Precise Asymptotics for Infinite Dimensional Itô-Lyons Maps of Brownian Rough Paths

Transcription

1 Precise Asymptotics for Infinite Dimensional Itô-Lyons Maps of Brownian Rough Paths Jointwork with Yuzuru INAHAMA (Nagoya Univ.) Hiroshi KAWABI (Department of Mathematics, Faculty of Science Okayama University, Japan) Newton Institute, Cambridge April 6, 2010

2 1 Introduction

3 Plan of This Talk: In this talk : : :, we discuss precise asymptotics for the laws of solutions of infinite dimensional formal Stratonovich type SDEs dx " t = ff(x " t ) "dw t (+B("; X " t )dt): We give a rigorous meaning of the solution through the Itô-Lyons map of RDEs in the rough path theory. (Rough path theory works on any Banach space!!) The main example is Malliavin s loop group-valued Brownian motion (, 1 2 Gross), and heat processes & heat kernel measures can be concrete examples to which rough path theory can be applied.

4 We present the following asymptotic expansion formula for the Laplace type functional integration Eˆ exp `` F (X " )=" 2 e`f Λ( )=" 2 ˆ` 0 + 1"+ + n" n +O(" n+1 ) as " & 0: In our proof, the Wentzell-Friedlin type LDP lim "&0 "2 log Eˆ exp `` F (X " )=" 2 = `F Λ ( ) and, a stochastic Taylor expansion in the sense of rough paths play important roles.

5 2 The Main Example (Heat Processes & Heat Kernel Measures)

6 X := L 0 (R r ) = fx 2 C([0; 1]; R r )j x(0)=x(1)=0g H := H 1 0 (R r ) j L 0 (R r ) : Cameron-Martin space := r 0 : pinned Wiener measure on L 0 (R r ) (L 0 (R r ); H 1 0 (R r ); r 0) is an abstract Wiener space. fw t g t 0 : X-valued BM with w 0 = 0 on a probability space (Ω; F; P), i.e., 1X w t ( ) = i(t) i ( ); f i g j H : CONS: i=1

7 {w t } t 0 = {w t (τ ) :0 τ 1} t 0 = {(w 1 t (τ ),..., wr t (τ )) : 0 τ 1} t 0 can be regarded as a collection of r-dim BMs with hw j (fi ); w ḱ (fi 0 )i t = t(fi ^ fi 0` fi fi 0 ) jk ; t 0 for 0 τ, τ 1, 1 j, k r. Following P. Malliavin (1990), we construct BM on L 0 (M) := fx 2 Xj x(fi ) 2 M; 0» fi» 1g. (M: compact smooth Riemannian manifold)

8 Step 1: Let " > 0. We consider a collection of the following fi (2 [0; 1])-wise SDEs on R r : 8 >< dx t " (fi ) = P r j=1 A j(x t " (fi )) "dw j t (fi ); (?) >: X 0 (fi ) = 0; (Scaled) Heat Process where A j := (a ij ) r i=1 2 C 1 b (R r ; R r ); 1» j» r. Malliavin (1990), Driver (1997), Fang-Zhang (2001),... Brzeźniak-Elworthy (2000): M-type 2 Banach space framework

9 9fX " (t; fi )g t 0;0»fi»1 : continuous modification by Kolmogorov-Totoki s criterion. (Malliavin, 90) fx t " ( )g t 0 fx 1 " 2 t ( )g t 0 =) X p T 1 ( ) XT 1 ( ) by " := p T ; t := 1. Definition "; T > 0, fa j g r j=1 : as above. (1) V " : the law of (scaled) heat process fx " t ( )g t 0 ( probability measure on P (L 0 (R r )) ) (2) T : the law of X 1 T ( ) (heat kernel measure). ( probability measure on L 0 (R r ) )

10 Step 2: M(j R r ) with 0 2 M. P x : R r! T x M; x 2 M; projection. fe j g r j=1 j R r : ONS A j (x) := P x [e j ] = P r i=1 i ) x X " (t; fi ) 2 M for all (t; fi ): ) fx " (t; )g t 0 : L 0 (M)-valued continuous process fx 1 (t; )g t 0 is BM on L 0 (M). i.e., 1 2 Gross : generator of fx1 (t; )g t 0

11 Example: M = S m j R m+1 (Nash s embedding) M is not Lie group if m 6= 1; 3: M := fx = (x 1 ; x m+1 ) j x x 2 m + (x m+1 ` 1) 2 = 1g ) Consider the case of r = m + 1. Set a i;j (x)= i;j ` u i u j for x 2 M; 1» i; j» m + 1; where u i := x i ` i;m+1 : ) fx t " ( )g t 0 is (scaled-)bm on L 0 (M):

12 3 Our Framework via Rough Path Theory

13 (X; H; ) : an abstract Wiener space Y : another Banach space P (X):=fx 2 C([0; 1]; X)j x 0 = 0g w = fw t g t 0 : X-valued BM with w 0 = 0 () P " : probability measure on P (X) induced by "w) 2 < p < 3 : roughness (fixed) X X : closure of X a X w.r.t. a tensor norm j j, i.e., jx yj» jxj X jyj X In this talk, we consider the projective tensor norm j j := j j _ (largest cross norm on X a X).

14 Assumption: (Exactness of X X w.r.t. ) 9C >0, 0» 9 < 1 s.t., 8N 2N, 8fG l g 2N l=1 of independent X-valued random variables with common distribution, it holds that Eh NX i G 2l`1 G 2l» CN l=1 Remark: dim(x) < 1 =) = 1=2 If we use the injective tensor norm, exactness condition automatically holds!

15 GΩ p (X) : the set of geometric RPs on X w(n) : n-th dyadic polygonal approximation of w w(n) := (1; w(n) 1 ; w(n) 2 ) is the smooth RP associated with w(n): w(n) 1 (s; t) := w(n) t ` w(n) s ; w(n) 2 (s; t) := Z t s `w(n)u` w(n) s dw(n)u Ledoux-Lyons-Qian (2002): Under exactness condition, Brownian RP exists!, i.e., 9 w = lim n!1 w(n) in GΩ p(x); P-a.s.

16 "w := (1; "w 1 ; " 2 w 2 ); " > 0. H := L 2;1 0 (H) j P (X): Cameron-Martin space () (P (X); H; P 1 ): an abstract Wiener space) Theorem 1 [Ledoux-Qian-Zhang ( 02), Inahama-K.( 06)] Under exactness, the law of "w satisfies LDP on GΩ p (X) with the good rate function 8 < 1 2 I(x) = khk2 H; if 9 h 2 H s.t. x = h; : 1; otherwise.

17 Now, we consider the following formal Stratonovich type SDE on Y : 8 < dx t " = ff(x t " ) "dw t ; (}) : X 0 " = 0(2 Y ); " > 0; where ff 2 Cb 1 (Y; L(X; Y )). We want to regard X " as a random variable on P (Y ). (i.e., Y -valued Wiener functional) =) How to formulate via the rough path theory?

18 We consider the RDE (} 0 ) dy t = ff(y t )dx t ; y 0 = 0; under ff 2 C 1 b (Y; L(X; Y )). By Lyons Universal Limit Theorem,... For all x 2 GΩ p (X), the RDE (} 0 ) has the unique solution ȳ 2 GΩ p (Y ). Moreover the Itô-Lyons map Φ : GΩ p (X) 3 x 7`! ȳ 2 GΩ p (Y ) is (locally Lipschitz) continuous. We define the solution to the formal Stratonovich type SDE (}) by X " t := Φ("w) 1 (0; t):

19 For h 2 BV(X), Φ("h) 1 (0; ) 2 BV(Y ) solves the following usual ODE on Y : dx " t = ff(x " t) "dh t ; x " 0 = 0 (2 Y ); " > 0: Here, we observe that this framework is applicable to the main example (heat processes on loop space). X = Y = L 0 (R r ); H = H 1 0 (R r ); = r 0 (We can show exactness for any tensor product.) Define the Nemytski map ff: X! L(X; Y ) by ff(y)[x]x (fi ) := a(y(fi ))[x(fi )] R r (2 R r ) for x; y 2 L 0 (R r ); a = (a ij ) 1»i;j»r.

20 =) ff 2 C 1 b `L0 (R r ); L(L 0 (R r ); L 0 (R r )) For each fixed " > 0 and 0» fi» 1, P`Φ("w) 1 (0; t)(fi ) = X fi " (t); 0» t» 1 = 1 [ Wong-Zakai s approximation ] For each t 0, Φ("w) 1 (0; t)( ) 2 L 0 (R r ), P-a.s. [ Lyons Universal Limit Theorem ] =) (t; fi ) 7! Φ("w) 1 (0; t)(fi ) is a bi-continuous modification of (t; fi ) 7! X t " (fi ).

21 4 Precise Asymptotics

22 We present the Wentzell-Freidlin type LDP for fx " g Theorem 2 [ Inahama-K. ( 06)] We have lim "&0 "2 log Eˆ exp `` F (X " )=" 2 = `C ; where F : P (Y )! R is continuous bounded, and C := infff (y) + Λ(y)j y 2 P (Y )g; 8 1 >< inf khk 2 2 H j y = Φ(h) 1 (0; ) Λ(y) = if 9 h 2 H s.t. y = Φ(h) 1 (0; ); >: 1; otherwise.

23 Corollary fv " g and f T g satisfies LDP. Corresponding rate functions I 1 and I 2 are given by 8 1 >< inf k k 2 2 H j = Φ( ) 1 (0; ) I 1 ( ) = if 9 2 H s.t. = Φ( ) 1 (0; ); >: I 2 (y) = 8 >< 1; otherwise. 1 inf k k 2 2 H j y = Φ( ) 1 (0; 1) if 9 2 H s.t. y = Φ( ) 1 (0; 1); >: 1; otherwise. ( 2 P (L 0 (R r )); y 2 L 0 (R r ))

24 S. Fang-T.S. Zhang ( 01): loop group-valued BM Combine Theorem 1 & the contraction principle (since the Itô-Lyons map Φ is continuous!) Laplace s Method: More precise asymptotics of Eˆ exp `` F (X " )=" 2? Schilder ( 66) BM Azencott ( 82), Ben Arous ( 88) finite-dim SDEs Kusuoka-Stroock(-Osajima) ( 91, 94, 08) Generalized Wiener functionals Takanobu-Watanabe ( 94) Rovira-Tindel ( 01) (Parabolic, Hyperbolic) SPDEs

25 Assumptions: (H1): F; G : P (Y )! R are continuous & bounded. (H2): F Λ (h) := F (Φ(h) 1 ) khk2 H; h 2 H attains its minimun C at a unique point 2 H. (=) C = F Λ ( ) ) (H3): F : (n + 3)-times Fréchet differentiable, G : (n + 1)-times Fréchet differentiable, on a nbd B(ffi) of ffi := Φ( ) 1, and fd i F g n+3 i=1 ; fdj Gg n+1 j=1, are bounded on B(ffi). (H4): A := D 2 [F (Φ(h) 1 ( ) is )] HˆH strictly larger than `Id H. i.e., (Ah; h) H > `khk 2 H.

26 Remark: (i) dffi t = ff(ffi t )d t ; ffi 0 = 0: (ii) A: self-adjoint & Hilbert-Schmidt operator on H. Theorem 3 [ Inahama-K. ( 07)] We have the following asymptotic expansion formula as " & 0: EˆG(X " ) exp `` F (X " )=" 2 e`f Λ( )=" 2` 0 + 1"+ + n" n +O(" n+1 ) : Here the explicit value of the coefficient 0 can be written in terms of G(ffi) and det 2 (Id H + A)`1=2.

27 Remark: If we deal with more general drift B("; X " t ), an additional term e`c( )=" appears in our asymptotic expansion formula. Remark: Quite recently, Inahama studies this topic in more general frameworks: more general coefficients case, i.e., dy " t = ff("; Y " t ) "dw t + B("; Y " t )dt fractional Brownian motion case on finite dimensions (3» p < 4 ) we need 3-rd level path estimates). (=) ICM Satellite@India, SPA conference@osaka)

28 5 Sketch of the Proof

29 In our proof of Theorem 3, we need... Fernique type theorem for Brownian RP Cameron-Martin type theorem for Brownian RP ) shifted Brownian RP: w + 2 GΩ p (X) for 2 H. (w + ) 1 (s; t) := w 1 (s; t) + ( t ` s ); (w + ) 2 (s; t) := w 2 (s; t) + + Z t s Z t ( u ` s ) w 1 (s; du) + s w 1 (s; u) d u ; Z t Wentzell-Freidlin type LDP (Theorem 2) s ( u ` s ) d u : (Stochastic) Taylor expansion in the sense of rough paths based on approaches of Azencott ( 82) & Ben Arous ( 88).

30 (Aida ( 07): finite-dim case with derivative equations) How to establish the expansion around 2 BV(X)? For simplicity, we consider the case of Drift=0. d ˆX " t =ff( ˆX " t )d("h + ) t ; for fixed 2 BV(X) i.e., ˆX " t =Φ("h + ) 1 (0; t); where Φ : GΩ p (X)!GΩ p (Y ) (Itô-Lyons map w.r.t. ff). We define ffi := ffi 0 by dffi t = ff(ffi t )d t, and set ffi := ˆX " ` ffi 0 = "ffi 1 + " 2 ffi " n ffi n +

31 ) d(ffi 0 + ffi) = ff(ffi 0 + ffi)d("h + ) X 1 1 n! rn ff(ffi 0 )[ ffi; : : : ; ffi; d("h + )] n=0 ) By picking up terms of " n, we have a series of linear ODEs for fffi n := ffi n (h; )g 1 n=1 : dffi n ` rff(ffi 0 )[ffi n ; d ] Example: = terms of (ffi 0 ; ffi 1 ; : : : ; ffi n`1 ; ; h) (2 Y ): dffi 1 ` rff(ffi 0 )[ffi 1 ; d ] = ff(ffi 0 )dh, dffi 2 ` rff(ffi 0 )[ffi 2 ; d ] = rff(ffi 0 )[ffi 1 ; dh] r2 ff(ffi 0 )[ffi 1 ; ffi 1 ; d ];

32 For given k 2 BV(Y ), we consider an ODE (]) dy t ` (rff)(ffi 0 )[y t ; d t ] = dk t ; y 0 = 0: Lemma [ An extension of Duhamel ] (1) Let fm t g be the solution of L(Y; Y )-valued ODE dm t [ ] Y = (dω t )[M t [ ] Y ] Y ; where dω t [ ] Y := rff(ffi 0 t )[ ; d t ] Y ˆY : Then y t := M t R t 0 M `1 s dk s solves the ODE (]). (2) The map k 2 BV(Y ) 7! y 2 BV(Y ) extends to a continuous map GΩ p (Y )! GΩ p (Y ).

33 Key-Proposition 1 (1) The map (h; ) 7! (h; ffi 0 ; ffi 1 ; : : : ; ffi n ) extends to a continuous map from GΩ p (X) ˆ BV(X) to GΩ p (X Y n+1 ). (2) Moreover, 9!=!(s; t): control function s.t 8h; 2 BV(X) with k k 1» r 0, k = 1; : : : ; n, jffi k t ` ffi k s j» (1 + kh 1 k p + kh 2 k 1=2 p=2 )k!(s; t) 1=p ; and!(0; 1) is dominated by a positive constant c = c(r 0 ) which may depend on n, but not on h; 2 BV(X) with k k 1» r 0.

34 We define the remainder term R n " := R n " (h; ) by R n " := ˆX "`ffi 0`"ffi 1`" 2 ffi 2` ` " n`1 ffi n`1 : Key-Proposition 2 (1) For each " > 0, the map (h; ) 7! ("h; ˆX " ; ffi 0 ; ffi 1 ; : : : ; ffi n`1 ; R n " ) extends to a continuous map from GΩ p (X) ˆ BV(X) to GΩ p (X Y n+2 ). (2) Moreover, 9!=!(s; t): control function s.t. 8 2 BV(X) with k k 1» r 0 and 8h 2 BV(X) with k"h 1 k p + k"h 2 k 1=2 p=2» r 1,

35 jr" n (h; ) t ` R" n (h; ) s j n!(s;» " + k"h 1 k p + k"h 2 kp=2 1=2 t) 1=p ; and!(0; 1) is dominated by a positive constant c = c(r 0 ; r 1 ) which depends only on r 0 ; r 1. By combining these theorems, we have the following asymptotic expansion formula: n+2 X F (Φ("w + ) 1 )= " m J (m) F (ffi)(w)+ R (n+3) F ("; ffi)(w) m=0 J (0) (1) F (ffi) = F (ffi); J F (ffi) = DF (ffi)[ffi1 (w)]; :::

36 6 Further Remarks and Problems

37 We compare RDEs (Stratonovich theory) with usual (Itô-)SDEs in an infinite dimensional framework. X = Y : an abstract Wiener space (B; H; ) ff 2 Cb 1 (B! L (2) (H; H)) ) Consider an (Itô-)SDE on B: dx t = A(X t )dw t ; A(z) := Id + ff(z): ) Moreover if we impose ff 2 Cb 1 (B; L(B; H)), dx t = A(X t ) dw t ` Q 2 (X t )dt; where Q 2 (z) := 1 1X (rff)(z)[ff(z)e i ; e i ] HˆH dt 2 i=1

38 ) In this case, we can understand RDEs, SDEs. Brzeźniak-Carroll ( 03): Wong-Zakai s approximation on M-type 2 Banach space framework? How should we understand Φ("w) 2? ( Driver-Gordina ( 08): 1-dim Heisenberg groups??)? Can we apply our results to Stochastic PDEs (RPDEs)? ( Caruana-Friz-Oberhauser, Gubinelli-Lejay-Tindel,...)? Stationay phase method?, i.e., precise asymptotics of h E exp ``F + p`1g (X 2 i " )="

39 Thank you very much for your attention!!