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 nprivaul@univ-lr.fr 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
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
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
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
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
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
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. 3.3. 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)
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
= 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
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
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