A Fourier analysis based approach of rough integration

Transcription

1 A Fourier analysis based approach of rough integration Massimiliano Gubinelli Peter Imkeller Nicolas Perkowski Université Paris-Dauphine Humboldt-Universität zu Berlin Le Mans, October 7, 215 Conference celebrating Vlad Bally s 6th birthday Typeset by FoilTEX

2 Integration Background: understand fdg for trajectories of stochastic processes; these are rough, e.g. only α-hölder continuous as for Brownian motion: α < 1 2 : rough path analysis f, g : [, 1] R; Riemann-Stieltjes theory: g of bounded variation with (signed) interval measure m g on the Borel sets of [, 1]: t f(s)dg(s) = t f(s)dm g (s). f of bounded variation with interval measure m f, integration by parts: t f(s)dg(s) = f(t)g(t) f()g() t g(s)dm f (s). Remark: obvious tradeoff between regularity of f and of g. Young s integral: f is α-, g β-hölder, and α + β > 1, fdg defined. Aim: Present Fourier based approach to Young s integral, embedded in new approach of rough paths. Typeset by FoilTEX 1

3 Application: rough SPDE Application goal: approach SPDE with rough path techniques in the spirit of Hairer s paper on regularity structures E. g.: on torus u(t, x) = Au(t, x)+g(u(t, x))du(t, x) + ξ(t, x), t with u : R + T d R n, A = ( ) σ fractional Laplacian with σ > 1 2, D spatial gradient, ξ space-time white noise Typeset by FoilTEX 2

4 Fourier decomposition and Young s integral Fourier decomposition of Hölder continuous functions f studied by Ciesielski (1961): f(t) = H pm, df G pm (t) p, m 2 p with piecewise linear G pm, p, m 2 p (Schauder functions). Then define t f(s)dg(s) = H pm, df H qn, dg p,m, q,n t G pm (s)dg qn (s). Lit: Baldi, Roynette 92: LDP; Ciesielski, Kerkyacharian, Roynette 93: calculus on Besov spaces; Roynette 93: BM on Besov spaces Typeset by FoilTEX 3

5 Haar and Schauder functions Define the Haar functions for p, 1 m 2 p and H = 1, H p =, p. H pm (t) = 2 p 1 [ m 1 2 p, 2m 1 2 p+1 ) (t) 2 p 1 [ 2m 1 2 p+1, m 2 p) (t) Haar functions form an orthonormal basis of L 2 ([, 1]). The primitives of the Haar functions G pm (t) = t H pm (s)ds, t [, 1], p, 1 m 2 p, are the Schauder functions. (H pm ) p, m 2 p is orthonormal basis. So if f = f(s)ds with f L 2 ([, 1]) (write f H) f(t) = t H pm, f H pm (s)ds = p, m 2 p H pm, f G pm (t) p, m 2 p Typeset by FoilTEX 4

6 Ciesielski s isomorphism Two observations: t pm = m 1 2 p, t 1 pm = 2m 1 2 p+1, t 2 pm = m 2 p ; then H pm, f = H pm df = 2 p [ 2f(t 1 pm) f(t pm) f(t 2 pm) ]. Hence H pm df c2 p(1 2 α) f α. Since G pm = 2 p/2 1, Schauder functions of one family with disjoint support 2 p ( ) H pm df G pm C2 αk f α. p K m= Thus series representation extends to closure of H w.r.t. α. This is C α, the space of α-hölder continuous functions. Typeset by FoilTEX 5

7 Ciesielski s isomorphism Define χ pm = 2 p 2 Hpm, ϕ pm = 2 p 2 Gpm, p, m 2 p. Then for p, m 2 p f = pm H pm, df G pm = pm 2 p χ pm, df ϕ pm = pm f pm ϕ pm, ϕ pm = 1 2, with f pm = 2 p χ pm, df = 2f(t 1 pm) f(t pm) f(t 2 pm). Since ϕ pm vanishes at t j pm for j =, 2, this implies that f p = q p 2 q m=1 f qm ϕ qn is the linear interpolation of f on the dyadic points t i pm, i =, 1, 2, m =,..., 2 p. Typeset by FoilTEX 6

8 Ciesielski s isomorphism According to observation on previous slide Ciesielski: Map f α = sup pm 2 pα f pm = sup pm 2 p(α 1 2 ) H pm, df f α. T α : C α l, f (2 pα f pm ) p,1 m 2 p isomorphism between a function space and a sequence space. Can be extended to Besov spaces B α p,q normed by α,p,q : for a function f : [, 1] R, < α < 1, 1 p, q, t [, 1] 1 ω p (t, f) = sup [ f(x + y) f(x) p dx] p, 1 f α,p,q = f p + [ y t 1 ( ω p(t, f) t α ) q1 t dt]1 q. Lit: Ciesielski, Kerkyacharian, Roynette 93: study of Brownian motion on Besov spaces, stochastic integral. Typeset by FoilTEX 7

9 Let now f C α, g C β. Then we may write Back to integration f = p,m f pm ϕ pm, g = p,m g pm ϕ pm. The Schauder functions are piecewise linear, thus of bounded variation. Therefore it is possible to define t f(s)dg(s) = = t f pm g qn ϕ pm (s)dϕ qn (s) p,m, q,n t f pm g qn ϕ pm (s)χ qn (s)ds. p,m, q,n To study the behaviour of the integrals on the rhs as functions of t we have to control for i, j p, m q, n 2 i χ ij, ϕ pm χ qn. Typeset by FoilTEX 8

10 Universal estimate, Paley-Littlewood packages Lemma 1 For i, p, q, j 2 i, m 2 p, n 2 q 2 i χ ij, ϕ pm χ qn 2 2(i p q)+p+q, except in case p < q = i, in which we have 2 i χ ij, ϕ pm χ qn 1. For f = pm f pmϕ pm as above let p f = 2 p m= f pm ϕ pm, S p f = q p q f. According to Ciesielski s isomorphism f C α iff f α = sup p (2 pα p f ) l <. Typeset by FoilTEX 9

11 Key lemma in Paley-Littlewood language Corollary 1 f, g continuous functions. For i, p, q, j 2 i, m 2 p, n 2 q i ( p f q g) 2 (i p q) i+p+q p f q g, except in case p < q = i, in which we have i ( p f q g) p f q g. For p > i or q > i we have i ( p f q g) =. Typeset by FoilTEX 1

12 Decomposition of the integral The corollary indicates that different components of the integral have different smoothness properties. We may write fdg = p,q p fd q g = p<q p fd q g + p q p fd q g = q S q 1 fd q g+ p p fd p g+ p p fds p 1 g. In view of the second part of Corollary 1, we expect the first part to be rougher. Integration by parts gives q S q 1 fd q g = S q 1 f q g q gds q 1 f q q = π < (f, g) q gds q 1 f. q Typeset by FoilTEX 11

13 Decomposition of the integral π < (f, g) : Bony paraproduct Defining further L(f, g) = p ( p fds p 1 g p gds p 1 f), (antisymmetric Lévy area) S(f, g) = p p fd p g = c p f p g p (symmetric part) we have fdg = π < (f, g) + S(f, g) + L(f, g). Typeset by FoilTEX 12

14 The Young integral In case the Hölder regularity coefficients of f and g are large enough, the three components of the integral behave well. According to the Corollary, we estimate for i i f i g i f i g 2 (α+β)i f α g β. This implies for any α, β ], 1[ S(f, g) α+β C f α g β. Similarly π < (f, g) β C f g β. and, but only if α + β > 1 L(f, g) α+β C f α g β. Typeset by FoilTEX 13

15 We can summarize the findings above. The Young integral Thm (Young s integral) Let α, β (, 1) be such that α + β > 1, and let f C α and g C β. Then I(f, dg) := p,q p fd q g C β and I(f, dg) β f α g β. Furthermore I(f, dg) π < (f, g) α+β f α g β. It is important to note that we get a version of the Lévy area only in case α + β > 1. If f and g arise in the context of Brownian motion, we usually have only α, β < 1 2, and Lévy area has to be given externally. Typeset by FoilTEX 14

16 Beyond Young s integral: an example The following example illustrates for α + β < 1 the Lévy area may fail to exist, and indicates what may be missing. Let f, g : [ 1, 1] R be given by f(t) := a k sin(2 k πt) and g(t) := k=1 a k cos(2 k πt), k=1 where a k := 2 αk and α [, 1]. For m N let f m, g m : [ 1, 1] R be the mth partial sum of the series. f m, g m are α-hölder continuous uniformly in m. For s, t [ 1, 1] and k N such that 2 k 1 s t 2 k with C independent of m by simple calculation: f m (t) f m (s) C t s α, g m (t) g m (s) C t s α. Hence also f, g α-hölder continuous. Typeset by FoilTEX 15

17 Beyond Young s integral: an example Lévy s area for (f m, g m ) given by 1 1 g m (s)df m (s) = = m k,l=1 m k,l= k π = a k a l 1 f m (s)dg m (s) 1 a k a l ( 2 l π 1 ( sin(2 k πs) sin(2 l πs)2 l π + cos(2 l πs) cos(2 k πs)2 k π ) ds (cos((2k 2 l )πs) cos((2 k + 2 l )πs))ds (cos((2 k 2 l )πs) + cos((2 k + 2 l )πs))ds ) m a 2 k2 k π= 2 k=1 m 2 (1 2α)k π. k=1 This diverges as m tends to infinity for α 1 2. Hence (f, g) possesses no Lévy area. Typeset by FoilTEX 16

18 Beyond Young s integral: an example Note that for 1 s t 1, and f g (s) R by trigonometry f(t) f(s) f g (s)(g(t) g(s)) = 2 a k sin(2 k 1 π(s t)) 1 + f g (s) 2 sin[2 k 1 π(s + t) + arctan((f g (s)) 1 )]. k=1 Hölder regularity for s =, t = 2 n, f g () > (f g () < analogous): quantity n 2 a k sin(2 k 1 n π) 1 + (f g ()) 2 sin[2 k 1 n π + arctan((f g ()) 1 )] k=1 2 αn sin ( π 2 + arctan((f g ()) 1 ) ) O( t s 2α ). Typeset by FoilTEX 17

19 Beyond Young s integral Hölder regularity at not better than α; hence f not controlled by g for α < 1 2 in the sense of following notion. (para)controlled path formalizes heuristics of fractional Taylor expansion. For α > let x C α. Then D α x = { f C α : f x C α s.t. f = f π < (f x, x) C 2α}. f D α x is called controlled by x, f x derivative of f w.r.t. x. On D α x define norm f x,α = f α + f x α + f 2α. If α > 1/3, then since 3α > 1 the term L(f π < (f x, x), x) is well defined. It suffices to make sense of L(π < (f x, x), x). This is done by commutator estimate: L(π < (f x, x), x) f x (s)dl(x, x)(s) 3α f x α x 2 α, and the integral is well defined provided L(x, x) exists. Typeset by FoilTEX 18

20 Beyond Young s integral Thm (Rough path integral) Let α (1/3, 1), α 1/2, α 2/3. Let x C α, f, g D α x. Assume that the Lévy area L(x, x) := lim N ( L(SN x k, S N x l ) ) 1 k d,1 l d converges uniformly, such that sup N L(S N x, S N x) 2α <. Then I(S N f, ds N g) = p N q N p f(s)d q g(s) converges in C α ε for all ε >. Denote the limit by I(f, dg). Then I(f, dg) D α x with derivative fg x, and I(f, dg) x,α f x,α ( 1 + g x,α )( 1 + x α + x 2 α + L(x, x) 2α ). Typeset by FoilTEX 19

21 Happy Birthday Vlad Typeset by FoilTEX 2

22 Rough quadratic variation and ergodic averages So far not clear what approach has to do with ergodic theory. Curious observation: For some < α < 1 let f C α. For p, 1 m 2 p, consider the sequence of dyadic intervals J pm = [ m 1 2 p, m 2 p [, and f pm = H pm, df. Then 2 p m=1 (f( m 1 2p) f(m 2 p )) 2 = 2 p p 1 2 q q= n=1 f 2 qn. Hence pathwise existence of quadratic variation reduces to Césaro summability of Ciesielski coefficients. In case of Brownian paths the Ciesielski coefficients are all i.i.d Gaussian variables W pm = H pm, dw, p, 1 m 2 p, and 2 p m=1 (W ( m 1 2p) W (m 2 p )) 2 = 2 p p 1 2 q q= n=1 W 2 qn, so that existence of quadratic variation is linked to law of large numbers. Typeset by FoilTEX 21

23 Rough quadratic variation and ergodic averages Goals: pathwise approach of quadratic variation via ergodic theory may need a pathwise concept of self similarity f self-similar e.g. if f pm = 1 for p, 1 m 2 p. Typeset by FoilTEX 22

24 An approximation of the Lévy area Aim: approximate Lévy s area by dyadic martingales. Filtration: F q = σ(χ 2 k +l : k q, l 2k 1), q. Martingales: M f q = p q N g q = p q χ pm, df χ pm, m<2 p χ pm, dg χ pm. m<2 p Rademacher functions: r q = 2 q/2 n<2 q χ qn and the associated martingale R q = p q r q. Typeset by FoilTEX 23

25 An approximation of the Lévy area Discrete time stochastic integral of X, Y : (X Y ) n = X k 1 Y k = X k 1 (Y k Y k 1 ). k n k n Then I(S k f, ds k g)= <q<k 2 q 2 E [ ] R q (N g q 1 M q f M f q 1 N q g ) f g + O(2 αk ) = 2 q 2 E [ [R, N g M f M f N g ] ] q <q<k f g + O(2 αk ). Expectation is w.r.t. Lebesgue measure on [, 1]. Typeset by FoilTEX 24

26 An approximation of the Lévy area A discrete analogue of Lévy s area appears naturally. Geometric interpretation: Proposition 1. M, N discrete time processes. Denote linear interpolation by X, Y respectively: X s = M k 1 + (s (k 1)) M k, s [k 1, k], similarly for Y. Then discrete time Lévy area 1 2 {((M M ) N) n ((N N ) M) n } equals the area between the curve {(X s, Y s ) : s n} and line chord from (M, N ) to (M n, N n ). Typeset by FoilTEX 25

27 Construction of the Lévy area For a d-dimensional process X = (X 1,..., X d ) we construct the area L(X, X) = L(X i, X j ) 1 i,j d. Assume the components are independent. Let R(s, t) = (E(X i sx j t )) 1 i,j d. Increment of R over rectangle [s, t] [u, v] R [s,t] [u,v] = R(t, v) + R(s, u) R(s, v) R(t, u) = (E(X i s,tx j u,v)) 1 i,j d. Let us make the following assumptions. ρ-var) There exist ρ [1, 2) and C > such that for all s < t 1 and for every partition s = t < t 1 < < t n = t of [s, t] n i,j=1 R [ti 1,t i ] [t j 1,t j ] ρ C t s. (HC) The process X is hypercontractive, i.e. for every m, n N and every p > 2 there exists C p,m,n > such that for every polynomial P : R n R of degree m, for all i 1,, i n, and for all t 1,..., t n [, 1] E( P (X i 1 t 1,..., X i n tn ) 2p ) C p,m,n E( P (X i 1 t 1,..., X i n tn ) 2 ) p. Typeset by FoilTEX 26

28 Construction of the Lévy area Lemma 3 Assume that the stochastic process Y : [, 1] R satisfies (ρ-var). Then for all p and for all M N 2 p N m 1,m 2 =M E(X pm1 X pm2 ) ρ (N M + 1)2 p. Lemma 4 Let Y, Z : [, 1] R be independent continuous processes, both satisfying (ρ-var) for some ρ [1, ]. Then for all i, p and all q < p, and for all j 2 i E 2 p 2 q m= n= 2 X pm Y qn 2 i χ ij, ϕ pm χ qn 2 (p i)(1/ρ 4) 2 (q i)(1 1/ρ) 2 i 2 p(4 3/ρ) 2 q/ρ Typeset by FoilTEX 27

29 Construction of the Lévy area Thm 2 Let X : [, 1] R d be a continuous stochastic process with independent components, and assume that X satisfies (ρ-var) for some ρ [1, 2) and (HC). Then for every α (, 1/ρ) almost surely N L(S N X, S N X) L(S N 1 X, S N 1 X) α <, and therefore the limit L(X, X) = lim N L(S N X, S N X) is almost surely an α-hölder continuous process. Condition (HC) is fulfilled by all Gaussian processes, also by all processes in fixed Gaussian chaos (Hermite processes), (ρ-var) by fractional Brownian motion or bridge of Hurst index H > 1 4. Typeset by FoilTEX 28