Letting p shows that {B t } t 0. Definition 0.5. For λ R let δ λ : A (V ) A (V ) be defined by. 1 = g (symmetric), and. 3. g

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1 4 Contents.1 Lie group p variation results Suppose G, d) is a group equipped with a left invariant metric, i.e. Let a := d e, a), then d ca, cb) = d a, b) for all a, b, c G. d a, b) = d e, a 1 b ) = a 1 b. For x C [, T ] G) and Π P s, t) let, τ x := x 1 τ x τ for τ Π. We then set V p x : Π) := d p ) ) 1/p ) 1/p x t, x t = t x p and V p x) := t Π sup V p x : Π). Π P,T ) We also define 1/p ) 1/p ρ x, y : Π) := d p t x, t y)) = t x) 1 t y p and set, t Π ρ x, y) := Then ρ x, e) = V p x) < and t Π t Π sup ρ x, y : Π)..1) Π P,T ) ρ x, y) V p x) + V p y) <. Definition.1. Let C,p [, T ], G) := {x C [, T ] G) : x ) = e and V p x) < }. Proposition.2. The function, ρ, defined in Eq..1) is a metric on C,p [, T ], G) and if G, d) is complete then so is C,p [, T ], G), ρ). Theorem.3 Kolmogorov s Continuity Criteria). Suppose that Ω, F, P ) is a probability space and X t : Ω S is a process for t [, T ] where S, ρ) is a complete metric space. Assume there exists positive constants, ε, β, and C, such that E[ρ X t, X s ) ε ] C t s 1+β.2) for all s, t [, T ]. Then for any α, β/ε) there is a modification, X, of X i.e. P X t = X ) t = 1 for all t) which is α Hölder continuous. Moreover, there is a random variable K α such that, and EK p α < for all p < β αε 1 α. ρx t, X s ) K α t s α for all s, t [, T ].3) Corollary.4. Let B t : WT R{ be } the projection map, B t ω) = ω t). Then there is a modifications, {B t } of Bt for which t B t is α Hölder continuous ν T almost surely for any α, 1/2). Proof. Applying Theorem.3 with ε := p { and } β := p/2 1 for any p 2, ) shows there is a modification {B t } t of Bt which is almost surely α Hölder continuous for any α, β/ε) =, p/2 1 ) =, 1/2 1/p). p Letting p shows that {B t } t is almost surely α Hölder continuous for all α < 1/2..2 Homogeneous Metrics on G V ) and G geo V ) We now go back to the specific case at hand. In our case the groups G and G geo are equipped with a dilation structure. Definition.5. For λ R let δ λ : A V ) A V ) be defined by δ λ α + a + A) := α + λa + λ 2 A where α R, a V, and A V V. We call δ λ the dilation isomorphism. Proposition.6. For each λ R, δ λ : A V ) A V ) is an isomorphism of algebras. Moreover δ λ restricts to a group isomorphism of G V ) and G geo V ) and to a Lie algebra isomorphism of Lie G and Lie G geo. Definition.7 Homogeneous Norm). A homogeneous norm on G or G geo ) is a continuous function, : G [, ) such that: 1. g G = iff g = 1, 2. δ λ g) = λ g homogeneous) for all λ R, 3. g 1 = g symmetric), and 4. gh g + h subadditive). Theorem.8. Let V be a Banach space and G = G 2 V ). For g = 1+g 1 +g 2 G = G V ), let Then; g G := γ g) + γ g 1) where γ g) := max g 1, ) 2 g 2. Page: 4 job: last_day macro: svmonob.cls date/time: 15-Mar-213/8:23

2 1. G is a homogeneous norm on G and by restriction on G geo. 2. γ g) g G 3γ g) for all g G. 3. d g, h) := g 1 h G for g, h G defines a left invariant homogeneous i.e. d δ λ g), δ λ h)) = λ d g, h)) metric on G. Theorem.9. Let X : G or G geo ) be a continuous multiplicative functional and Y C [, T ] G) be associated to X via, Y t) := X,t or equivalently by, X st = Y s) 1 Y t) for all s, t). Then X is a p rough path iff Y has finite p variation relative to d. Theorem.1 Enhanced Brownian Motion). Let {B t } t be a R d valued Brownian motion. Then for all α, 1/2), there exists a X st = 1 + B t B s + X 2 st G geo R d ) such that B t B s + X 2 st 1/2 Cα t s α a.s., where C α is a an a.s. positive finite random constant, i.e. C α is independent of s, t but not of the sample point ω. Proof. Let Y t := 1 + B t + B τ db τ = Y t := 1 + B t + B τ db τ + tc where C := d i=1 e i e i which we assume to be chosen to be continuous. We then define X st := Ys 1 Y t. Since we have, X st = Therefore, Y 1 s 1 B s s = 1 + B t B s + = 1 + B t B s + = 1 + B t B s + = 1 B s s B τ db τ + B 2 s s s B τ db τ B τ db τ + B 2 s ) 1 + B t + s B τ db τ B s B t B s ) B τ B s ) db τ a.s. B τ db τ ) B τ db τ + B 2 s B s B t.2 Homogeneous Metrics on G V ) and G geo V ) 5 E [d p Y s, Y t )] = E [ d p 1, Ys 1 )] Y t = E [d p 1, X st )] [ ) 1/2 p] CE B t B s + B τ B s ) db τ. Let b σ := B s+σ B s a new Brownian motion and T := t s, then the above equation may be written as, T 1/2 p E [d p Y s, Y t )] CE b T + b σ db σ s [ T 1 1/2 ) p] = CE b1 + T b σ db σ = C p, n) T p/2 = C p, n) t s p/2. For the second line we have used Lemma.11 below and the Brownian scaling, d = T b T 1 ), to conclude that b and therefore, bσ + b σ+ ) bσ+ b σ ) d= T bt 1 σ + b T 1 σ + ) bt 1 σ + b T 1 σ) T 1 d b σ db σ = T b σ db σ. As p is arbitrary, it now follows by an application of Kolmogorov s continuity criteria Theorem.3 as in the proof of Corollary.4, that almost surely, d Y s, Y t ) C α t s α where α can be chosen to be any point in, 1/2). Lemma.11. Suppose that {a t } t and {b t } t are independent real valued Brownian motions and let M t := a τ db τ. Then for all p < there is a constant c p < such that E M T p c 2 pt p <. In fact, there exists ε > [ such that E e ε M T 2] < but we do not prove this here. Proof. Let Π be a partition of [, T ] and let S Π := τ Π a τ bτ b τ ). Then freezing a we are going to use Fubini s theorem), S Π is a Gaussian random variable. Therefore by the scaling arguments used above, Page: 5 job: last_day macro: svmonob.cls date/time: 15-Mar-213/8:23

3 E b S Π p = c p E b S Π 2) p/2 and therefore, ) p/2 = c p a τ 2 τ = c p T p/2 a τ 2 τ T τ Π τ Π c p T p/2 a τ p τ T τ Π E S Π p = E a E b S Π p = c p T p/2 1 τ Π E a aτ p τ = c p T p/2 1 τ Π c p τ p/2 τ c 2 pt p. ) p/2 The result now follows by Fatou s lemma and the fact that M T = lim Π S Π where the limit is taken in L 2 or in probability. Remark: another proof of this result could be based on hypercontractivity arguments.

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