Quantum stochastic calculus applied to path spaces over Lie groups

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1 Quantum stochastic calculus applied to path spaces over Lie groups Nicolas Privault Département de Mathématiques Université de La Rochelle Avenue Michel Crépeau 1742 La Rochelle, France Abstract Quantum stochastic calculus is applied to the proof of Skorokhod and Weitzenböck type identities for functionals of a Lie group-valued Brownian motion. In contrast to the case of R d -valued paths, the computations use all three basic quantum stochastic differentials. Key words: Quantum stochastic calculus, Lie group-valued Brownian motion. Mathematics Subject Classification. 6H7, 81S25, 58J65, 58C35. 1 Introduction Quantum stochastic calculus [4, [7, and anticipating stochastic calculus [6 have been linked in [5, where the Skorokhod isometry is formulated and proved using the annihilation and creation processes. On the other hand, a Skorokhod type isometry has been constructed in [3 for functionals on the path space over Lie groups. This isometry yields in particular a Weitzenböck type identity in infinite-dimensional geometry. We refer to [1, [2 for the case of path spaces over Riemannian manifolds. We will prove such a Skorokhod type isometry formula on the path space over a Lie group, using the conservation operator which is usually linked to stochastic calculus for jump processes. In this way we will recover the Weitzenböck formula established 1

2 in [3. This provides a link between the non-commutative settings of Lie groups and of quantum stochastic calculus. This paper is organised as follows. In Sect. 2 we recall how the Skorokhod isometry can be derived from quantum stochastic calculus in the case of R d -valued Brownian motion. In Sect. 3 the gradient and divergence operators of stochastic analysis on path groups are introduced, and the Skorokhod type isometry of [3 is stated. The proof of this isometry is given in Sect. 4 via quantum stochastic calculus on the path space over a Lie group. Sect. 4 ends with a remark on the links between vanishing of torsion and quantum stochastic calculus. 2 Skorokhod isometry on the path space over R d In this section we recall how the Skorokhod isometry is linked to quantum stochastic calculus. Let (B(t)) t R+ denote an R d -valued Brownian motion on the Wiener space W with Wiener measure µ. Let S = {G = g(b(t 1 ),..., B(t n )) : g Cb ((R d ) n ), t 1,..., t n > }, and { } U = u i G i : G i S, u i L 2 (R + ; R d ), i = 1,..., n, n 1. Let D : L 2 (W ) L 2 (W R + ; R d ) be the closed operator given by D t G = 1 [,ti (t) i g(b(t 1 ),,..., B(t n )), t, for G = g(b(t 1 ),..., B(t n )) S, and let δ denote its adjoint. Thus E[Gδ(u) = E[ DG, u, G Dom(D), u Dom(δ), where Dom(D) and Dom(δ) are the respective domains of D and δ. We let, denote the scalar product in both L 2 (R + ; R d ) and L 2 (W R + ; R d ), and let (, ) denote the scalar product on R d. Since D is a derivation, we have the divergence relation δ(ug) = Gδ(u) u, DG, G S, u U. 2

3 Given u L 2 (R + ; R d ), the quantum stochastic differentials da (t) and da + (t) are defined from and a u G = a + v G = They satisfy the Itô table u(t)da (t)g = DG, u, G Dom(D), (2.1) v(t)da + (t)g = δ(vg), G Dom(D). (2.2) dt da v (t) da + v (t) dt da + u (t) da u (t) (u(t), v(t))dt with da u (t) = u(t)da (t) and da + v (t) = v(t)da + (t). Using the Itô table we have and a u a + v = a + v a u = da + v (s)da u (t) + da u (s)da + v (t) + which implies the canonical commutation relation This relation and its proof can be abbreviated as da u (s)da + v (t) + da + v (s)da u (t), u(t)v(t)dt a u a + v = u, v + a + v a u. (2.3) d[a u, a + v (t) = [da u (t), da + v (t) = (u(t), v(t))dt, where [, denotes the commutator of operators. Relation (2.3) is easily translated back to the Skorokhod isometry: E [δ(uf )δ(vg) = a + u F, a + v G = F, a u a + v G = u F, v G + F, a + v a u G = u F, v G + a v F, a u G [ = E[ u, v F G + E (u(t) D s F, D t G v(s))dsdt, 3

4 F, G S, u, v L 2 (R + ; R d ), which implies E[δ(h) 2 = E[ h 2 L 2 (R + ;R d ) + E [ (D s h(t), (D t h(s)) )dsdt, h U, where (D t h(s)) denotes the adjoint of D t h(s) in R d R d. In this note we carry over this method to the proof of a Skorokhod type isometry on the path space over a Lie group, using the calculus of all the annihilation, creation and gauge (or conservation) process and the Itô table where (q(t)) t R+ defined from dt da v (t) da + v (t) q(t)dλ(t) dt da + u (t) da u (t) (u(t), v(t))dt q (t)u(t)da (t) p(t)dλ(t) p(t)v(t)da + (t) p(t)q(t)dλ(t) is a (bounded) measurable operator process on R d and q(t)dλ(t) is q(t)dλ(t)f = δ(q( )D F ), for F Dom(D) such that (q(t)d t F ) t R+ Dom(δ). 3 Skorokhod isometry on the path space over a Lie group Let G be a compact connected d-dimensional Lie group with associated Lie algebra G identified to R d and equipped with an Ad-invariant scalar product on R d G, also denoted by (, ). The commutator in G is denoted by [,. Let ad(u)v = [u, v, u, v G, with Ad e u = e adu, u G. The Brownian motion (γ(t)) t R+ on G is constructed from (B(t)) t R+ via the Stratonovich differential equation { dγ(t) = γ(t) db(t) γ() = e, where e is the identity element in G. Let IP(G) denote the space of continuous G- valued paths starting at e, with the image measure of the Wiener measure by I : (B(t)) t R+ (γ(t)) t R+. Let S = {F = f(γ(t 1 ),..., γ(t n )) : f C b (G n )}, 4

5 and { } Ũ = u i F i : F i S, u i L 2 (R + ; G), i = 1,..., n, n 1. Definition 3.1 For F = f(γ(t 1 ),..., γ(t n )) S, f C b (Gn ), we let DF L 2 (W R + ; G) be defined as DF, v = d ( dε f γ(t 1 )e ε 1 v(s)ds,..., γ(t n )e ε ) tn v(s)ds ε=, v L 2 (R +, G). In other terms, D acts as a natural gradient on the cylindrical functionals on IP(G) with D t F = i f(γ(t 1 ),..., γ(t n ))1 [,ti (t), t. Let δ denote the adjoint of D, that satisfies E[F δ(v) = E[ DF, v, F S, v L 2 (R + ; G), (3.1) (that δ exists and satisfies (3.1) can be seen as a consequence of Lemma 4.1 below). Given v L 2 (R + ; G) we define q v (t) = ad(v(s))ds, t >. Definition 3.2 ([3) The covariant derivative of u L 2 (R + ; G) in the direction v L 2 (R + ; G) is the element v u of L 2 (R + ; G) defined as follows: v u(t) = q v (t)u(t) = ad(v(s))u(t)ds, t >. In the following we will distinguish between v which acts on L 2 (R + ; G) and q v (t) which acts on G and is needed in the quantum stochastic integrals to follow. The operators q v (t) and v are antisymmetric on G and L 2 (R + ; G) respectively, because (, ) is Ad-invariant. The Skorokhod isometry on the path space over G holds for the covariant derivative. The definition of v extends to Ũ, as v (uf )(t) = u(t) DF, v + F q v (t)u(t), t >, F S, u L 2 (R + ; G). 5

6 Let s (uf )(t) G G be defined as e i e j, s (uf )(t) = u(t), e j e i, D s F + 1 [,t (s)f e j, ad(e i )u(t), i, j = 1,..., d. In this context the following isometry has been proved in [3. Theorem 3.3 ([3) We have for h Ũ: [ E[ δ(h) 2 = E[ h 2 L 2 (R + ;G) + E ( s h(t), ( t h(s)) )dtds. (3.2) The proof in [3 is clear and self-contained, however its calculations involve a number of coincidences which are apparently not related to each other. In this paper we provide a short proof which offers some explanation for these. Let the analogs of (2.1)-(2.2) be defined as ã u F = and ã + u F = dã u (t)f = DF, u, dã + u (t)f = δ(uf ), u L 2 (R + ; G), i.e. dã u (t)f = (u(t), D t F )dt. Our proof relies on F S, F S, a) the relation dã u (t) = da u (t) + q u (t)dλ(t), t >, u L 2 (R + ; G), (3.3) see Lemma 4.1 below, b) the commutation relation between ã u and ã + v which is analogous to (2.3) and is proved via quantum stochastic calculus in the following lemma. Lemma 3.4 We have on S: ã u ã + v ã + v ã u = u, v + ã + vu ã+, uv (3.4) u, v L 2 (R + ; G). 6

7 Proof. Using the quantum Itô table, Relation (3.3), Lemma 4.2 below and the fact that q v(t) = q v (t), we have dã u (t) dã + v (t) dã + v (t) dã u (t) = ( da u (t) + q u (t)dλ(t) ) (da + v (t) q v (t)dλ(t) ) ( da + v (t) q v (t)dλ(t) ) (da u (t) + q u (t)dλ(t) ) = (u(t), v(t))dt + q v (t)u(t)da (t) q u (t)q v (t)dλ(t) +q u (t)v(t)da + (t) + q v (t)q u (t)dλ(t) = (u(t), v(t))dt + v u(t)da (t) + q vu(t)dλ(t) + u v(t)da + (t) q uv(t)dλ(t) = (u(t), v(t))dt + dã vu (t) + dã+ uv (t). This commutation relation can be interpreted to give a proof of the Skorokhod isometry (3.2): Proof of Th Applying Lemma 3.4 we have E [ δ(uf ) δ(vg) = ã + u F, ã + v G = F, ã u ã + v G = u F, v G + ã v F, ã u G + F v u, DG + DF, G u v [ = E[ u, v F G + E (F s u(t) + D s F u(t), G( t v(s)) + v(s) D t G)dsdt [ = E[ u, v F G + E s (uf )(t), ( t (vg)(s)) dsdt, F, G S, u, v L 2 (R + ; G). We mention that a consequence of Th. 3.3 is the following Weitzenböck type identity, cf. [3, which extends the Shigekawa identity [8 to path spaces over Lie groups: Theorem 3.5 ([3) We have for u Ũ: E[ δ(u) 2 + E [ [ du 2 L 2 (R + ;G) L 2 (R + ;G) = E[ u 2L 2(R+) + E u 2 L 2 (R + ;G) L 2 (R + ;G). The next section is devoted to two lemmas that are used to prove (3.3). 7 (3.5)

8 4 Quantum stochastic differentials on path space The following expression for D using quantum stochastic integrals can be viewed as an intertwining formula between D, D and I. Lemma 4.1 We have for v L 2 (R + ; G): Proof. dã v (t) = da v (t) + q v (t)dλ(t), t >. The process t γ(t)e v(s)ds satisfies the following stochastic differential equation in the Stratonovich sense: d (γ(t)e ) t v(s)ds = γ(t)e ( t v(s)ds Ade ) t v(s)ds db(t) + v(t)dt = γ(t)e v(s)ds ( e qv(t) db(t) + v(t)dt ), t >. Let I 1 (u) denote the Wiener integral of u L 2 (R + ; R d ) with respect to (B(t)) t R+, and let G = g (I 1 (u 1 ),..., I 1 (u n )) S, and F = f(γ(t 1 ),..., γ(t n )) S. Since exp(q v (t)) : G G is isometric, we have from the Girsanov theorem: E [ ( f γ(t 1 )e 1 v(s)ds,..., γ(t n )e ) [ tn v(s)ds G = E F e I 1(v) 1 2 v 2 Θ v G, where ( Θ v G = g u 1 (s)e qv(s) db(s) u 1, v,..., ) u n (s)e qv(s) db(s) u n, v. From the derivation property of D and the divergence relation δ(vg) = Gδ(v) v, DG we have d dε Θ εvg ε= = = = i g (I 1 (u 1 ),..., I 1 (u n )) d dε Θ εvi 1 (u i ) ε= ( i g (I 1 (u 1 ),..., I 1 (u n )) v, u i + ) qv(s)u i (s)db(s) i g (I 1 (u 1 ),..., I 1 (u n )) ( v, DI 1 (u i ) + δ( v DI 1 (u i ))) = v, Dg (I 1 (u 1 ),..., I 1 (u n )) i g (I 1 (u 1 ),..., I 1 (u n )) δ( v DI 1 (u i )) 8

9 = v, DG 1 2 δ( i g (I 1 (u 1 ),..., I 1 (u n )) v DI 1 (u i )) j i g (I 1 (u 1 ),..., I 1 (u n )) ( u j, v u i + u i, v u j ) i,j=1 = v, DG δ( v DG), since v is antisymmetric. Hence E[ DF, v G = d ( [f dε E γ(t 1 )e ε 1 h(s)ds,..., γ(t n )e ε ) tn h(s)ds Using the identity δ(v) = I 1 (v) we have = d [F dε E e εi 1(v) 1 2 ε2 v 2 Θ εv G ε= = E [F (GI 1 (v) v, DG δ( v DG)). E[ DF, v G = E [F (Gδ(v) v, DG δ( v DG)) G ε= = E [F (δ(vg) δ( v DG)) [ ( ) = E F a + v q v (t)dλ(t) [ ( ) = E G a v + q v (t)dλ(t) F G. It follows from the proof of Lemma 4.1 that D admits an adjoint δ that satisfies δ(uf ) = a + u F and q u (t)dλ(t)f, F S, E[F δ(u) = E[ DF, u, F S, u Ũ. Letting we have ã + u F = dã + u (t)f = δ(uf ), dã + u (t) = da + u (t) q u (t)dλ(t). The following Lemma shows that [ v, u = vu uv. 9

10 This means that the Lie bracket {u, v} associated to the gradient on L 2 (R + ; G) via [ u, v = {u,v} satisfies {u, v} = u v v u, i.e. the connection defined by on L 2 (R + ; G) has vanishing torsion. Lemma 4.2 We have [q u (t), q v (t) = q uv(t) q vu(t), t >, u, v L 2 (R + ; G). Proof. The Jacobi identity on G shows that [ [q u (t), q v (t) = ad(u(s))ds, ad(v(s))ds = = s ad([u(τ), v(s))dτds ad(q u (s)v(s))ds = q uv(t) q vu(t). ([ = ad s ad(q v (s)u(s))ds u(s)ds, ad([v(τ), u(s))dτds ) v(s)ds The Lie derivative on IP(G) in the direction u L 2 (R + ; G), introduced in [3, can be written ã u in our context. Finally we show that the vanishing of torsion discovered in [3 can be obtained via quantum stochastic calculus. Precisely, the Lie bracket {u, v} associated to ã v via [ã u, ã v = ã {u,v} satisfies {u, v} = uv v u, i.e. the connection defined by on IP(G) also has a vanishing torsion. Proposition 4.3 We have on S: u, v L 2 (R + ; G). ã u ã v ã v ã u = ã vu ã uv. Proof. Using Lemma 4.2, the quantum Itô table implies dã u (t) dã v (t) dã v (t) dã u (t) = ( da u (t) + q u (t)dλ(t) ) (da v (t) + q v (t)dλ(t) ) ( da v (t) + q v (t)dλ(t) ) (da u (t) + q u (t)dλ(t) ) = q u (t)v(t)da (t) + q u (t)q v (t)dλ(t) q v (t)u(t)da (t) q v (t)q u (t)dλ(t) 1

11 = u v(t)da (t) q uv(t)dλ(t) ( v u(t)da (t) q vu(t)dλ(t)) = dã uv (t) dã vu (t), from Lemma 4.1. References [1 A.B. Cruzeiro and P. Malliavin. Renormalized differential geometry on path space: Structural equation, curvature. J. Funct. Anal., 139: , [2 B. Driver. A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold. J. Funct. Anal., 11(2): , [3 S. Fang and J. Franchi. Platitude de la structure riemannienne sur le groupe des chemins et identité d énergie pour les intégrales stochastiques [Flatness of Riemannian structure over the path group and energy identity for stochastic integrals. C. R. Acad. Sci. Paris Sér. I Math., 321(1): , [4 R.L. Hudson and K. R. Parthasarathy. Quantum Ito s formula and stochastic evolutions. Comm. Math. Phys., 93(3):31 323, [5 J. M. Lindsay. Quantum and non-causal stochastic calculus. Probab. Theory Related Fields, 97:65 8, [6 D. Nualart and E. Pardoux. Stochastic calculus with anticipative integrands. Probab. Theory Related Fields, 78: , [7 K.R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkäuser, [8 I. Shigekawa. de Rham-Hodge-Kodaira s decomposition on an abstract Wiener space. J. Math. Kyoto Univ., 26(2):191 22,