C 1 -path x t x 1 (f(x u), dx u ) rough path analyi p-variation (1 < p < 2) rough path 2 Introduction f(x) = (fj i(x)) 1 i n,1 j d (x R d ) (n, d) Cb

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1 Rough path analyi 1 x t ( t 1) R d path f(x) = t (f 1 (x),, f d (x)) R d R d - C x t 1 (f(x u), dx u ) Stieltje path x t p-variation norm (1 < p < 2) x p := { up D } 1/p N x ti x ti 1 p (D = { = t < < t N = 1} ) lim m(d) i=1 N (f(x i 1 ), x ti x ti 1 ) (t i 1 i 1 t i ) (1.1) i=1 Young integral Path γ- Hölder γ > 1 p p-variation norm Brown path 2 < p < 3 p-variation norm Young integral (1.1) iterated integral (x(u) x()) dx(u) Rd R d Terry Lyon rough path analyi 1. Young integral path x t rough path iterated integral (x(u) x()) dx(u) 2. x t path Young integral iterated integral (x(u) x()) dx(u) rough path x t R d R d x 2 (, t), t x t = y t x 2 (, t) y 2 (, t) 3. x t x 2 (, t) (Chen ) (Young integral implicit x t x ) 4. Brown path rough path analyi Brown path rough path Chen rough path 1

2 C 1 -path x t x 1 (f(x u), dx u ) rough path analyi p-variation (1 < p < 2) rough path 2 Introduction f(x) = (fj i(x)) 1 i n,1 j d (x R d ) (n, d) Cb R d path x t ( t 1) Riemann-Stieltje I,t (x) := f(x u )dx u = f i j(x u )dx j u, (, t [, 1]) (2.1) R n - path x t Brown T.Lyon ( [8, 9]) rough path analyi x t path(c 1 path) I,t (x) Ĩ,t (x) := f(x )(x t x ), (2.2) ( ) J,t (x) := f(x )(x t x ) + ( f)(x ) (x u x ) dx u. (2.3) a = d i=1 ai e i, b = d i=1 bi e i, c = d i=1 ci e i, e i = t (,..., i 1,..., ) [( f)(x)(a b)] i = [ ( )] i ( f)(x ) (x u x ) dx u = [ ( 2 f)(x)(a b c) ] i = 1 j,k d 1 j,k d 1 k,l d f i j x k (x)a k b j (2.4) f i j x k (x ) ([ ] i i ) Ĩ,t (x) I,t (x) J,t (x) I,t (x) = = + [ { 1 } ] f(x ) + ( f)(x + θ(x u x ))dθ (x u x ) dx u [f(x ) + ( f)(x )(x u x )] dx u { 1 } ( f)(x + θ(x u x ))dθ ( f)(x ) (x u x )dx u = J,t (x) + { 1 ( θ (x k u x k )dx j u (2.5) 2 f i j x l x k (x)a l b k c j (2.6) ) } ( 2 f)(x + r(x u x ))dr dθ (x u x ) (x u x )dx u =: J,t (x) + R,t (x) (2.7) 2

3 I,t (x) J,t (x) C x u x 2 ẋ u du. (2.8) [, 1] D := { = t < t 1 < < t N = 1} D, t ( ) t i Ĩ,t (x, D) := Ĩ ti 1,t i (x) (2.9) J,t (x, D) := I,t (x) = t i 1,t i t,t i 1,t i D t i 1,t i t,t i 1,t i D t i 1,t i t,t i 1,t i D J ti 1,t i (x) (2.1) I ti 1,t i (x) (2.11) I,t (x) = lim Ĩ,t (x, D) (2.12) m(d) I,t (x) = lim m(d) J,t(x, D), (2.13) m(d) D (2.12) Stieltje (2.13) T.Lyon Taylor f(x u ) 2 (2.13) p-variation norm Definition 2.1 path {x t } t 1 = {(, t) t 1} R d R d R d x 1 (, t) = x(t) x() (2.14) x 2 (, t) = (x(u) x()) dx(u) (2.15) ψ(, ) : V (V ) q-variation norm ψ q = up D { n 1 } 1/q ψ(t i, t i+1 ) q i= (2.16) D := { = t < t 1 < < t N = 1} ψ = x 1, x 2 [, t] q-variation norm ψ q,[,t] Theorem < p < 3 x t, y t R d path t 1 f(x u )dx u max { x 1 p, ȳ 1 p, x 2 p/2, ȳ 2 p/2 } R < (2.17) max { x 1 ȳ 1 p, x 2 ȳ 2 p/2 } ε (2.18) C(R, p, f) R, p, f 3 upnorm f(y u )dy u ε C (R, p, f), (2.19) 3

4 d Brown path w(t) Theorem 2.3 (P n w)(t) { k 2 } n 2n k= w(t) path path (P n w) 1, (P n w) 2 w lim max{ (P nw) n,m 1 (P m w) 1 p, (P n w) 2 (P m w) 2 p/2 } = (2.2) Brown path w Theorem2.3 path (P n w) 1, (P n w) 2 Cauchy Theorem 2.2 (, t) lim n I,t (P n w) I,t (w) = f(w u)dw u (P n w) 1, (P n w) 2 p-variation norm, p/2-variation norm w 1, w 2 Theorem 2.2 x 1, x 2, ȳ 1, ȳ 2 w 1, w 2, η 1, η 2 ( η 1, η 2 Brown path η ) Theorem 2.4 w lim I,t(P n w) = n lim (P nw) n 2 (, t) = Stratonovich 1. I,t (w) Stratonovich 2. w 2 (, t) Stratonovich f(w u ) dw u (2.21) (w(u) w()) dw(u) (2.22) f(w u ) dw u w (w(u) w()) dw(u) path p-variation, p/2-variation norm Theorem 2.2 (2.13) 3 I,t (x) path x [, t] [, t] variation norm control function T.Lyon Definition 3.1 ω(, ) : [, ) control function u t 1 Example 3.2 path x t q > ω(, u) + ω(u, t) ω(, t). (3.1) ω(, t) = x 1 q q,[,t] + x 2 q/2 q/2,[,t] (3.2) (3.1) ω q = p control function 4

5 3.1 f(x u)dx u Theorem 3.3 path x t control function ω, t x 1 (, t) ω(, t) 1/p (3.3) x 2 (, t) ω(, t) 2/p (3.4) ( I,t (x) = f(x u )dx u C(f, p) ω(, t) 1/p + ω(, t) 2/p + ω(, t) 3/p), (3.5) C(f, p) f 2 up-norm p 1 2 iterated integral variation norm Lemma 3.4 N 2 D = { = t < t 1 < < t N = t} t i 1 2ω(, t) ω(t i 1, t i+1 ) N 1. (3.6) Proof. (N 1) min i ω(t i 1, t i+1 ) = N 1 i=1 ω(t i 1, t i+1 ) j,2j+2 N ω(t 2j, t 2j+2 ) + l,2l+3 N ω(t 2l+1, t 2l+3 ) 2ω(, t). (3.7) Proof of Theorem 3.3 D = D {, t} J,t (x, D) = J,t (x, D) D = { = t < < t N = t} N 2 (3.6) i D 1 := D \ {t i } J,t (x, D) J,t (x, D 1 ) J,t (x, D) J,t (x, D 1 ) = J ti 1,t i (x) + J ti,t i+1 (x) J ti 1,t i+1 (x) = ( f(x ti ) f(x ti 1 ) ) ( x ti+1 x ti ) < u < t x 2 (, t) = = = u u + f(x ti 1 ) x 2 (t i 1, t i ) + f(x ti ) x 2 (t i, t i+1 ) f(x ti 1 ) x 2 (t i 1, t i+1 ). (3.8) (x(r) x()) dx(r) (x(r) x()) dx(r) + (x(r) x()) dx(r) + u u (x(r) x()) dx(r) (x(r) x(u)) dx(r) + (x(u) x()) (x(t) x(u)) = x 2 (, u) + x 2 (u, t) + x 1 (, u) x 1 (u, t) (3.9) 5

6 J,t (x, D) J,t (x, D 1 ) = f(x ti 1 ) (x(t i ) x(t i 1 )) ( ) x ti+1 x ti [ 1 { } ] + ( f)(x ti 1 + θ(x ti x ti 1 )) ( f)(x ti 1 ) dθ (x(t i ) x(t i 1 )) ( ) x ti+1 x ti + f(x ti ) x 2 (t i, t i+1 ) f(x ti 1 ) x 2 (t i, t i+1 ) f(x ti 1 ) x 1 (t i 1, t i ) x 1 (t i, t i+1 ) = R(f, x, t i 1, t i+1 ) [ x 1 (t i 1, t i ) x 1 (t i 1, t i ) x 1 (t i, t i+1 )] +S(f, x, t i 1, t i ) [ x 1 (t i 1, t i ) x 2 (t i, t i+1 )], (3.1) R(f, x, t i 1, t i+1 ) = S(f, x, t i 1, t i ) = 1 1 ( θ ( 2 f) ( x ti 1 + τ(x ti x ti 1 ) ) ) dτ dθ (3.11) ( ) ( 2 f) x ti 1 + θ(x ti x ti 1 ) dθ. (3.12) t i J,t (x, D) J,t (x, D 1 ) { (2ω(, ) t) 3/p C 2 f + N 1 ( 2ω(, t) N 1 ) 1/p ( ) } 2ω(, t) 2/p N 1 C ( 2ω(, t) N 1 ) 3/p 2 f. (3.13) J,t (x, D 1 ) t i D 2 J,t (x, D) (f(x )x 1 (, t) + f(x )x 2 (, t)) [ N ( ) ] 2ω(, t) 3/p C 2 f.(3.14) k 1 lim m(d) J,t (x, D) = I,t (x) 2 < p < 3 Remark 3.5 (1) lim m(d) J,t (x, D) I,t (x) (2.8), (2.11) D = { = t < < t N = 1}, D [, 1] D J,t (x, D) J,t (x, D ) C 2 f k 1 k=2 1 k 3/p max ω(t i 1, t i ) 3 p 1 ω(, 1). (3.15) 1 i N lim m(d) J,t (x, D) path x t rough path Young (2) Young (2.12) x t γ = 1+ε 2 -Hölder (ε > ) x 1 p < (1 < p < 2) m(d) Young x p-variation norm ω(, t) := x 1 p p,[,t] Hölder ω(, t) = C t, x 1 (, t) ω(, t) γ 6

7 3.2 Theorem 2.2 I,t (x) I,t (y) Theorem 2.2 control function Theorem 3.6 path x t, y t control function ω, t max { x 1 (, t), y 1 (, t) } ω(, t) 1/p (3.16) max { x 2 (, t), y 2 (, t) } ω(, t) 2/p (3.17) x 1 (, t) y 1 (, t) εω(, t) 1/p (3.18) x 2 (, t) y 2 (, t) εω(, t) 2/p. (3.19) f(x u )dx u f(y u )dy u εc(f, ω(, 1), p)ω(, t) 1/p. (3.2) C(f, ω(, 1), p) ω(, 1), p, f 3 up-norm Proof. N 2 D = { = t < < t N = t} D k ω x t, y t J,t (x, D) J,t (y, D) N 2 k= { J,t (x, D k ) J,t (x, D k 1 ) } { J,t (y, D k ) J,t (y, D k 1 ) } + J,t (x) J,t (y). (3.21) (3.1) { J,t (x, D k ) J,t (x, D k 1 ) } { J,t (y, D k ) J,t (y, D k 1 ) } ( ) 2ω(, t) 3/p ( C ε 2 f + 3 ) f. (3.22) N k 1 J,t (x) J,t (y) m(d) Theorem 3.6 Theorem 2.2 Proof of Theorem 2.2 Control function ω ω(, t) = x 1 p p,[,t] + ȳ 1 p p,[,t] + x 2 p/2 p/2,[,t] + ȳ 2 p/2 p/2,[,t] + ( ε 1 x 1 ȳ 1 p,[,t] ) p + ( ε 1 x 2 ȳ 2 p/2,[,t] ) p/2. (3.23) Theorem 3.6 Theorem 2.2 7

8 4 x path f(x u)dx u pathx t, y t 1 x dy 1 f(t, w)dw(t) Itô 1. d = 1 driving path x t 1 f(x) = t (f 1 (x),..., f d (x)) ((F i ) (x) = f i (x)) F i (x t ) F i (x ) = f i (x u )dx u x t 2 (x u x ) dx u = (x u x )dx u = (x t x ) 2 p/2-variation norm x p-variation norm 2 driving path 2. Itô path x t f(x t ) = f(x ) + 2 (4.1) ( f)(x )dx (4.2) x t Brown path Stratonovich ( path ) Itô Theorem path x t T 2 (R d ) = R R d (R d R d ) x(, t) = (1, x 1 (, t), x 2 (, t)) x mooth rough path (mooth rough path ) x 1, x 2 (2.14), (2.15) x 1 firt level path, x 2 econd level path T 2 (R d ) (a, a 1, a 2 ) (b, b 1, b 2 ) = (a b, a b 1 + a 1 b, a 2 b + a b 2 + a 1 b 1 ) (4.3) (R d truncated tenor algebra) < u < t x(, t) = x(, u) x(u, t) (4.4) Chen(K.T.Chen), x 1 (, t) = x 1 (, u) + x 1 (u, t) ( < u < t) (3.9) t < t 1 < < t N 1 x 1 (t, t N ) = x 2 (t, t N ) = N 1 i= N 1 i= x 1 (t i, t i+1 ) (4.5) N 1 x 2 (t i, t i+1 ) + i=1 x 1 (t, t i ) x 1 (t i, t i+1 ) (4.6) 8

9 J,t (x, D) (2.13) I,t (x) Theorem 3.3 Chen x 1, x 2 path (3.3), (3.4) x 2 path iterated integral (4.4), (3.3), (3.4) roughne p rough path (2 < p < 3) (3.3), (3.4) control function Chen C( T 2 (R d )) x = ( x 1, x 2 ) x 1 p-variation norm x 2 p/2-variation norm Roughne p rough path Ω p (R d ) x, ȳ Ω p (R d ) d p ( x, ȳ) = x 1 ȳ 1 p + x 2 ȳ 2 p/2 Ω p (R d ) Smooth rough path rough path d p rough path geometric rough path 4. Almot rough path y 1 (, t) := J,t (x) (4.7) y 2 (, t) := f(x ) f(x ) (x 2 (, t)) (4.8) f(x) f(x)(a b) = (f(x)a) (f(x)b) y 1 (, t), y 2 (, t) Chen ȳ = (1, ȳ 1, ȳ 2 ) almot rough path < u < t ȳ 1 (, u) + ȳ 1 (u, t) ȳ 1 (, t) Cω(, t) θ (4.9) ȳ 2 (, u) + ȳ 2 (u, t) + ȳ 1 (, u) ȳ 1 (u, t) ȳ 2 (, t) Cω(, t) θ (4.1) (θ = 3 p > 1) C, u, t (4.9) (3.1) I,t (x) ȳ 2 z 2 (, t) := I,u (x) di,u (x) { N 1 = lim m(d) i= ȳ 2 (t i, t i+1 ) + N 1 i=1 ) } ȳ 1 (t k, t k+1 ) ȳ 1 (t i, t i+1 ) ( i 1 k= (4.11) rough path rough path x rough path z = (1, z 1, z 2 ) almot rough path rough path econd level path (I,t (x) ) x rough path x z 5. x path ODE ż t = g(z t )ẋ t (4.12) z = a R d (4.13) g( ) C b (RN, M N,d ) (M N,d N d ) driving path x t 9

10 Theorem 4.1 (4.12) I(x) t Driving path y t I(y) t x, y Theorem 3.6 (I(x) t I(x) ) (I(y) t I(y) ) εc(r, p, f)ω(, t) 1/p. (4.14) C(R, p, f) R, p, f 3 up-norm ω Theorem 2.2 Theorem 2.2 d p Theorem 4.1 path x, y rough path Ω p (R d ) rough path ODE Picard ( ) ( ) ( ) ẋt 1 ẋt = (4.15) g(z t ) ẑ t = t (x t, z t ) R d+n ż t f(x, z) = ( 1 g(z) ż t ) M (d+n),(d+n) ẑ t ẑ t = f(ẑ t ) ẑ t Picard ẑ t (n) = t (a, ) + f (ẑ u (n 1)) dẑ u (n 1) (4.16) ẑ t () = (x t, ) (4.17) lim n ẑ t (n) ẑ t ẑ t rough path rough path ẑ ( 1, I,t (ẑ), I,u(ẑ) di,u (ẑ) ) 6. H 1 p-variation Smooth rough path H 1 ([, 1] R d ) C p Propoition 4.2 path x t x 1 p x H 1, x 2 p/2 x 1 p x H Theorem 2.3 Brown path w t < γ < 1/2 γ- Hölder w w 1 p < (2 < p < 3) (w(u) w()) dw(u) w (, t) p/2-variation norm w 2 (, t) 1

11 Lemma 4.3 κ > p 2 1 w t n k = k 2 n. w 2 p/2 p/2 C p,κ n=1 n κ 2n k=1 ( w 1 (t n k 1, tn k ) p + w 2 (t n k 1, tn k ) p/2). (4.18) Chen T 2 (R d ) Theorem roughne p (n < p < n + 1) rough path 9. p-variation norm Hölder norm Support theorem, Large deviation, Laplace method, Wiener rough path C 3 ( [2] potential function ), weak Poincaré inequality ( [3] path ( ) weak Poincaré, Weak Poincaré inequality S. Kuuoka [1], [1] ) A.Lejay lejay/rough.html rough path, Reference [1] S. Aida, Uniform Poitivity Improving Property, Sobolev Inequality and Spectral Gap, J. Funct.Anal., 158 (1998) no.1, [2] S. Aida, Semiclaical limit of the lowet eigenvalue of a Schrödinger operator on a Wiener pace, J.Funct.Anal. 23 (23), no.2, [3] S. Aida, Weak Poincaré inequalitie on domain defined by Brownian rough path, to appear in the Annal of Probability. [4] P. Friz, Continuity of the Ito-Map for Hölder rough path with application to the upport theorem in Hölder norm, in [5] M. Ledoux, T. Lyon and Z. Qian, Lévy area of Wiener procee in Banach pace, The Annal of Probability, 3. (22), No.2, [6] M. Ledoux, Z. Qian and T. Zhang, Large deviation and upport theorem for diffuion via rough path, Stochatic procce and their application, 12 (22), No.2, [7] A. Lejay, An Introduction to Rough Path, Séminaire de probabilité XXXVII, Lecture Note in Mathematic (Springer-Verlag), (23). [8] T. Lyon, Differential equation driven by rough ignal, Rev.Mat.Iberoamer., 14 (1998),

12 [9] T. Lyon and Z. Qian, Sytem control and rough path, (22), Oxford Mathematical Monograph. [1] M. Röckner and F-Y. Wang, Weak Poincaré inequalitie and L 2 -Convergence Rate of Markov Semigroup, J.Funct.Anal. 185 (21), no.2,

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull

Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and