y ), (Y,e Ye ) ∈ BT,K we have

0 0 X MT (Y,Y ) − MT (Y,e Ye ) . kf(Y ) − f(Ye)kα + k k2αkf(Y ) − f(Ye)kα α,X

f(Y )−f(Ye ) + kXkα kf(Y ) − f(Ye)kα + R 2α

1 0 0 Y Ye . kY − Yekα + kY − Ye kα + R − R 2 2α

1 0 0 = k(Y,Y ) − (Y,e Ye )kα,X 2 for T = T (X, X , α, β) small enough. The unique solution of the rough differential equation is thus found as the unique ﬁxed point of α MT in CX . Thanks to the boundedness assumption on f and its derivatives, T can be chosen uniformly small with respect to the initial condition and therefore the argument can be iterated to obtain a global solution on [0, 1]. At last notice notice that

|Ys,t| 0 |Xs,t| Y 2α−β ≤ Y + R |t − s| < ∞ |t − s|β ∞ |t − s|β 2α thus Y ∈ Cβ and Y 0 ∈ Cβ as well because (Y,Y 0) is a ﬁxed point (Y 0 = f(Y )). Finally

Y 0 R(s,t) = Ys,t − Ys Xs,t

Z t

= f(Yr) dXr − f(Ys)Xs,t s 0 X 3α ≤ Y ∞ | s,t| + o |t − s| Y 2β 0 β q by item (i) in Gubinelli’s theorem, and therefore R ∈ C2 and (Y,Y ) ∈ CX . ^ Recalling the discussion right after equation (17), it is not difﬁcult to obtain the following result: i i β i Theorem 6.3. Let β ∈ (1/2, 1/2], X , X ∈ C , ξ ∈ V , i = 1, 2 and let 1/3 < α < β, and 3 i i 0 f ∈ Cb . Denote by Y , (Y ) , i = 1, 2, the two solutions to Z t i i i i Yt = ξ + f Yr dXr , i = 1, 2 . 0 1 2 Then, with M = X β ∨ X β < ∞ (recall notation (8)), we have 9 9 9 9 ˆ 1 2 1 2 1 1 2 2 dα Y ,Y ≤ C ξ − ξ + ρβ X , X , X , X 1 2 for a suitable constant C, and the same bound holds for Y − Y α.

24 Therefore the Itô-Lyons map is continuous in the rough path metric: the additional information given by the second order function X allows to recover continuity of the solution map (once the notion of solution has been modiﬁed using the rough path integral in order to be able to use such additional information).

7 Stochastic processes as rough paths

We have so far built a pathwise integration for rough functions. We want to see what happens when such functions are paths of a stochastic process. We will associate to a stochastic process X a random rough path, enhancing the process with a second-order component X . As for deterministic functions the choice for X is not unique; nevertheless stochastic integration theory suggests some canonical choices, at least in many important situations. This enhancement deﬁnes a map Ψ: X 7→ (X, X ) that allows to factorise the Itô solution map S to the SDE dYt = f(Yt) dXt into Φ ◦ Ψ = S, where Φ is the Itô-Lyons solution map. The map Ψ will be in general only measurable, but it depends only on X and on the choice of the stochastic integration in the SDE, not on f; on the other hand the map Φ is continuous in the sense of the results of the previous section.

d We will deal first with the example of a Brownian motion B in R . The idea is to define B s,t R t as an iterated stochastic integral s Bs,r ⊗ dBr; here the choice of different stochastic integration methods leads to different lifts B . We will mainly discuss Itô and Stratonovich integration; the fact that both Itô and Stratonovich integrals of Brownian increments against Brownian motion are well defined is given here as well known.

To show that the couple (B, B ) is a (random) rough path we will need a version of Kol- mogorov’s continuity criterion that can be obtained by an easy modiﬁcation of the proof of the classical Kolmogorov criterion via dyadic approximations. The Lq-type norms below will always refer to integration with respect to P.

Theorem 7.1 (Kolmogorov criterion for rough paths). Consider a probability space (Ω, F, P).Let q ≥ 2, β > 1/q and let X : [0,T ] → E and X : [0,T ]2 → E ⊗ E satisfy Chen’s relation and be such that, for some constant C,

β 2β kXs,tk ≤ C|t − s| , kX s,tk q ≤ C|t − s| . q 2

Then for every α ∈ [0, β − 1/q) there exists a modiﬁcation (X, X ) and two random variables q q Kα ∈ L , K α ∈ L 2 such that

α 2α |Xs,t| ≤ Kα|t − s| , |X s,t| ≤ K α|t − s| .

Thus if 1/3 < α < β − 1/q (X, X ) belongs to C α (almost surely).

25 Notice that just applying the classical Kolmogorov criterion to the process X 0,t we would not get the right 2α-Hölder estimate: one has to adapt the proof considering X and X at the same time.

Now, given a Brownian motion B in Rd define Z t B I 0,t := B0,r ⊗ dBr 0 where the integral is to be intended in Itô sense. Since both B· and B 0,· are almost surely continuous, we define B I B I B I s,t = 0,t − 0,s − Bs ⊗ Bs,t which is continuous in (s, t). By additivity of the Itô integral we have almost surely Z t B I s,t = Bs,r ⊗ dBr . s B I Using the fact that 0,1 belongs to the second Wiener chaos or also by explicit computations, B I it can be shown that it has finite moments of all orders; the same holds for s,t by Brownian p scaling, yielding a factor |t − s|. Since B has finite moments as well and kBs,tkq ≤ C |t − s|, we can apply theorem 7.1 with any q > 2 and β = 1/2, obtaining that for any α ∈ (1/3, 1/2) with probability one (B, B I ) belongs to C α([0,T ]; Rd) for any T > 0. Notice that h i B I 1 B i,j B j,i 1 1 Sym s,t = s,t + s,t = Bs,t ⊗ Bs,t − Id(t − s) 2 i,j 2 2 because, by Itô formula,

Bi dBj + Bj dBi = d(BiBj) − hBi,Bji dt . B S B I 1 1 B S Now deﬁne s,t = s,t+ 2 Id(t−s). Since 2 Id(t−s) is symmetric, we have that Anti s,t = B I B S 1 Anti s,t (Anti denotes the antisymmetric part of a tensor) and Sym s,t = 2 Bs,t ⊗ Bs,t. B S B I 1 1 s,t and s,t have obviously the same regularity, so we know that, for any α ∈ ( /3, /2), with B S α Rd probability one (B, ) ∈ Cg ([0,T ]; ) for any T > 0. Moreover we have that almost surely B S R t s,t coincides with the Stratonovich integral s Bs,r ⊗ ◦ dBr. In both the Itô and the Stratonovich case the fact that the second order process satisﬁes Chen’s relation is an immediate consequence of the properties of the integrals. The process Anti(B s,t) is the so-called Lévy stochastic area.

Consider now the dyadic piecewise linear approximation to B on [0, 1], that can be realised as (n) h ni B := E B B k , 0 ≤ k ≤ 2 . 2n R · (n) h S i Then B ⊗ dB(n) = E B B k n , as can be easily checked component-wise. By 0 2n ,0≤k≤2 (n) R · (n) B S uniform integrability it follows that B → B and 0 B ⊗ dB(n) → almost surely.

26 The bounds given by Kolmogorov’s criterion for B and B S hold true for B(n) and R B(n) ⊗ dB(n) as well via conditioning. We can thus apply Doob’s maximal inequality to show that (n) R · (n) (n) kB kα and 0 B ⊗ dB 2α are ﬁnite uniformly in n. Proposition 3.8 allows now to obtain the convergence Z · B(n), B(n) ⊗ dB(n) → B, B S 0 in C α. This provides a rough path counterpart of the Wong-Zakai principle. However if we consider in dimension d = 2 the paths Z · B(n) + X(n), B(n) ⊗ dB(n) + X (n) 0 where X(n), X (n) is the rough path considered in example 3.6, that is

2 Z t (n) 1 cos 2πn t X (n) (n) (n) Xt = 2 , s,t = Xs,r ⊗ dXr , n sin 2πn t s we obtain a smooth approximation to B that will converge with respect to the rough path metric A ρα to a process B, B where

0 1 B A = B S + π(t − s) . −1 0

Loosely speaking, if we add to B oscillations at small scales that in the limit produce an antisymmetric term in the iterated integrals, the Wong-Zakai principle does not hold anymore.

There is also a canonical way to associate a rough path to a Q-Wiener process in a separable P (k) Hilbert space H. Let (ek) be an orthonormal basis for H and let Xt = λkβt ek be a Q- 2 P 2 Wiener process, with (λk) ∈ ` and Qy = λkhy, ekiek. Then the series

∞ Z t X X (j) (k) s,t = λjλk βs dβs ej ⊗ ek j,k=1 s converges in L2 and almost surely (uniformly on compacts of H); the integral can be chosen to be either Itô or Stratonovich integral. Then (X, X ) ∈ C α, where X takes values in the closure of H ⊗ H with respect to the Hilbert-Schmidt norm. Brownian motion with values in Banach spaces can be also seen as a rough path, but we will not discuss this here; a hint of the general method developed in [Ledoux et al., 2002] is given in exercise 3.17 in [Friz and Hairer, 2014].

With the two ﬁnite-dimensional enhanced Brownian motions built above we can now un- dertake the task of comparing rough integrals with stochastic integrals and solution to rough differential equations with solutions to stochastic differential equations. To this end let N1 ⊂ Ω

27 B I α C 2 be a null set such that B(ω), (ω) ∈ C ∀ω ∈ N1 , and consider a Cb function f. Then, α thanks to example 5.3 (iii), there exists a null set N2 such that f B(ω) , Df B(ω) ∈ CB(ω) C C C ∀ω ∈ N2 . Gubinelli’s theorem implies that for any ω ∈ N3 = (N1 ∪ N2) the rough integral Z t I X B I f(Br) dBr = lim f(Bu)Bu,v + Df(Bu) u,v (22) |Π|→0 0 [u,v]∈Π exists. Fix a sequence Πn of partitions of [0, t] such that |Πn| → 0. Then, in probability, Z t X f(Br) dItôBr = lim f(Bu)Bu,v (23) 0 n→∞ [u,v]∈Πn and, switching to a subsequence, convergence holds outside some null set N4. C C Thus we have that on N5 = (N3 ∪ N4) the limit X I lim Df(Bu)B (24) n→∞ u,v [u,v]∈Πn exists and equals the difference between (22) and (23). Denote by t(n) ≤ t(n) ≤ · · · ≤ t(n) 1 2 N(Πn) the partition points of Πn. Then 2 2 N(Πn)−1 X I X I Df(B )B = Df B (n) B u u,v t (n) (n) j tj ,tj+1 j=1 [u,v]∈Πn L2 L2 2 N(Πn)−1 2 X B I ≤ kDfk∞ (n) (n) tj ,tj+1 j=1 L2

N(Π )−1 n 2 2 X (n) (n) . kDfk∞ tj+1 − tj j=1 and this converges to 0 when n → ∞ (that is, when |Πn| → 0), therefore the rough integral and C the Itô integral coincide on N5 . A more careful look at the proof reveals that we can conclude the same if we substitute the 0 α C integrand f(Bu) with any controlled process (Y (ω),Y (ω) ∈ CB(ω) ∀ω ∈ N2 such that 0 0 sup C kY (ω)k ≤ M < ∞ and both Y , and Y are adapted to the ﬁltration generated ω∈N2 ∞ by B. The general case of unbounded Y follows by localisation.

B S B I 1 0 Now, since s,t = s,t + 2 (t − s) Id, we have that for Y , Y adapted processes such that 0 α C (Y (ω),Y (ω)) ∈ CB(ω) ∀ω ∈ N2 , Z t S X 0B S Ys dBs = lim YuBu,v + Y u,v |Π|→0 0 [u,v]∈Π

28 exists and Z t Z t Z t S I 0 1 Ys dBs = Ys dBs + Ys d s Id , 0 0 0 2 thanks to remark 5.7. To conclude that

Z t Z t S Ys dBs = Ys ◦ dBs 0 0 we just need to show that Z t 0 1 1 Ys d s Id = [Y,B]t . 0 2 2

This easily follows writing

X X 0 Y Yu,vBu,v = Yu,v (Bu,v ⊗ Bu,v) + Ru,vBu,v , [u,v]∈Π [u,v]∈Π

B I recalling that Bu,v ⊗ Bu,v = 2 Sym u,v + Id(v − u) and repeating the computation used for (24) above with B I substituted by Sym(B I ). With some more effort it can be shown that a similar result holds for the backward Itô integral: if we set I t B B B I s,t = Bs,r dBr = s,t + (t − s) Id , s where the symbol H denotes backward Itô integration, then for any (Y,Y 0) almost surely con- 0 T trolled by B and such that Yt, Yt are measurable with respect to Ft (the backward ﬁltration of R T H T B), r Ys dBs exists almost surely as a rough integral and equals r Ys dBs.

It now arises as a natural question whether Itô formula can be obtained at the rough path level. This can actually be easily done and rough paths are seen to obey a second order change of variables rule, in the same spirit as semimartingales. It turns out that the “right” deﬁnition of the bracket is the following: given (X, X ) ∈ C α we set

[X]s,t := Xs,t ⊗ Xs,t − 2 Sym(X s,t) .

2α It is immediately checked that [X] belongs to C2 . The above deﬁnition is motivated by the fact that for a smooth function f the second differential D2f is always symmetric and therefore it does not interact with Anti(X s,t). For such an f we have

1 f(X ) − f(X ) = Df(X )X + hD2f(X )X ,X i + O |X |3 . (25) v u u u,v 2 u u,v u,v u,v

29 Now 1 1 X hD2f(X )X ,X i = ∂ f(X )Xi Xj 2 u u,v u,v 2 ij u u,v u,v i,j

1 X = ∂ f(X ) 2 Sym(X )i,j + [X] 2 ij u u,v u,v i,j

X 1 = ∂ f(X ) X i,j + [X]i,j ij u u,v 2 u,v i,j hence (25) equals   X X 1 X ∂ f(X )Xi + ∂ f(X )X i,j + ∂ f(X )[X]i,j + O |X |3 ,  i u u,v i,j u u,v 2 ij u u,v u,v i i,j i,j where we can recognise the Riemann-sum approximation to a rough integral. Using the facts 2α that [X]s,t = [X]0,t − [X]0,s and (t 7→ [X]0,t) ∈ C and the assumption α > 1/3, we can sum along partitions and pass to the limit obtaining

1 1 X α 3 Theorem 7.2 (Itô formula for rough paths). For α ∈ ( /3, /2], (X, ) ∈ C and f ∈ Cb the following formula holds true: Z t Z t 1 2 f(Xt) − f(X0) + Df(Xs) dXs + D f(Xs) d[X]s , 0 2 0 where the last integral is well deﬁned in Young sense.

8 Stochastic differential equations

We briefly discuss the relations between rough and stochastic differential equations. We will focus on the case when the drift coefficient is identically 0. The general case of Lipschitz drift can be handled as well but it requires a (not difficult) modification in the argument used to show existence and uniqueness of solutions to rough differential equations; the interested reader can consult [Friz and Victoir, 2010] or [Friz and Hairer, 2014]. We begin from Stratonovich SDEs. The convergence of the piecewise linear approximation of B studied in the previous section implies that the enhancing map

B 7→ B, B S

0,α d with values in Cg [0, t]; R is measurable for any ﬁxed t. 3 Rn Rd Rn Rn Let f ∈ Cb ; L( ; ) and ξ ∈ . We know that, on a set of full probability, there exists 0 α a unique solution (Y,Y ) ∈ CB to the rough differential equation S dY = f(Y ) dB ,Y0 = ξ . (26)

30 B 0 0 Since the Itô-Lyons map (B, ) 7→ (Y,Y ) is continuous, it follows that Yt and Yt are adapted B S B to Ft = σ (Br,s, r,s): 0 ≤ r ≤ s ≤ t . By construction we have that Ft = Ft and therefore, as discussed above, the rough integral in (26) coincides with the Stratonovich integral almost surely. Thus Y is a strong solution to the SDE

dY = f(Y ) ◦ dB,Y0 = ξ . B I B S 1 B I The identity s,t = s,t − 2 (t − s) Id provides measurability of and therefore we can repeat the argument to obtain strong solutions to Itô SDEs. We have (more or less) proved the following: 3 Rn Rn×d Rn Theorem 8.1. Let b be Lipschitz, σ ∈ Cb ; and ξ ∈ . Then on a set of full 0 α probability there exists a controlled process (Y,Y ) ∈ CB that solves the RDE dY = b(Y ) dt + I σ(Y ) dB , Y0 = ξ. The process Y is a strong solution of the ITô SDE dYt = b(Yt) dt + σ(Yt) dBt. S If we consider the RDE dY = b(Y ) dt + σ(Y ) dB , Y0 = ξ, the same holds but Y is then a strong solution of dYt = b(Yt) dt + σ(Yt) ◦ dBt. From these results is now easy to obtain a proof of the classical Wong-Zakai principle for SDEs.

9 Applications to a stochastic partial differential equation

Up to now we have only interpreted classical results in stochastic analysis in the framework of rough paths. We aim now at giving a hint of how rough paths can be used to obtain new results that could not be obtained (or, at least, not that easily) with classical arguments. We will show how the stochastic Burgers equation can be solved with the help of rough paths theory. We will outline the strategy used in [Hairer, 2011], but will omit most of the proofs and the details. We first consider a Gaussian centred process X taking values in Rd such that its components X(j), j = 1, . . . , d are independent. To realise X as a rough path we need to define X ; this can R t (i) (j) be done in a non-unique way, but making sense of the integrals 0 X0,s dXs could possibly lead to a canonical choice for X . Nevertheless the paths of X will in general have not enough regularity for R X(i) dX(j) to be defined as a Young integral and we do not assume any semimartingale structure on X, so that classical stochastic integration theory is of no help. It turns out that the task can be accomplished if the covariance of X is regular enough. Set 1   p  p s, t E  X u, v  R := Xs,t ⊗ Xs,¯ t¯ , kRkp,[s,t]2 := sup R . s,¯ t¯  ¯ u,¯ v¯  Π,Π [u,v]∈Π  [¯u,v¯]∈Π¯ The idea is to define Z t Z (i) (j) (i) (j) X (i) (j) X0,r dXr := lim X0,r dXr = lim X0,uXu,v (27) |Π|→0 |Π|→0 0 Π [u,v]∈Π

31 whit the limits taken in L2, and to show that this object actually exist. The choice of increments of the form X0,u (analogue of left-point evaluation in [u, v] plays no particular role here). Existence of the limit object in (27) can be shown with some effort in a way similar in spirit (but quite different in details, see [Friz and Victoir, 2011]) to the proof of Gubinelli’s theorem thanks to the following uniform bound ([Towghi, 2002]).

Lemma 9.1 (Towghi’s maximal inequality). Let X(1), X(2) be two continuous centred Gaussian (1) (2) (i) processes, independent and with covariances R , R such that R p,[0,t]2 < ∞, i = 1, 2, for some p < 2. Then " # Z 2 sup E X(1) dX(2) ≤ C R(1) R(2) . 0,r r 2 2 Π Π 2,[0,t] 2,[0,t] s, t 1/p α Since Kolmogorov’s criterion implies that if R |t−s| then |Xs,t| ≤ Kα|t−s| s, t . q for every α < 1/p and some Kα ∈ ∩q<∞L , we have that the interesting cases are those for which p ∈ [1, 2) due to the assumptions in Towghi’s lemma, the case p < 1 falling into the realm of Young integration when it comes to deﬁne (27). The mentioned existence result and Kolmogorov’s continuity criterion yield easily the following theorem:

Theorem 9.2. Let X(i) be a d-dimensional continuous centred Gaussian process with independent components whose covariances satisfy the bound

1 R(i) ≤ C|t − s| p ∀i , ∀0 ≤ s ≤ t ≤ T p,[s,t]2

3 X i,j R (i) (j) 1 1 X for some p ∈ [1, /2). Set s,t = limΠ→0 Π Xs,r dXr . Then for any α ∈ ( /3, /2p)(X, ) α belongs to Cg almost surely. Consider now the stochastic Burgers equation

∂ u = ∂2u + g(u)∂ u + ξ t x x (28) u|t=0 = u0 where u = (u1, . . . , ud) : [0, 1] × [0, 2π] → Rd, g : Rd → Rd×d is smooth, ξ is space-time α d white noise and u0 ∈ C [0, 2π]; R . The idea here is to look for something like a “mild solution”: u should (formally) satisfy

Z t u(t, x) = ψˆ(t, x) + St−s g u(s, ·) ∂xu(s, ·) (x) ds (29) 0 where St is the heat semigroup and ψˆ is the solution to the stochastic heat equation

( ˆ 2 ˆ ∂tψ = ∂xψ + ξ ˆ (30) ψ|t=0 = u0

32 on [0, 1] × [0, 2π]. If g were a gradient function, g = ∂xG, we might formally write g(u)∂xu = ∂xG(u) and use R t the properties of St to write the integral in (29) as 0 ∂xSt−s G u(s, ·) (x) ds. Existence and uniqueness of a solution would then follow from a fixed point argument (and this would be enough to cover the case d = 1). In the general case, denoting by p the heat kernel, we have Z 2π St−s g u(s, ·) ∂xu(s, ·) (x) = pt−s(x − y)g u(s, y) ∂xu(s, y) dy (31) 0 and we expect u to have at most the same regularity in space as ψˆ, so that the integral above is not well defined in a classical sense, as we shall see in a while. Therefore we might hope to use the theory of rough integration in order to give a well defined concept of solution to (28). To this end, we begin by studying some properties of the solution to a slight modification of the stochastic PDE (30) and by realising it as a rough path. Consider the equation 2 2 ∂tψ = ∂xψ − ψ + ξ = ∂x − 1 ψ + ξ ˆ (32) ψ|t=0 = u0 on [0, 1] × [0, 2π]. Denoting by W a cylindrical Wiener process over L2([0, 2π]) (where we consider periodic boundary conditions), we can rewrite (32) as 2 dψt = ∂x − 1 ψt dt + dWt

(we will use the two notations ψt(x) and ψ(t, x) interchangeably in what follows). It is the easily seen that a stationary§ (in time) solution to this SDE is given by X k ψ(t, x; ω) = Y (t; ω)ek(x) , k∈Z

n 1 o n 1 o n 1 o where (ek)k∈Z = √ sin(kx) ∪ √ cos(kx) ∪ √ is the trigonometric or- π k>0 π k<0 2π 2 2 2 k thonormal basis for L ([0, 2π]) , so that ∂x − 1 ek = (−k − 1)ek for all k and the Y ’s solve k 2 k k dYt = (−k − 1)Yt dt + dBt , k k (B )k∈Z being independent standard Brownian motions in R. The Y ’s are then independent Ornstein-Uhlenbeck processes, and by standard covariance calculations it can be computed that

1 2 E Y kY k = e(k +1)|t−s| . s t 2(k2 + 1) It follows that h i X 1 E |ψ (x) − ψ (y)|2 ≤ C k2α|x − y|2α t t 2(k2 + 1) k∈Z for any α ∈ [0, 1], therefore, by the classical Kolmogorov criterion, for every ﬁxed t the function ¶ ψt is almost surely α-Hölder in the variable x for every α < 1/2 . The same of course holds for §The additional linear term −ψ appearing in equation (32) is introduced precisely to have a solution that is stationary. ¶ Similar computations show that ψ is almost surely α-Hölder in time for any α < 1/4.

33 ψˆ, and since we expect our solution u to have the same space regularity as ψˆ, we see that the integral on the right hand side of (31) is indeed ill-deﬁned. However the covariance calculation above implies that E [ψ(t, x)ψ(t, y)] = K (|x − y|) for a smooth function K (that can be explicitly given, see [Hairer, 2011]). It can be shown that any covariance function R such that R(x, y) = K (|x − y|) for some C2 function K has 1-variation norm over [x, y] bounded by |x−y| times a constant. In our situation this means that

Rψt(·) ≤ C |x − y| 1,[x,y]2 where C depends only on K thanks to the stationarity in time of ψ. By theorem 9.2 we can asso- 1 d i i ciate to the map x 7→ ψt (x), . . . , ψt (x) a rough path (ψ, Ψ) = ψ i=1,...,d , Ψ i=1,...,d , seen as a function of the space variable x. This will allow us to interpret (31) as a rough integral. α Rd Moreover it can be shown that the process t 7→ (ψt, Ψt) with values in Cg [0, 2π]; admits a continuous modiﬁcation. An alternative way to construct a rough path (in space) associated to ψ is to set Z y (n) X k (n) (n) (n) (n) ψt = Yt ek(x) , Ψx,y = ψt (z) − ψt (x) dψt (z) , |k|≤n x and to show that Ψ(n) converges almost surely in L2. Using the Kolmogorov criterion for rough paths it is then easily shown that the limit object (ψ, Ψ) almost surely belongs to C α for any α < 1/2. We now write our function u that should solve in some sense the Burgers equation as u(t, x) = ψ(t, x) + v(t, x) for some function v; then equation (28) reads 2 2 ∂tψ + ∂tv = ∂xψ + ∂xv + g(ψ + v)(∂xψ + ∂xv) + ξ so that v should solve 2 ∂tv = ∂xv + g(ψ + v)(∂xψ + ∂xv) + ξ (33) (with initial condition v(0, ·) = 0), which in mild form is written as Z t Z t v(t, x) = St−s g ψ(t, ·) + v(t, ·) ∂xψ(t, ·) + ∂xv(t, ·) (x) dt + St−sψ(t, ·)(x) dt 0 0 Z t Z 2π = pt−s(x − y)g u(s, y) ∂xψ(s, y) dy ds (34) 0 0 Z t Z 2π + pt−sg u(s, y) ∂xv(s, y) dy ds (35) 0 0 Z t Z 2π + pt−s(x − y)ψ(s, y) dy ds . (36) 0 0

34 From (33) we may infer that, if we can make sense of the equation, v(t, ·) should belong to Cα+1 1 2 for any α < /2 since ∂xv should have the same regularity of ∂xψ; therefore the integral in (35) should be a well defined Young integral Z t Z 2π pt−sg u(s, y) dvs(y) ds . 0 0 Thanks to the construction of (ψ, Ψ) given above we can interpret the inner integral in (34) as the rough integral in the variable y Z 2π pt−s(x − y)g u(s, y) dψs(y) . (37) 0 It can be easily shown that for every x the map y 7→ pt−s(x − y)g ψ(s, y) + v(s, y) belongs to α Cψ and its Gubinelli derivative is pt−s(x − y)Dg ψ(s, y) + v(s, y) , so that the integral is well defined in the rough path sense. We can therefore say by definition that a function u solves the stochastic Burgers equation (28) if v := u − ψ is a mild solution to equation(33) in the sense just specified. Note that it can be proved that v is a mild solution if and only if it is a weak solution (in the PDE sense). To solve the stochastic Burgers equation we now have to prove existence and uniqueness of a 1 1 Rd mild solution to equation (33). To this end let T ≤ 1 and set CT := C [0,T ]; C [0, 2π]; . 1 1 We shall see that the map MT,ψ : CT → CT given by Z t Z 2π (MT,ψv)(t, x) = pt−s(x − y)g ψs(y) + vs(y) dvs(y) ds 0 0 Z t Z 2π + pt−s(x − y)g ψs(y) + vs(y) dψs(y) ds 0 0

1 2 =: MT,ψv(t, x) + MT,ψv(t, x) has a unique ﬁxed point (we can forget about the integral in (36) because it does not depend on v). 1 Let BK be the ball of radius K in C (equipped with the norm kvk 1 := sup kvtk 1 Rd ). T CT t∈[0,T ] C ([0,2π]; ) 1 For MT,ψ we have the bound Z t M 1 ≤ sup kS [g (ψ + v ) ∂ v ]k ds T,ψ C1 t−s s s x s C1 T t∈[0,T ] 0

Z t ≤ sup kSt−skL∞→C1 kg(ψs + vs)∂xvskL∞ ds t∈[0,T ] 0

Z t 1 ≤ sup √ kg(ψs + vs)kL∞ k∂xvskL∞ ds t∈[0,T ] 0 t − s √ ≤ TC

35 1 where C depends on ψ and K and we have used the well known fact St as a map from C into C satisﬁes 1 kStk ∞ 1 ≤ C √ ∀t ≤ 1 . L →C t

Therefore M 1 v ≤ K for T sufﬁciently small. With similar computations it is not difﬁcult T,ψ 1 CT to show that for v, v¯ ∈ BK

Z t 1 1 1 M v − M v¯ ≤ sup √ [kg(ψ + v ) − g(ψ +v ¯ )k ∞ k∂ v k ∞ T,ψ T,ψ C1 s s s s L x s L T t∈[0,T ] 0 t − s

+ kg(ψs +v ¯s)kL∞ k∂xvs + ∂xv¯skL∞ ] ds √ ≤ TC kv − v¯k 1 , CT

1 hence we see that MT,ψ is a contraction for T small enough. 2 At last we consider MT,ψ. Set

Z x Z(t, x) := g ψt(y) + vt(y) dψt(y) 0 so that Z t 2 MT,ψv(t, x) = ∂x [St−sZ(s, x)] ds , 0 which is well deﬁned by standard semigroup properties. It follows that

Z t 2 M v ≤ sup k∂ S k α 1 kZk α ds . T,ψ C1 x t−s C →C C T t∈[0,T ] 0

Using the bounds on the rough integral in Gubinelli’s theorem and the fact that St as a map from Cα into C1 satisﬁes the inequality

1 − 2 (2−α) kStkCα→C1 ≤ Ct

(this can be shown by hands with an explicit computation or using the theory of analytic semi- groups) we ﬁnd that α 2 2 MT,ψv 1 ≤ CT CT where C again depends on ψ and on K (and obviously on y). Contractivity is now shown with the same kind of estimates, using the bound for the difference of rough integrals given in 5.6. This yields existence of a unique ﬁxed point v for the map MT,ψ 1 in the space CT ; v + ψ is then the unique solution of the stochastic Burgers equation (28).

36 References

[Baudoin, 2013] Baudoin, F. (2013). Rough paths theory. http://www.math.purdue. edu/~fbaudoin/Rough.pdf. [Chen, 1958] Chen, K.-T. (1958). Integration of paths—a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math. Soc., 89:395–407. [Friz and Shekhar, 2013] Friz, P. and Shekhar, A. (2013). Doob-Meyer for rough paths. Bull. Inst. Math. Acad. Sin. (N.S.), 8(1):73–84. [Friz and Victoir, 2011] Friz, P. and Victoir, N. (2011). A note on higher dimensional p- variation. Electron. J. Probab., 16:no. 68, 1880–1899. [Friz and Hairer, 2014] Friz, P. K. and Hairer, M. (2014). A course on rough paths. Universitext. Springer, Cham. With an introduction to regularity structures. [Friz and Victoir, 2010] Friz, P. K. and Victoir, N. B. (2010). Multidimensional stochastic processes as rough paths, volume 120 of Cambridge Studies in Advanced Mathematics. Cam- bridge University Press, Cambridge. Theory and applications. [Gubinelli, 2010] Gubinelli, M. (2010). Ramiﬁcation of rough paths. J. Differential Equations, 248(4):693–721. [Hairer, 2011] Hairer, M. (2011). Rough stochastic PDEs. Comm. Pure Appl. Math., 64(11):1547–1585. [Hairer and Kelly, 2015] Hairer, M. and Kelly, D. (2015). Geometric versus non-geometric rough paths. Ann. Inst. Henri Poincaré Probab. Stat., 51(1):207–251. [Ledoux et al., 2002] Ledoux, M., Lyons, T., and Qian, Z. (2002). Lévy area of Wiener processes in Banach spaces. Ann. Probab., 30(2):546–578. [Lyons, 1991] Lyons, T. (1991). On the nonexistence of path integrals. Proc. Roy. Soc. London Ser. A, 432(1885):281–290. [Lyons and Victoir, 2007] Lyons, T. and Victoir, N. (2007). An extension theorem to rough paths. Ann. Inst. H. Poincaré Anal. Non Linéaire, 24(5):835–847. [Lyons, 1998] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215–310. [Lyons et al., 2007] Lyons, T. J., Caruana, M., and Lévy, T. (2007). Differential equations driven by rough paths, volume 1908 of Lecture Notes in Mathematics. Springer, Berlin. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean Picard. [Towghi, 2002] Towghi, N. (2002). Multidimensional extension of L. C. Young’s inequality. JIPAM. J. Inequal. Pure Appl. Math., 3(2):Article 22, 13 pp. (electronic).