From random walks to rough paths
aa r X i v : . [ m a t h . P R ] O c t FROM RANDOM WALKS TO ROUGH PATHS
EMMANUEL BREUILLARD, PETER FRIZ AND MARTIN HUESMANN
Abstract.
Donsker’s invariance principle is shown to hold for random walksin rough path topology. As application, we obtain Donsker-type weak limittheorems for stochastic integrals and differential equations. Introduction
Consider a random walk in R d , given by the partial sums of a sequence of inde-pendent random-variables ( ξ i : i = 1 , , , . . . ), identically distributed, ξ i D = ξ withzero-mean and unit variance. Donsker’s theorem (e.g. [9]) states that the rescaledand piecewise-linearly-connected random-walk W ( n ) t = 1 n / (cid:16) ξ + · · · + ξ [ tn ] + ( nt − [ nt ]) ξ [ nt ]+1 (cid:17) converges weakly to d -dimensional Brownian motion B in the sense that E (cid:16) f (cid:16) W ( n ) (cid:17)(cid:17) → E ( f ( B )) as n → ∞ , for all continuous, bounded functionals f on C (cid:0) [0 , , R d (cid:1) with uniform topology;we shall use the shorter notation W ( n ) = ⇒ B on C (cid:0) [0 , , R d (cid:1) to decribe this type of convergence. It was observed by Lamperti in [5] that thisconvergence takes place in α -H¨older topology, i.e. W ( n ) = ⇒ B on C α -H¨ol (cid:0) [0 , , R d (cid:1) for α < ( p − / p provided E (cid:16) | ξ | p (cid:17) < ∞ , p > α and p is sharp. In particular, for convergence in α -H¨older topology for any α < / t W ( n ) t is a (random) Lipschitz path, it can be canonically lifted bycomputing iterated integrals up to any given order, say N . The resulting, lifted,path is denoted by t S N ( W ( n ) t ) ≡ (cid:18) , W ( n ) t , Z t W ( n ) s ⊗ W ( n ) s , . . . (cid:19) and can be viewed as a (random) path with values in the step- N free nilpotentgroup, realized as the subset G N (cid:0) R d (cid:1) of the tensor-algebra R ⊕ R d ⊕ · · · ⊕ (cid:0) R d (cid:1) ⊗ N .See [2] and [6, 7, 8] for background on iterated integrals and rough paths. One canask if S N (cid:16) W ( n ) · (cid:17) converges weakly to a limit. The obvious candidate is Brownianmotion B on the step- N free nilpotent group i.e. the symmetric diffusion with Key words and phrases.
Donskers’s theorem, rough paths. generator given by the sub-Laplacian on the stratified Lie group G N (cid:0) R d (cid:1) associatedto the covariance matrix of ξ . In uniform topology, the answer is affirmative andfollows from work of Stroock–Varadhan [10], see also [11].There is motivation from rough path theory to work in stronger topologies thanthe uniform one. Indeed, various operations of SDE theory and stochastic integra-tion theory are continuous functions of enhanced Brownian motion (i.e. Brown-ian motion plus L´evy’s area, or equivalently: Brownian motion on the step-2 freenilpotent group) in rough path sense. More specifically, any G (cid:0) R d (cid:1) -valued path x ( · ) has canonically defined path increments, denoted by x s,t := x − s x t , so that k x s,t k = d ( x s , x t ) where k·k and d are the (Euclidean) Carnot-Caratheodory normand metric on G (cid:0) R d (cid:1) . In its simplest (non-trivial) setting, and under mild regu-larity conditions on the vector fields V , . . . , V d , rough path theory asserts that theODE solution to(1.1) dy = d X i =1 V i ( y ) dx i , y (0) = y ∈ R e , depends continuously (uniformly on bounded sets) on the driving signal x ( · ) ∈ C ∞ (cid:0) [0 , , R d (cid:1) with respect to ˜ d α -H¨ol ( x, x ′ ) := d α -H¨ol ( S ( x ) , S ( x ′ )) , α ∈ (1 / , / , where x := S ( x ) and d α -H¨ol ( x , x ′ ) = sup s,t ∈ [0 , d (cid:0) x s,t , x ′ s,t (cid:1) | t − s | α It is easy to see that C α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) := { x : d α -H¨ol ( x , ) < ∞} , is a complete metric space for d α -H¨ol (up to constants : d α -H¨ol ( x , x ′ ) = 0 iff x − t x ′ t ≡ constant). It follows that the very meaning of the ODE (1.1) can be extended, ina unique and continuous fashion, to the d α -H¨ol -closure of lifted smooth paths in C α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) . This closure is a Polish space for the metric d α -H¨ol and isdenoted by C ,α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) . If α ∈ (1 / , /
2) it is known as the space of geometric α -H¨older rough paths . Itincludes almost every realization of enhanced Brownian motion B , B ≡ (cid:18) , B, Z B ⊗ ◦ dB (cid:19) where ◦ dB denotes the Stratonovich differential of B . The resulting “generalized”ODE solution driven by B can then be identified as the classical StratonovichSDE solution, [6, 7, 8]. This provides an essentially deterministic approach to SDEtheory with numerous benefits when it comes to regularity questions of the Itˆo map,construction of stochastic flows, etc. Another property of such rough paths is that It is crucial to have α > /
3. The solution map x y would not be continuous in d α -H¨ol for α ≤ / denotes the trivial path identically equal to the unit element ∈ G ` R d ´ The same construction applies when G ` R d ´ is replaced by G N ` R d ´ in which case onerequires α ∈ (1 / ( N + 1) , /N ) in order to speak of geometric α -H¨older rough paths . ROM RANDOM WALKS TO ROUGH PATHS 3 there is a unique (again: modulo constants) lift of x ∈ C ,α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) toa path of similar regularity in the step- N group for all N ≥ ,S N ( x ) ∈ C ,α -H¨ol (cid:0) [0 , , G N (cid:0) R d (cid:1)(cid:1) . (See [8, Thm 3.7.] for instance.) In the case of the enhanced Brownian motion x = B ( ω ), the process S N ( B ) identifies as Brownian motion B plus all iteratedStratonovich integrals up to order N . In particular, S N ( B ) is then realization ofBrownian motion on the step- N free nilpotent group.Our main result is Theorem 1.
Assume E (cid:16) | ξ | p (cid:17) < ∞ for some real number p ≥ . Then S ( W ( n ) t ) = ⇒ B in C ,α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) provided (1.2) α ∈ (cid:18) , p ∗ − p ∗ (cid:19) with p ∗ = min ([ p ] , p/ In particular, if E (cid:16) | ξ | p (cid:17) < ∞ for all p < ∞ then the weak convergence holds forany α < / . Let us point out that Theorem 1 implies, of course, α -H¨older convergence for all α < p ∗ − p ∗ but only for α > / S N ( W ( n ) t ) for all N is a (deterministic) consequence of Theorem 1, cf. Corollary3 below.Although reminiscent of Lamperti’s sharp upper bound on the H¨older exponent, p − p , the actual “coarsened” form of our upper bound in (1.2), with p replaced by p ∗ , i.e. the largest even integer smaller or equal to p, comes from our argument (whichrequires us to work with integer powers). To handle the case of “integrabilitylevel” p ∈ (1 ,
4) it is clear that the step-2 setting will not be sufficient. Indeed,from Lamperti’s bound, we would have to work at least in the step- N group with N ∼ p/ ( p − α -H¨oldertopology for G N (cid:0) R d (cid:1) -valued paths with1 / ( N + 1) < α < ( p − / p. Any result of the form S N ( W ( n ) t ) = ⇒ S N ( B ) in C ,α -H¨ol (cid:0) [0 , , G N (cid:0) R d (cid:1)(cid:1) would then be equally interesting as it would constitute a “step- N ” convergenceresult in rough path topology with similar corollaries as those described below.Unfortunately, as explained in Section 4, the “coarsened” H¨older exponents thatwe obtain in the step- N setting are not bigger than 1 / ( N + 1) in general. Althoughwe suspect this to be an artefact of our proof, we currently do not know how tobypass this difficulty in order to handle p ∈ (1 , p ≥ ξ i ’s) . EMMANUEL BREUILLARD, PETER FRIZ AND MARTIN HUESMANN
Corollary 1 (Donsker-Wong-Zakai type convergence) . Assume V = ( V , . . . , V d ) is a collection of C -bounded vector fields on R e and let ( Y n ) denote the family of(random) ODE solutions to dY n = V ( Y n ) dW ( n ) , Y = y ∈ R e . Then, with α as in (1.2), Y n = ⇒ Y in C α -H¨ol ([0 , , R e ) where Y is the (up to indistinguishability) unique continuous solution to the StratonovichSDE dY = V ( Y ) ◦ dB, Y = y ∈ R e . Remark 1.
The regularity assumptions of the ( V i ) can be slightly weakened. Onecan also add a drift vector field (only assumed C -bounded say) and in fact theweak convergence can be seen to hold in sense of flows of C -diffeomorphisms (andthen C k -flows for k ∈ N , provided additional smoothness assumptions are made on V ). Indeed, all this follows from the appropriate (deterministic) continuity resultsof rough path theory, cf. for instance, combined with stability of weak convergenceunder continuous maps. Corollary 2 (Weak convergence to stochastic integrals) . Assume ϕ = ( ϕ , . . . , ϕ d ) is a collection of C b -bounded functions from R d to R e . Then Z · ϕ (cid:16) W ( n ) (cid:17) dW ( n ) = ⇒ Z · ϕ ( B ) ◦ dB. Corollary 3 (Convergence to BM on the free step- N nilpotent group) . Assume N ≥ . Then S N ( W ( n ) t ) = ⇒ ˜B in C ,α -H¨ol (cid:0) [0 , , G N (cid:0) R d (cid:1)(cid:1) with α as in (1.2) and ˜B a Brownian motion on G N (cid:0) R d (cid:1) . Acknowledgement 1.
The second author is partially supported by a LeverhulmeResearch Fellowship and EPSRC grant EP/E048609/1. Donsker’s theorem for enhanced Brownian motion and randomwalks on groups
We first discuss the case of a random walk for finite moments of all orders.
Theorem 2 (Donsker’s theorem for enhanced Brownian motion) . Assume E ξ = 0 and E ( | ξ | p ) < ∞ for all p ∈ [1 , ∞ ) and α < / . Then S ( W ( n ) · ) = ⇒ B in C ,α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) where B is a ( G (cid:0) R d (cid:1) -valued) enhanced Brownian motion. In fact, we shall prove a more general theorem that deals with random walks ongroups. More precisely, by a theorem of Chen [2] we have S (cid:16) W ( n ) (cid:17) t = δ n − / (cid:16) e ξ ⊗ · · · ⊗ e ξ [ nt ] ⊗ e ( nt − [ nt ]) ξ [ nt ]+1 (cid:17) ROM RANDOM WALKS TO ROUGH PATHS 5 where δ denotes dilation on G (cid:0) R d (cid:1) and e v = (cid:0) , v, v ⊗ / (cid:1) ∈ G (cid:0) R d (cid:1) , v ∈ R d .Observe that ( ξ i ) := (cid:0) e ξ i (cid:1) is a sequence of independent, identically distributed G (cid:0) R d (cid:1) -valued random variables, centered in the sense that E ( π ( ξ i )) = E ξ i = 0 , where π is the projection from G (cid:0) R d (cid:1) → R d . Let us also observe that the shortestpath which connects the unit element 1 ∈ G (cid:0) R d (cid:1) with e ξ i is precisely e tξ i so thatpiecewise linear interpolation on R d lifts to geodesic interpolation on G (cid:0) R d (cid:1) . Weshall thus focus on the following Donsker-type theorem. Theorem 3.
Let ( ξ i ) be a centered IID sequence of G (cid:0) R d (cid:1) -valued random vari-ables with finite moments of all orders, ∀ q ∈ [1 , ∞ ) : E ( k ξ i k q ) < ∞ and consider the rescaled random walk defined by W ( n )0 = 1 and W ( n ) t = δ n − / (cid:16) ξ ⊗ · · · ⊗ ξ [ tn ] (cid:17) for t ∈ (cid:8) , n , n , . . . (cid:9) , piecewise-geodesically-connected in between (i.e. W ( n ) t | [ in , i +1 n ] is a geodesic connecting W ( n ) i/n and W ( n )( i +1) /n ). Then, for any α < / , W ( n ) converges weakly to B , in C ,α -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) . Proof of theorem 3
Following a standard pattern of proof, weak convergence follows from conver-gence of the finite-dimensional-distributions and tightness (here in α -H¨older topol-ogy).Step 1: (Convergence of the finite-dimensional-distributions) This is an imme-diate consequence of the : Theorem 4. (Central limit theorem for centered i.i.d variables on a nilpotent Liegroup) Let N be a simply connected nilpotent Lie group and ξ , ..., ξ n , ... be i.i.d.random variables with values in N which we assume centered (i.e. their projection π ( ξ i ) on the abelianization of N has zero mean) and with a finite second moment,i.e. E (cid:16) k ξ i k (cid:17) < ∞ . Then we have the following convergence in law: δ n − / ( ξ ⊗ · · · ⊗ ξ n ) → B where B is the time value of the Brownian motion on N associated to ξ (i.e.the symmetric diffusion on N with infinitesimal generator the left invariant sub-Laplacian P a ij X i X j where ( a ij ) is the covariance matrix of π ( ξ i ) ). This theorem is a straightforward consequence of the main result of Wehn’s(unpublished) 1962 thesis, cf. [4], [1, Thm 3.11] or [3]. It also follows a fortiorifrom the much stronger Stroock-Varadhan Donsker-type theorem in connected Liegroups [10].Step 2: (Tightness) We need to find positive constants a, b, c such that for all u, v ∈ [0 ,
1] sup n E h d (cid:16) W ( n ) v , W ( n ) u (cid:17) a i ≤ c | v − u | b , EMMANUEL BREUILLARD, PETER FRIZ AND MARTIN HUESMANN then we can apply Kolmogorov’s tightness criterion to obtain tightness in γ -H¨oldertopology, for any γ < b/a . Using basic properties of geodesic interpolation, we seethat it is enough to consider u, v ∈ (cid:8) , n , n , . . . (cid:9) and then (by independence ofincrements and left invariance of d ) there is no loss of generality in taking [ u, v ] =[0 , k/n ] for some k ∈ { , . . . , n } . It follows that what has to be established reads ∃ a, b, c : 1 n a/ E [ k ξ ⊗ · · · ⊗ ξ k k a ] ≤ c (cid:12)(cid:12)(cid:12)(cid:12) kn (cid:12)(cid:12)(cid:12)(cid:12) b , uniformly over all n ∈ N and 0 ≤ k ≤ n , and such that b/a can be taken arbitrarilyclose to 1 /
2. To this end, it is enough to show that for all p ∈ { , , . . . } ( ∗ ) : E h k ξ ⊗ · · · ⊗ ξ k k p i = O (cid:0) k p (cid:1) since we can then take a = 4 p, b = 2 p − b/a = (2 p − / (4 p ) ↑ / p ↑ ∞ . Thus, the proof is finished once we show ( ∗ ) and this is the content ofthe last step of this proof.Step 3: Let P be a polynomial function on G ( R d ) , i.e. a polynomial in a i , a ij where a = (cid:0) a i , a ij ; 1 ≤ i ≤ d, ≤ i < j ≤ d (cid:1) ∈ g (cid:0) R d (cid:1) is the log-chart of G ( R d ), g a = log ( g ). We define the degree d ◦ P by agreeingthat monomials of form (cid:0) a i (cid:1) α i (cid:0) a ij (cid:1) α i,j have degree P α i +2 P α i,j . If ξ is a G ( R d )-valued random variable with momentsof all orders, then T P : g E ( P ( g ⊗ ξ )) − P ( g )is well defined and is another polynomial function. Moreover if ξ is centered, aneasy application of the Campbell-Baker-Hausdorff formula reveals that T P is ofdegree ≤ d o P −
2. (For instance, P ( a ) := (cid:0) a ij (cid:1) m has degree 2 m ; then T P is seento contain terms of the form (cid:0) a ij (cid:1) m − and (cid:0) a ij (cid:1) m − (cid:0) a k (cid:1) etc. all of which areindeed of degree 2 m − p ∈ { , , . . . } , k e a k p ∼ X i (cid:12)(cid:12) a i (cid:12)(cid:12) p + X i But the function T P : g E ( P ( g ⊗ ξ )) − P ( g ) is a polynomial function of degreeat most d ◦ P − p − 2. Hence d ◦ T l P ≤ d ◦ P − l = 2 (2 p − l )and the above sum contains only a finite number of terms, more precisely E [ P ( ξ ⊗ · · · ⊗ ξ k )] = p X l =0 (cid:18) kl (cid:19) T l P (1) . Since each of these terms is O ( k p ), as k → ∞ , we are done.4. Extension to finite moments The question remains what happens if we weaken the moment assumption to E (cid:16) k ξ i k p (cid:17) < ∞ for some p > . where, for now, ξ i ∈ G (cid:0) R d (cid:1) . If we could prove that(4.1) E h k ξ ⊗ · · · ⊗ ξ k k p i = O ( k p )then the arguments of the previous section apply line-by-line to obtain tightness(and hence weak convergence) in C ,γ -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) , for any γ < p − p . In the case that p − p > / γ ∈ (1 / , / 2) since then C ,γ -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) is a genuine rough path space . Otherwise, i.e. if γ < p − p ≤ / 3, tightness in C ,γ -H¨ol (cid:0) [0 , , G (cid:0) R d (cid:1)(cid:1) is worthless (from the point ofview of rough path applications). However, we can still ask for the smallest integer N such that p − p > N + 1and consider G N (cid:0) R d (cid:1) -valued random variables ξ i with finite (2 p )-moments. Again,if we could show that (4.1) holds, all arguments extend and we would obtain tight-ness in C ,γ -H¨ol (cid:0) [0 , , G N (cid:0) R d (cid:1)(cid:1) where γ can be chosen to be in (cid:16) N +1 , N (cid:17) so thatwe have tightness in a ”step- N rough path topology”. Unfortunately, the ”poly-nomial” proof of the previous section does not allow to obtain (4.1) but only thefollowing slightly weaker result. Proposition 1. Let ( ξ i ) be a centered IID sequence of G N (cid:0) R d (cid:1) -valued randomvariables with finite (2 p ) -moments, p real p > E (cid:16) k ξ i k p (cid:17) < ∞ . Then (4.3) E h k ξ ⊗ · · · ⊗ ξ k k q i = O ( k q ) The integer part of 1 /γ must match the level of nilpotency: [1 /γ ] = 2. EMMANUEL BREUILLARD, PETER FRIZ AND MARTIN HUESMANN for all q ≤ q ( p, N ) ≤ p where q ( p, N ) may be taken to be q ( p, N ) = min m =1 ,...,N m h pm i . Proof. Set q m = m [ p/m ]. The conclusion is equivalent to |k ξ ⊗ · · · ⊗ ξ k k| L q = O (cid:16) k / (cid:17) and will follow, with q = min { q m : m = 1 , . . . , N } from (cid:12)(cid:12)(cid:12) | π m (log ( ξ ⊗ · · · ⊗ ξ k )) | /m (cid:12)(cid:12)(cid:12) L q ≤ (cid:12)(cid:12)(cid:12) | π m (log ( ξ ⊗ · · · ⊗ ξ k )) | /m (cid:12)(cid:12)(cid:12) L qm = O (cid:16) k / (cid:17) provided we can show the O (cid:0) k / (cid:1) -estimate of the last line, for all m ∈ { , . . . , N } .To this end, consider P m ( e a ) given by X i ...i m ∈{ ,...,d } (cid:0) a m ; i ...i m (cid:1) p/m ] and observe that it has homogenous degree 2 m [ p/m ] ≤ p . Now observe thatthe condition of existence of moments of order 2 p for ξ , i.e. (4.2), implies (usingH¨older’s inequality) that the expectation of any polynomial in ξ of homogeneousdegree at most 2 p is finite. Hence the operator T defined in Step 3 of the proof ofTheorem 3 applies to P m and the argument there (see also [3, Lemma 2.4] showsthat it will reduce its degree by two. The same arguments as earlier will then giveus the O (cid:0) k / (cid:1) -estimate with q m determined by2 q m /m = 2 [ p/m ] or q m = m [ p/m ] . (cid:3) As in the previous section, (4.3) with q = q implies α -H¨older tightness/convergencefor any α < ( q − / q . In particular, when q − q > N + 1 or q = min m =1 ,...,N m h pm i > N + 1 N − N − α > N +1 and thus have ”rough path convergence”.Case N = 2: In this case, the above condition reduces to min([ p ] , p/ > p ≥ 4. Indeed, for p = 4 we have q (4 , N ) = 4 which implies α -H¨oldertightness/convergence for any α < q − q = 38and since 3 / ∈ (1 / , / 2) we can indeed pick α ∈ (1 / , / p < q ( p, N ) ≤ α < − 16 = 13in which case α -H¨older topology is not a rough path topology on the space of G N (cid:0) R d (cid:1) = G (cid:0) R d (cid:1) -valued paths.Case N = 3: In this case, the above condition reduces to min([ p ] , p/ , p/ > p ≥ 4. But by assuming p ≥ ROM RANDOM WALKS TO ROUGH PATHS 9 group.Case N ≥ 4: In this case, min([ p ] , p/ , p/ , p/ , . . . ) > h N − i = 1 sothat m [ p/m ] > ≥ m ∈ { , . . . , N } . In particular we can take m = 4 and see 4 [ p/ ≥ ⇒ [ p/ ≥ ⇒ p ≥ 4. Then again, we can make the remark that under the as-sumption p ≥ Corollary 4. Assume E (cid:16) | ξ | p (cid:17) < ∞ for some real p ≥ . Then the rescaled (step-2) lift of the (rescaled, piecewise linearly connected) random walk W ( n ) t convergesin α -H¨older for any < α < p ∗ − p ∗ ≡ α ∗ ( p ) where p ∗ = min([ p ] , p/ . 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