aa r X i v : . [ m a t h . P R ] F e b HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS
TAI MELCHER
Abstract.
This paper studies Brownian motion and heat kernel measure ona class of infinite dimensional Lie groups. We prove a Cameron-Martin typequasi-invariance theorem for the heat kernel measure and give estimates on the L p norms of the Radon-Nikodym derivatives. We also prove that a logarithmicSobolev inequality holds in this setting. Contents
1. Introduction 12. Preliminaries 22.1. Abstract Wiener spaces 22.2. Extensions of Lie algebras 33. Semi-infinite Lie algebras and groups 53.1. Examples 73.2. Hilbert-Schmidt norms 103.3. Length and distance 123.4. Ricci curvature 164. Brownian motion 184.1. Multiple Itˆo integrals 184.2. Brownian motion and finite dimensional approximations 235. Heat kernel measure 315.1. Quasi-invariance and Radon-Nikodym derivative estimates 325.2. Logarithmic Sobolev inequality 33References 341.
Introduction
We define Brownian motion on a class of infinite dimensional Lie algebras whichwe call semi-infinite Lie algebras.
We then prove a Cameron-Martin type quasi-invariance result for the associated heat kernel measure, as well as a logarithmicSobolev inequality. A particular example of these semi-infinite Lie algebras wastreated in [10], and we build on the methods used there.We briefly describe here the main results and give an outline of the paper; seeSections 2 and 3 for definitions. Let (
W, H, µ ) be an abstract Wiener space and v bea finite dimensional Lie algebra equipped with an inner product. Let g = W ⊕ v be aLie algebra extension of W by v , and we will call g CM = H ⊕ v the Cameron-Martin Mathematics Subject Classification.
Primary 60J65 28D05; Secondary 58J65 22E65.
Key words and phrases.
Heat kernel measure, infinite dimensional Lie group, quasi-invariance,logarithmic Sobolev inequality.
Lie subalgebra of g . If g is nilpotent, we may define an explicit group operationon g via the Baker-Campbell-Hausdorff-Dynkin formula, and W ⊕ v equipped withthis group operation will be denoted by G . Similarly, G CM = H ⊕ v with the samegroup operation is called the Cameron-Martin subgroup of G , and we equip G CM with the left invariant Riemannian metric which agrees with the inner product h ( A, a ) , ( B, b ) i g CM = h A, B i H + h a, b i v on g CM ∼ = T e G CM .In Section 2, we set the notation and give some standard facts needed aboutabstract Wiener spaces and extensions of Lie algebras. In Section 3, we constructthe semi-infinite Lie algebras and give some examples. We make some additionalrequirements so that the Lie bracket on g is continuous, making g into a BanachLie algebra. In Section 3.2, this gives bounded Hilbert-Schmidt norms for the Liebracket, and, in Section 3.4, lower bounds on the Ricci curvature of G and a uniformlower bound on certain finite dimensional approximations of G .In Section 4, we define Brownian motion on G as the solution to a stochasticdifferential equation with respect to a Wiener process on g . Let B t denote Brownianmotion on g . Then, Brownian motion on G is the solution to the Stratonovichstochastic differential equation δg t = g t δB t := L g t ∗ δB t , with g = e = (0 , . For t >
0, let ∆ n ( t ) denote the simplex in R n given by { s = ( s , · · · , s n ) ∈ R n : 0 < s < s < · · · < s n < t } . Let S n denote the permutation group on (1 , · · · , n ), and, for each σ ∈ S n , let e ( σ ) denote the number of “errors” in the ordering ( σ (1) , σ (2) , · · · , σ ( n )), that is, e ( σ ) = { j < n : σ ( j ) > σ ( j + 1) } . Then the Brownian motion on G may bewritten as g t = r − X n =1 X σ ∈S n (cid:18) ( − e ( σ ) (cid:30) n (cid:20) n − e ( σ ) (cid:21)(cid:19) Z ∆ n ( t ) [[ · · · [ δB s σ (1) , δB s σ (2) ] , · · · ] , δB s σ ( n ) ] , where this sum is finite since g is assumed to be nilpotent. In Section 4, we show thatthese stochastic integrals are well-defined and each may be expressed as a sum ofiterated Itˆo integrals. We also show that g t may be realized as a limit of Brownianmotions living on the finite dimensional approximations to G . In particular, weshow in Proposition 4.9 that this convergence holds in L p , for all p ∈ [1 , ∞ ).In Theorem 5.3, we apply the previous results and a theorem from [11] to provethat ν t = Law( g t ) is invariant under (right or left) translation by elements of G CM . Moreover, this theorem gives good bounds on the L p -norms of the Radon-Nikodym derivatives. These results are important for future applications to spacesof holomorphic functions on G , as in [12]. We also show in Theorem 5.7 that alogarithmic Sobolev inequality holds for polynomial cylinder functions on G .For heat kernel analysis, quasi-invariance results, and logarithmic Sobolev in-equalities in related infinite dimensional settings, see [1, 17].2. Preliminaries
Abstract Wiener spaces.
In this section, we summarize several well knownproperties of Gaussian measures and abstract Wiener spaces that are required for
EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 3 the sequel. For proofs of these results, see Section 2 of [10]. Also see [6, 19] formore on abstract Wiener spaces and some particular examples.Suppose that W is a real separable Banach space and B W is the Borel σ -algebraon W . Definition 2.1.
A measure µ on ( W, B W ) is called a (mean zero, non-degenerate) Gaussian measure provided that its characteristic functional is given by(2.1) ˆ µ ( u ) := Z W e iu ( x ) dµ ( x ) = e − q ( u,u ) , for all u ∈ W ∗ , for q = q µ : W ∗ × W ∗ → R a symmetric, positive definite quadratic form. That is, q is a real inner product on W ∗ . Theorem 2.2.
Let µ be a Gaussian measure on a real separable Banach space W .For ≤ p < ∞ , let (2.2) C p := Z W k w k pW dµ ( w ) . For w ∈ W , let k w k H := sup u ∈ W ∗ \{ } | u ( w ) | p q ( u, u ) and define the Cameron-Martin subspace H ⊂ W by H := { h ∈ W : k h k H < ∞} . Then (1)
For all ≤ p < ∞ , C p < ∞ . (2) H is a dense subspace of W . (3) There exists a unique inner product h· , ·i H on H such that k h k H = h h, h i H for all h ∈ H , and H is a separable Hilbert space with respect to this innerproduct. (4) For any h ∈ H , k h k W ≤ √ C k h k H . (5) If { k j } ∞ j =1 is an orthonormal basis of H and ϕ is a bounded linear mapfrom W to a real Hilbert space C , then (2.3) k ϕ k H ∗ ⊗ C := ∞ X j =1 k ϕ ( k j ) k C = Z W k ϕ ( w ) k C dµ ( w ) < ∞ . A simple consequence of (2.3) is that(2.4) k ϕ k H ∗ ⊗ C ≤ k ϕ k W ∗ ⊗ C Z W k w k W dµ ( w ) = C k ϕ k W ∗ ⊗ C . Extensions of Lie algebras.
Suppose v is a Lie algebra and Der( v ) is the setof derivations on v . That is, Der( v ) consists of all linear maps ρ : v → v satisfyingLeibniz’s rule: ρ ([ X, Y ] v ) = [ ρ ( X ) , Y ] v + [ X, ρ ( Y )] v . Der( v ) forms a Lie algebra with Lie bracket defined by the commutator:[ ρ , ρ ] = ρ ρ − ρ ρ , for ρ , ρ ∈ Der( v ) . Der( v ) is a subset of linear maps on v , so if v is a normed vector space, one mayequip Der( v ) with the usual norm(2.5) k ρ k = sup {k ρ ( X ) k v : k X k v = 1 } . TAI MELCHER
Now suppose that h and v are Lie algebras, and that there is a linear mapping α : h → Der( v )and a skew-symmetric bilinear mapping ω : h × h → v , satisfying, for all X, Y, Z ∈ h ,(B1) [ α X , α Y ] − α [ X,Y ] h = ad ω ( X,Y ) and(B2) X cyclic ( α X ω ( Y, Z ) − ω ([ X, Y ] h , Z )) = 0 . Then, one may verify that, for X + V , X + V ∈ h ⊕ v ,[ X + V , X + V ] g := [ X , X ] h + ω ( X , X ) + α X V − α X V + [ V , V ] v defines a Lie bracket on g := h ⊕ v , and we say g is an extension of h over v . Thatis, g is the Lie algebra with ideal v and quotient algebra g / v = h . The associatedexact sequence is 0 → v ι −→ g π −→ h → , where ι is inclusion and π is projection. In fact, the following theorem (see, forexample, [2]) states that these are the only extensions of h over v . Theorem 2.3.
Isomorphism classes of extensions of h over v (that is, short exactsequences of Lie algebras → v → g → h → ) modulo the equivalence described bythe commutative diagram of Lie algebra homomorphisms −−−−→ v −−−−→ g −−−−→ h −−−−→ id y ϕ y id y −−−−→ v −−−−→ g ′ −−−−→ h −−−−→ , correspond bijectively to equivalence classes of pairs of linear maps α : h → Der( v ) and skew-symmetric bilinear maps ω : h × h → v satisfying (B1) and (B2), where ( α, ω ) ≡ ( α ′ , ω ′ ) if there exists a linear b : h → v such that α ′ X = α X + ad b ( X ) , and ω ′ ( X, Y ) = ω ( X, Y ) + α X b ( Y ) − α Y b ( X ) − b ([ X, Y ]) + [ b ( X ) , b ( Y )] v . The corresponding isomorphism ϕ : g → g ′ is given by ϕ ( X + V ) = X − b ( X ) + V . When v = V is an abelian Lie algebra, these pairs consist of a Lie algebrahomomorphism α : h → gl ( V ) and ω ∈ H ( h , V ) is a Chevalley cohomology classwith coefficients in the h -module V (see [16], Chapter 1, Sections 3.1 and 4.5).For definitions and details on extensions of Lie algebras, see Section XIV.5 of [7].Reference [2] also gives a nice (although unpublished) summary. Reference [26]gives some conditions under which the extension of h over v is nilpotent (when h and v are nilpotent); [22] gives a characterization of extensions of a Lie algebra overa Heisenberg Lie algebra. EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 5 Semi-infinite Lie algebras and groups
Throughout the rest of this paper (
W, H, µ ) will denote a real abstract Wienerspace, and v will denote a Lie algebra with dim( v ) = N < ∞ , equipped with aninner product h· , ·i v and a continuous Lie bracket [ · , · ] v . Note that this implies thatthere exists a constant c < ∞ such that k [ X, Y ] k v ≤ c k X k v k Y k v , for all X, Y ∈ v . For simplicity, we will assume that c ≡
1. Also, Der( v ) willdenote the derivations of v , equipped with the norm defined in (2.5). Definition 3.1.
Let (
W, H, µ ) be an abstract Wiener space and v a finite dimen-sional Lie algebra. Then g = W ⊕ v endowed with a Lie bracket satisfying(1) [ g , g ] ⊂ v , and(2) [ · , · ] : g × g → g is continuous,will be called a semi-infinite Lie algebra .Motivated by the discussion in Section 2.2, we may consider W as an abelian Liealgebra and construct extensions of W over v . So suppose there is a skew-symmetriccontinuous bilinear mapping ω : W × W → v and a continuous linear mapping α : W → Der( v )such that α and ω satisfy (B1) and (B2), which in this setting become(C1) [ α X , α Y ] = ad ω ( X,Y ) and(C2) α X ω ( Y, Z ) + α Y ω ( Z, X ) + α Z ω ( X, Y ) = 0 , for all X, Y, Z ∈ W . Then we may define a Lie algebra structure on g := W ⊕ v viathe Lie bracket[( X , V ) , ( X , V )] g := (0 , ω ( X , X ) + α X V − α X V + [ V , V ] v ) . The vector space g is also a Banach space in the norm k ( w, v ) k g := k w k W + k v k v , and g CM := H ⊕ v is a Hilbert space with respect to the inner product h ( A, a ) , ( B, b ) i g CM := h A, B i H + h a, b i v . The associated Hilbertian norm on g CM is given by k ( A, a ) k g CM := q k A k H + k a k v . Notation 3.2.
Let k ω k := sup {k ω ( w , w ) k v : k w k W = k w k W = 1 } and k α k := sup {k α w v k v : k w k W = k v k v = 1 } be the uniform norms of ω and α , which are finite by their assumed continuity. TAI MELCHER
It will be useful to note that(3.1) k [ · , · ] k := sup {k [ g , g ] k v : k g k g = k g k g = 1 } ≤ k ω k + 2 k α k + 1 < ∞ , and similarly(3.2) C := C ( ω, α ) := sup {k [ h, k ] k v : k h k g CM = k k k g CM = 1 } ≤ k [ · , · ] k < ∞ . Thus, for all ℓ = 1 , · · · , r − k ad ℓh k k v ≤ C ℓ k h k ℓ g CM k k k g CM . If v is nilpotent, ω and α may be chosen so that g is a nilpotent Lie algebra (seeSection 3.1 for some examples). For g nilpotent of step r , the Baker-Campbell-Hausdorff-Dynkin formula implies thatlog( e A e B ) = A + B + r − X k =1 X ( n,m ) ∈I k a kn,m ad n A ad m B · · · ad n k A ad m k B A, for all A, B ∈ g , where(3.3) a kn,m := ( − k ( k + 1) m ! n !( | n | + 1) , I k := { ( n, m ) ∈ Z k + × Z k + : n i + m i > ≤ i ≤ k } , and for each multi-index n ∈ Z k + , n ! = n ! · · · n k ! and | n | = n + · · · + n k , see, for example, [15]. Since g is nilpotent of step r ,ad n A ad m B · · · ad n k A ad m k B A = 0 if | n | + | m | ≥ r. for A, B ∈ g . In particular, one may verify that g · h = g + h + r − X k =1 X ( n,m ) ∈I k a kn,m ad n g ad m h · · · ad n k g ad m k h g (3.4)defines a group structure on g . Note that g − = − g and the identity e = (0 , Definition 3.3.
When we wish to emphasize the group structure on g , we willdenote g by G . Similarly, when we wish to view g CM as a subgroup of G , it will bedenoted by G CM and will be called the Cameron-Martin subgroup .(Since g is simply connected and nilpotent, the exponential map is a globaldiffeomorphism (see, for example, Theorems 3.6.2 of [25] or 1.2.1 of [9]), and wemay identify g and G under exponential coordinates. In particular, we may view g as both a Lie algebra and Lie group.) Lemma 3.4.
The Banach space topologies on g and g CM make G and G CM intotopological groups. Proof.
Since g and g CM are topological vector spaces, g g − = − g and( g , g ) g + g are continuous by definition. The map ( g , g ) [ g , g ] iscontinuous in both the g and g CM topologies by the estimates in equations (3.1)and (3.2). It then follows from (3.4) that ( g , g ) g · g is continuous as well. EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 7
Examples.
In this section, we give a few simple examples of semi-infinite Liealgebras.
Example . If v is a finite dimensional inner product space, we may consider v as an abelian Lie algebra, and taking α ≡ Example . Suppose v is an N -dimensional nilpotent Lie algebra. One standardway to construct Lie algebra extensions is as follows. Let β : W → v be a continuouslinear map, and define α : W → Der( v ) as the inner derivation α X := ad β ( X ) .In this case, (C1) and (C2) are both satisfied if ω : W × W → v is given by ω ( X, Y ) := [ β ( X ) , β ( Y )] v . Thus, g has Lie bracket[( X, V ) , ( Y, U )] g = (0 , [ β ( X ) , β ( Y )] v + [ β ( X ) , U ] v − [ β ( Y ) , V ] v + [ V, U ] v ) , and, if v is nilpotent Lie algebra of step r , then g is nilpotent of step r .One should note for this construction that, since β is linear, we have the decom-position W = Nul( β ) ⊕ Nul( β ) ⊥ , where dim(Nul( β ) ⊥ ) ≤ dim( v ) = N . Thus, for X = X + X , Y = Y + Y ∈ W , ω ( X + X , Y + Y ) = [ β ( X + X ) , β ( Y + Y )] = [ β ( X ) , β ( Y )] , and ω is a map on Nul( β ) ⊥ × Nul( β ) ⊥ . Thus, [Nul( β ) , Nul( β )] = { } and similarly[Nul( β ) , v ] = { } . So g = W ⊕ v = Nul( β ) ⊕ Nul( β ) ⊥ ⊕ v is in a sense just an extension of the finite dimensional subspace Nul( β ) ⊥ by v . Example . One can generalize the previous example by taking a linear map β : W → h , where h is nilpotent Lie algebra, and constructing an extension of h bya nilpotent Lie algebra. For the sake of a concrete example, consider the following.Let W = W ( R ) = { σ : [0 , → R : σ is continuous and σ (0) = 0 } and H = (cid:26) σ ∈ W : σ is absolutely continuous and Z k ˙ σ ( s ) k ds < ∞ (cid:27) , so that ( W, H ) is standard Wiener space. Let v = R be an abelian Lie algebra.Let ¯ σ = R σ ( s ) ds = (¯ σ , ¯ σ , ¯ σ ), and define ω : W × W → R by ω ( σ, τ ) = (¯ σ ¯ τ − ¯ τ ¯ σ , ¯ σ ¯ τ − ¯ τ ¯ σ , α σ : R → R by α σ ( x, y, z ) = (0 , , ¯ σ y − ¯ σ x ) . Then α σ α τ = 0 and (C1) is trivially satisfied. Using that α κ ω ( σ, τ ) = (0 , , ¯ κ (¯ σ ¯ τ − ¯ τ ¯ σ ) − ¯ κ (¯ σ ¯ τ − ¯ τ ¯ σ ))one may verify that (C2) is satisfied. Thus, the Lie bracket for this extension g = W ⊕ R is given by[( σ, v ) , ( τ, u )] = (0 , ¯ σ ¯ τ − ¯ τ ¯ σ , ¯ σ ¯ τ − ¯ τ ¯ σ , ¯ σ u − ¯ σ u + ¯ τ v − ¯ τ v ) , [( κ, w ) , [( σ, v ) , ( τ, u )]] = (0 , , , ¯ κ (¯ σ ¯ τ − ¯ τ ¯ σ ) − ¯ κ (¯ σ ¯ τ − ¯ τ ¯ σ )) , and all higher order brackets are 0. TAI MELCHER
Note that this construction corresponds to the extension g = R ⊕ R , the 4 × U = R and V = v = R , and define ω ′ : U × U → V by ω (( a, b, c ) , ( a ′ , b ′ , c ′ ))= a b
00 0 0 c a ′ b ′
00 0 0 c ′ − a ′ b ′
00 0 0 c ′ a b
00 0 0 c = ab ′ − ba ′
00 0 0 bc ′ − cb ′ , and α ′ : U → gl ( V ) by α ( a,b,c ) ( x, y, z )= a b
00 0 0 c x z y − x z y a b
00 0 0 c = ay − cx . Then ω = ω ′ ◦ β and α = α ′ ◦ β where β : W → U is given by β ( σ ) = (¯ σ , ¯ σ , ¯ σ ). Example . Consider v = R n ⊕ R as an abelian Lie algebra. For ω : W × W → R n ,we may write ω = ( ω , · · · , ω n ), where ω i : W × W → R are bilinear, anti-symmetric,continuous maps. Similarly, for α : W × R n → R , we have α i ( · ) = α · e i , where { e i } ni =1 is the standard basis for R n . Thus, α w ( a , . . . , a n ) = n X i =1 a i α i ( w ) . Then α and ω satisfy (C2) as long as α ∧ ω + · · · + α n ∧ ω n = 0 . In the case n = 1, this is not very interesting, since α ∧ ω = 0 implies that ω = α ∧ β for some β ∈ W ∗ .For n = 2, we have v = R ⊕ R . Let Ω : W × W → R be bilinear, antisymmetric,and continuous, and γ : W → R be linear and continuous. Then define ω : W × W → R by ω = (Ω , Ω) and α : W × R → R by α = γ and α = − γ , so that, for any u, w ∈ W and v = ( v , v ) ∈ R , ω ( w, u ) = (Ω( w, u ) , Ω( w, u )) and α w v = γ ( w )( v − v ) . Note that, for any w, u, h ∈ W , ω and α satisfy α h ω ( w, u ) = α h (Ω( w, u ) , Ω( w, u )) = γ ( h )(Ω( w, u ) − Ω( w, u )) = 0 . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 9
Thus, for any ( w, v, x ) , ( w ′ , v ′ , x ′ ) , ( w ′′ , v ′′ , x ′′ ) ∈ W ⊕ v ,[( w, v, x ) , ( w ′ , v ′ , x ′ )] = (0 , ω ( w, w ′ ) , α w v ′ − α w ′ v )= (0 , (Ω( w, w ′ ) , Ω( w, w ′ )) , γ ( w )( v ′ − v ′ ) + γ ( w ′ )( v − v )) , [( w ′′ , v ′′ , x ′′ ) , [( w, v, x ) , ( w ′ , v ′ , x ′ )]] = (0 , , α w ′′ ω ( w, w ′ )) = 0 , and g is a step 2 Lie algebra. The group operation is given by( w, v, x ) · ( w ′ , v ′ , x ′ ) = ( w + w ′ , v + v ′ + 12 (Ω( w, w ′ ) , Ω( w, w ′ )) ,x + x ′ + 12 ( γ ( w )( v ′ − v ′ ) + γ ( w ′ )( v − v )) . As an example of a particular appropriate Ω and γ , again let W = W ( R ) and H be as in Example 3.7. Suppose ϕ is an anti-symmetric bilinear form on R , ρ : R → R is a linear map, and let η be finite measure on [0 , σ, τ ) = Z ϕ ( σ ( s ) , τ ( s )) dη ( s )and γ ( σ ) = Z ρ ( σ ( s )) dη ( s ) . Example . Here we make a slight modification on the previous example to con-struct a stratified step 3 Lie algebra. Let v = R = R ⊕ R ⊕ R be an abelian Liealgebra. Let Ω and γ be as in the previous example. Define ω : W × W → R by ω ( w, u ) = (Ω( w, u ) , Ω( w, u ) , Ω( w, u ))and α : W × v → v by α w (( v , v , v ) , ( x , x ) , y ) = (0 , ( γ ( w )( v − v ) , γ ( w )( v − v )) , γ ( w )( x − x ))(so α w is a particular element of the 6 × α w α u = α u α w and so α satisfies (C1), and also α v ω ( w, u ) = (0 , ( γ ( v )(Ω( w, u ) − Ω( w, u )) , γ ( v )(Ω( w, u ) − Ω( w, u ))) ,
0) = 0 , so α and ω satisfy (C2) trivially. The Lie bracket is given by[( w, v, x, y ) , ( w ′ , v ′ , x ′ , y ′ )] = (0 , ω ( w, w ′ ) , α w v ′ − α w ′ v, α w x ′ − α w ′ x ) , or, more explicitly, this may be written componentwise as[( w, v, x, y ) , ( w ′ , v ′ , x ′ , y ′ )] = (Ω( w, w ′ ) , Ω( w, w ′ ) , Ω( w, w ′ )) ∈ R , [( w, v, x, y ) , ( w ′ , v ′ , x ′ , y ′ )] = ( γ ( w )( v ′ − v ′ ) − γ ( w ′ )( v − v ) , γ ( w )( v ′ − v ′ ) − γ ( w ′ )( v − v )) ∈ R , and [( w, v, x, y ) , ( w ′ , v ′ , x ′ , y ′ )] = γ ( w )( x ′ − x ′ ) − γ ( w ′ )( x − x ) ∈ R . Thus,[( w ′′ , v ′′ , x ′′ , y ′′ ) , [( w, v, x, y ) , ( w ′ , v ′ , x ′ , y ′ )]]= (0 , , α w ′′ ω ( w, w ′ ) , α w ′′ ( α w v ′ − α w ′ v ))= (0 , , , α w ′′ α w v ′ − α w ′′ α w ′ v )= (0 , , , γ ( w ′′ ) γ ( w )( v ′ − v ′ ) − γ ( w ′′ ) γ ( w ′ )( v − v )) , and all higher order brackets are 0. So for g = ( w, v, x, y ) and g ′ = ( w ′ , v ′ , x ′ , y ′ ),the group operation is given by( g · g ′ ) = w + w ′ ( g · g ′ ) = v + v ′ + 12 ω ( w, w ′ )( g · g ′ ) = x + x ′ + 12 ( α w v ′ − α w ′ v )( g · g ′ ) = y + y ′ + 12 ( α w x ′ − α w ′ x ) + 112 ( α w v ′ + α w ′ v − α w α w ′ ( v − v ′ )) . Clearly, this example may be further modified to make nilpotent Lie algebras ofarbitrary step.3.2.
Hilbert-Schmidt norms.
In this section, we will show that the assumedcontinuity of ω and α makes the Lie bracket into a Hilbert-Schmidt operator on g CM . This result will be needed later in guaranteeing that our stochastic integralsare well-defined. Notation 3.10.
Let H , . . . , H n and V be Hilbert spaces, and let { h ij } dim( H i ) j =1 denote an orthonormal basis for each H i . If ρ : H × · · · × H n → V is a multilinearmap, then the Hilbert-Schmidt norm of ρ is defined by k ρ k := k ρ k H ∗ ⊗···⊗ H ∗ n ⊗ V = X j ,...,j n k ρ ( h j , . . . , h nj n ) k V . In particular, for H an infinite dimensional Hilbert space with orthonormal basis { h i } ∞ i =1 , ρ : H ⊗ n → V is Hilbert-Schmidt if k ρ k = k ρ k ( H ∗ ) ⊗ n ⊗ V = ∞ X j ,...,j n =1 k ρ ( h j , . . . , h j n ) k V < ∞ . One may verify directly that these norms are independent of the chosen bases.
Proposition 3.11.
For all w ∈ W and x ∈ v , (3.5) k α w · k v ∗ ⊗ v ≤ N k α k k w k W and k α · x k H ∗ ⊗ v ≤ C k α k k x k v , where C is as in equation (2.2). Also, (3.6) k ω ( w, · ) k H ∗ ⊗ v ≤ C k ω k k w k W . Furthermore, k α k ≤ N C k α k < ∞ and k ω k ≤ C k ω k < ∞ . Proof.
Let { e i } Ni =1 be an orthonormal basis of v . Then, for any w ∈ W , k α w · k v ∗ ⊗ v = N X i =1 k α w e i k v ≤ N X i =1 k α k k w k W k e i k v = N k α k k w k W . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 11
For fixed x ∈ v , α · x : W → v is a continuous linear map. Thus, equation (2.3) gives k α · x k H ∗ ⊗ v = Z W k α w x k v dµ ( w ) ≤ Z W k α k k w k W k x k v dµ ( w ) = C k α k k x k v . Similarly, for fixed w ∈ W and ω ( w, · ) : W → v , k ω ( w, · ) k H ∗ ⊗ v = Z W k ω ( w, w ′ ) k v dµ ( w ′ ) ≤ Z W k ω k k w k W k w ′ k W dµ ( w ′ ) = C k ω k k w k W . Since w α w is a continuous linear map from W to v ∗ ⊗ v , it follows fromequations (2.3) and (2.4) that k α k = Z W k α w · k v ∗ ⊗ v dµ ( w ) ≤ Z W N k α k k w k W dµ ( w ) = N C k α k , and since w ω ( w, · ) is a continuous linear map from W to H ∗ ⊗ v , k ω k = k h ω ( h, · ) k H ∗ ⊗ ( H ∗ ⊗ v ) = Z W k ω ( w, · ) k H ∗ ⊗ v dµ ( w ) ≤ Z W C k ω k k w k W dµ ( w ) = C k ω k . This proposition easily gives the following result.
Corollary 3.12.
For all m ≥ , [[[ · , · ] , . . . ] , · ] : g ⊗ mCM → v is Hilbert-Schmidt. Proof.
For m = 2, this follows from the previous proposition and the continuityof the Lie bracket on v , since taking { h i } ∞ i =1 = { k i } ∞ i =1 ∪ { e j } Nj =1 , where { k i } ∞ i =1 and { e j } Nj =1 are orthonormal bases of H and v , respectively, gives k [ · , · ] k = k [ · , · ] k g ∗ CM ⊗ g ∗ CM ⊗ v = ∞ X i ,i =1 k [[ h i , h i ] k v = ∞ X i ,i =1 k ω ( k i , k i ) k v + ∞ X i =1 N X j =1 k α k i e j k v + ∞ X i =1 N X j =1 k α k i e j k v + N X j ,j =1 k [ e j , e j ] k v = k ω k + 2 k α k + N < ∞ . Now assume the statement is true for all m = 2 , . . . , ℓ . Consider m = ℓ + 1. Writing[[ h i , h i ] , · · · , h i ℓ ] ∈ v in terms of the orthonormal basis { e j } Nj =1 and using multiple applications of the Cauchy-Schwarz inequality gives k [[[ · , · ] , . . . ] , · ] k = k [[[ · , · ] , . . . ] , · ] k ( g ∗ CM ) ⊗ ℓ +1 ⊗ v = ∞ X i ,...,i ℓ +1 =1 k [[[ h i , h i ] , · · · , h i ℓ ] , h i ℓ +1 ] k v = ∞ X i ,...,i ℓ +1 =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 [ e j , h i ℓ +1 ] h e j , [[ h i , h i ] , · · · , h i ℓ ] i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v ≤ N ∞ X i ,...,i ℓ +1 =1 N X j =1 k [ e j , h i ℓ +1 ] k v |h e j , [[ h i , h i ] , · · · , h i ℓ ] i| ≤ N ∞ X i ℓ +1 =1 N X j =1 k [ e j , h i ℓ +1 ] k v ∞ X i ,...,i ℓ =1 N X j =1 |h e j , [[ h i , h i ] , · · · , h i ℓ ] i| ≤ N k [ · , · ] k g ⊗ CM ⊗ v · k [[[ · , · ] , . . . ] , · ] k g ⊗ ℓCM ⊗ v , where in the penultimate inequality we have used that all terms in the sums arepositive. The last line is finite by the induction hypothesis.3.3. Length and distance.
In this section, we define the Riemannian distanceon G CM and show that the topology induced by this metric is equivalent to theHilbert topology induced by k · k g CM .For g ∈ G, let L g : G → G and R g : G → G denote left and right multiplicationby g , respectively. As G is a vector space, to each g ∈ G we can associate thetangent space T g G to G at g , which is naturally isomorphic to G . Notation 3.13.
For f : G → R a Frech´et smooth function and v, x ∈ G and h ∈ g ,let f ′ ( x ) h := ∂ h f ( x ) = ddt (cid:12)(cid:12)(cid:12)(cid:12) f ( x + th ) , and let v x ∈ T x G denote the tangent vector satisfying v x f = f ′ ( x ) v . If σ ( t ) is anysmooth curve in G such that σ (0) = x and ˙ σ (0) = v (for example, σ ( t ) = x + tv ),then L g ∗ v x = ddt (cid:12)(cid:12)(cid:12)(cid:12) g · σ ( t ) . Notation 3.14.
Let
T > C ([0 , T ] , G CM ) denote the collection of C -paths g : [0 , T ] → G CM . The length of g is defined as ℓ CM ( g ) := Z T k L g − ( s ) ∗ g ′ ( s ) k g CM ds. The Riemannian distance between x, y ∈ G CM then takes the usual form d CM ( x, y ) := inf { ℓ CM ( g ) : g ∈ C ([0 , T ] , G CM ) such that g (0) = x and g ( T ) = y } . Note that the value of T in the definition of d CM is irrelevant since the lengthfunctional is invariant under reparameterization. EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 13
Proposition 3.15.
For g, x ∈ G and v x ∈ T x G , L g ∗ v x = v + r − X k =1 X ( n,m ) ∈I k a kn,m × X j ∈ { , . . . , k } m j > m j − X ℓ =0 ad n g ad m x · · · ad n j g ad ℓx ad v ad m j − ℓ − x ad n j − g · · · ad n k g ad m k x g, (3.7) where a kn,m are the coefficients in the group multiplication given in equation (3.3). Proof.
The proof is a simple computation. Let x ( t ) = x + tv , and first notethat ddt (cid:12)(cid:12)(cid:12)(cid:12) ad n g ad m x ( t ) · · · ad n k g ad m k x ( t ) g = X j ∈ { , . . . , k } m j > m j − X ℓ =0 ad n g ad m x · · · ad n j g ad ℓx ad v ad m j − ℓ − x ad n j − g · · · ad n k g ad m k x g. Then using (3.4) and plugging this into L g ∗ v x = ddt (cid:12)(cid:12)(cid:12)(cid:12) g · x ( t )= ddt (cid:12)(cid:12)(cid:12)(cid:12) g + x ( t ) + r − X k =1 X ( n,m ) ∈I k a kn,m ad n g ad m x ( t ) · · · ad n k g ad m k x ( t ) g yields the desired result. Example . When r = 3, the group operation is g · h = g + h + 12 [ g, h ] + 112 ([ g, [ g, h ]] + [ h, [ h, g ]]) . Thus, L g ∗ v x = ddt (cid:12)(cid:12)(cid:12)(cid:12) g · x ( t )= ddt (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) g + x ( t ) + 12 [ g, x ( t )] + 112 ([ g, [ g, x ( t )]] + [ x ( t ) , [ x ( t ) , g ]]) (cid:19) = v + 12 [ g, v ] + 112 ([ g, [ g, v ]] + [ v, [ x, g ]] + [ x, [ v, g ]]) . Proposition 3.17.
There exists K = K ( a ∧ b ) < ∞ (for a, b ≥ ) such that K (0) = 0 and, for all x, y ∈ G CM , d CM ( x, y ) ≤ (1 + K ( k x k g CM ∧ k y k g CM )) k y − x k g CM + o (cid:0) k y − x k g CM (cid:1) . Proof.
For notational simplicity, let T = 1. If g ( s ) is a path in C CM for0 ≤ s ≤
1, then, by equation (3.7), taking g = g − ( s ), x = g ( s ), and v g ( s ) = g ′ ( s ), ℓ CM ( g ) = Z (cid:13)(cid:13)(cid:13)(cid:13) g ′ ( s ) + r − X k =1 X ( n,m ) ∈I k a kn,m X m j > m j − X ℓ =0 ad n g − ( s ) ad m g ( s ) · · · ad n j g − ( s ) ad ℓg ( s ) ad g ′ ( s ) ad m j − ℓ − g ( s ) · · · ad n k g − ( s ) ad m k g ( s ) g − ( s ) (cid:13)(cid:13)(cid:13)(cid:13) g CM ds = Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g ′ ( s ) + r − X k =1 X ( n,m ) ∈I k ( − | n | { m k > } a kn,m ad | m | + | n | g ( s ) g ′ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ds = Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g ′ ( s ) + r − X ℓ =1 d ℓ ad ℓg ( s ) g ′ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ds, (3.8)where(3.9) d ℓ := ℓ X k =1 X ( n, m ) ∈ I k | m | + | n | = ℓ ( − | n | { m k > } a kn,m . Taking g ( s ) = x + s ( y − x ) for 0 ≤ s ≤
1, this gives d CM ( x, y ) ≤ ℓ CM ( g )= Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( y − x ) + r − X ℓ =1 d ℓ ad ℓx + s ( y − x ) ( y − x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ds = Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( y − x ) + r − X ℓ =1 d ℓ X ( n, m ) ∈ I ℓ | m | + | n | = ℓ s | n | ad m x ad n y − x · · · ad m ℓ x ad n ℓ y − x ( y − x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ds. Splitting off all terms in the sum of order two or higher and evaluating the integralgives d CM ( x, y ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( y − x ) + r − X ℓ =1 d ℓ ad ℓx ( y − x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) r − X ℓ =1 d ℓ X ( n, m ) ∈ I ℓ | m | + | n | = ℓ {| n | > } | n | + 1 ad m x ad n y − x · · · ad m ℓ x ad n ℓ y − x ( y − x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ≤ r − X ℓ =1 X ( n, m ) ∈ I ℓ | m | + | n | = ℓ | d ℓ | C ℓ k x k ℓ g CM k y − x k g CM + o (cid:0) k y − x k g CM (cid:1) , where C = C ( ω, α ) is as defined in (3.2). Interchanging the roles of x and y in g ( s ),and thus in this inequality, completes the proof. Notation 3.18.
Let τ denote the norm topology on G CM and τ d denote the topol-ogy induced by d CM . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 15
Proposition 3.19.
For any y ∈ G and W ∈ τ such that y ∈ W , there exists U ∈ τ d such that y ∈ U ⊂ W . Proof.
First we will show that, there exists ε > x, y ∈ G CM and ε ∈ (0 , ε / d CM ( x, y ) < ε , then k x − y k g CM < ε . Then we will show thatthe continuity of the map x
7→ k x − y k g CM (for fixed y ) suffices to complete theproof.Let d ℓ be as in equation (3.9) and C = C ( ω, α ) be as in equation (3.2). Let κ := r − X ℓ =1 | d ℓ | C ℓ , and take ε := 1 / κ ∧
1. Let B ε := { x ∈ g CM : k x k g CM ≤ ε } . Suppose y ∈ B ε ,and let g : [0 , → G CM be a C -path such that g (0) = e and g (1) = y . Further,let T ∈ [0 ,
1] be the first time that g exits B ε , with the convention that T = 1 if g ([0 , ⊂ B ε . Then, by equation (3.8), ℓ CM ( g − ) ≥ ℓ CM ( g | [0 ,T ] ) ≥ Z T k g ′ ( s ) k g CM − r − X ℓ =1 | d ℓ | (cid:13)(cid:13)(cid:13) ad ℓg ( s ) g ′ ( s ) (cid:13)(cid:13)(cid:13) g CM ds ≥ − r − X ℓ =1 | d ℓ | C ℓ ε ℓ ! Z T k g ′ ( s ) k g CM ds ≥ (1 − κε ) k g ( T ) k g CM ≥ k y k g CM . Taking the infimum over g implies that d CM ( e , y ) ≥ k y k g CM , for all y ∈ B ε . Now, if y / ∈ B ε , then the path g would have had to exit B ε and ℓ CM ( g ) ≥k g ( T ) k g CM / ε / d CM ( e , y ) ≥ ε /
2. Thus, d CM ( e , y ) ≥
12 min( ε , k y k g CM ) , for all y ∈ G CM . By the left invariance of d CM , this implies that, for any x, y ∈ G CM , d CM ( x, y ) = d CM ( e , x − y ) ≥
12 min( ε , k x − y k g CM ) . So if d CM ( x, y ) < ε /
2, then k x − y k g CM ≤ d CM ( x, y ).Now let W ∈ τ (non-empty) and fix y ∈ W . Recall that Lemma 3.4 implies thatthe map x
7→ k x − y k g CM is τ -continuous, and clearly k x − y k g CM = 0 if and onlyif x = y . Thus, A n ( y ) := (cid:26) x : k x − y k g CM < n (cid:27) ↓ { y } , and there exists N sufficiently large that 1 /N < ε / A N ( y ) ⊂ W . Then B N ( y ) := (cid:26) x : d CM ( x, y ) < N (cid:27) ∈ τ d satisfies B N ( y ) ⊂ A N ( y ), since x ∈ B N ( y ) implies that d CM ( x, y ) < / N < ε / k x − y k g CM ≤ d CM ( x, y ) < /N . Thus, B N ( y ) ⊂ W .In particular, taking W = { x : k y − x k g CM < δ } for some δ > N such that d CM ( x, y ) < / N impliesthat k y − x k g CM < δ . Propositions 3.17 and 3.19 give the following corollary. Corollary 3.20.
The topologies τ and τ d are equivalent. Ricci curvature.
In this section, we compute the Ricci curvature of certain fi-nite dimensional approximations of G and show that it is bounded below uniformly.This result will be used in Section 5.1 to give L p -bounds on Radon Nikodym deriva-tives of ν t . It will also be applied in Section 5.2 to prove a logarithmic Sobolevinequality for ν t . First we must define the appropriate approximations.Let i : H → W be the inclusion map, and i ∗ : W ∗ → H ∗ be its transpose. Thatis, i ∗ ℓ := ℓ ◦ i for all ℓ ∈ W ∗ . Also, let H ∗ := { h ∈ H : h· , h i H ∈ Range( i ∗ ) ⊂ H } . That is, for h ∈ H , h ∈ H ∗ if and only if h· , h i H ∈ H ∗ extends to a continuouslinear functional on W , which we will continue to denote by h· , h i H . Because H is adense subspace of W , i ∗ is injective and thus has a dense range. Since h
7→ h· , h i H as a map from H to H ∗ is a conjugate linear isometric isomorphism, it follows that H ∗ ∋ h
7→ h· , h i H ∈ W ∗ is a conjugate linear isomorphism also, and so H ∗ is adense subspace of H .Now suppose that P : H → H is a finite rank orthogonal projection such that P H ⊂ H ∗ . Let { k j } mj =1 be an orthonormal basis for P H . Then we may extend P to a (unique) continuous operator from W → H (still denoted by P ) by letting(3.10) P w := m X j =1 h w, k j i H k j for all w ∈ W . Notation 3.21.
Let Proj( W ) denote the collection of finite rank projections on W such that P W ⊂ H ∗ and P | H : H → H is an orthogonal projection, that is, P has the form given in equation (3.10). Further, let G P := P W ⊕ v (a subgroup of G CM ), and we equip G P with the left invariant Riemannian metric induced fromthe restriction of the inner product on g CM = H ⊕ v to Lie( G P ) = P H ⊕ v =: g PCM .Let Ric P denote the associated Ricci tensor at the identity in G P . Proposition 3.22.
For X = ( A, a ) ∈ g PCM , h Ric P X, X i g PCM = 14 kh a, [ · , · ] ik g PCM ) ∗ ⊗ ( g PCM ) ∗ − k [ · , X ] k g PCM ) ∗ ⊗ v , where ( g PCM ) ∗ = ( P H ) ∗ ⊗ v ∗ . Proof.
For g any nilpotent Lie algebra with orthonormal basis Γ,(3.11) h Ric
X, X i = 14 X Y ∈ Γ k ad ∗ Y X k − X Y ∈ Γ k ad Y X k , for all X ∈ g ; see for example Theorem 7.30 and Corollary 7.33 of [5].So let Γ m := { h i } m + Ni =1 = { ( k i , } mi =1 ∪ { (0 , e j ) } Nj =1 be an orthonormal basis of g PCM = P H ⊕ v , where { k i } mi =1 and { e j } Nj =1 are orthonormal bases of P H and v ,respectively. Then, for Y ∈ g PCM ,ad ∗ Y X = X h i ∈ Γ m h ad ∗ Y X, h i i g CM h i = X h i ∈ Γ m h X, ad Y h i i g CM h i . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 17
Thus, X h i ∈ Γ m k ad ∗ h i X k g CM = X h i ∈ Γ m X h j ∈ Γ m h X, ad h i h j i g CM = X h i ,h j ∈ Γ m h X, [ h i , h j ] i g CM . Plugging this into (3.11) gives h Ric P X, X i g PCM = 14 X h i ,h j ∈ Γ m h X, [ h i , h j ] i g CM − X h i ∈ Γ m k [ h i , X ] k g CM = 14 X h i ,h j ∈ Γ m h a, [ h i , h j ] i v − X h i ∈ Γ m k [ h i , X ] k v . Corollary 3.23.
Let K := −
12 sup n k [ · , X ] k g ∗ CM ⊗ v : k X k g CM = 1 o . Then
K > −∞ and K is the largest constant such that h Ric P X, X i g PCM ≥ K k X k g PCM , for all X ∈ g PCM , holds uniformly for all P ∈ Proj( W ) . Proof.
The first assertion is simple, since K ≥ − k [ · , · ] k > −∞ , by Corollary 3.12. Now, for P ∈ Proj( W ) as in Notation 3.21, Proposition 3.22implies that h Ric P X, X i g PCM ≥ − k [ · , X ] k g PCM ) ∗ ⊗ v . Thus, h Ric P X, X i g PCM k X k g PCM ≥ − k [ · , X ] k g PCM ) ∗ ⊗ v k X k g PCM ≥ −
12 sup n k [ · , X ] k g PCM ) ∗ ⊗ v : k X k g PCM = 1 o =: K P . (3.12)Noting that the infimum of K P over all P ∈ Proj( W ) is K completes the proof. Remark . Of course, one can compute the Ricci curvature for g = W ⊕ v justas in Proposition 3.22. Choose an orthonormal basis Γ = { h i } ∞ i =1 = { ( k i , } ∞ i =1 ∪{ (0 , e j ) } Nj =1 of g CM = H ⊕ v , where { k i } ∞ i =1 is an orthonormal basis of H , and { e j } Nj =1 is an orthonormal basis of v . Then, for all X = ( A, a ) ∈ g CM , h Ric
X, X i g CM = 14 ∞ X i,j =1 h a, [ h i , h j ] i v − ∞ X i =1 k [ h i , X ] k v = 14 kh a, [ · , · ] ik g ∗ CM ⊗ g ∗ CM − k [ · , X ] k g ∗ CM ⊗ v ≥ K k X k g CM . Brownian motion
Suppose that B t is a smooth curve in g CM with B = 0, and consider thedifferential equation ˙ g t = L g t ∗ ˙ B t , with g = e . The solution g t may be written as follows (see [24]): For t >
0, let ∆ n ( t ) denotethe simplex in R n given by { s = ( s , · · · , s n ) ∈ R n : 0 < s < s < · · · < s n < t } . Let S n denote the permutation group on (1 , · · · , n ), and, for each σ ∈ S n , let e ( σ ) denote the number of “errors” in the ordering ( σ (1) , σ (2) , · · · , σ ( n )), that is, e ( σ ) = { j < n : σ ( j ) > σ ( j + 1) } . Then(4.1) g t = r X n =1 X σ ∈S n (cid:18) ( − e ( σ ) (cid:30) n (cid:20) n − e ( σ ) (cid:21)(cid:19) × Z ∆ n ( t ) [ · · · [ ˙ B s σ (1) , ˙ B s σ (2) ] , . . . , ] ˙ B s σ ( n ) ] ds. For n ∈ { , · · · , r } and σ ∈ S n , let F σn : g ⊗ nCM → v be the linear map given by(4.2) F σn ( k ⊗ · · · ⊗ k n ) := [[ · · · [ k σ (1) , k σ (2) ] , · · · ] , k σ ( n ) ] . Recall that F σn is Hilbert-Schmidt by Corollary 3.12. Then we may write(4.3) g t = r − X n =1 X σ ∈S n c σn F σn Z ∆ n ( t ) ˙ B s ⊗ · · · ⊗ ˙ B s n ds ! . Using this as our motivation, we first explore stochastic integral analogues of equa-tion (4.3) where the smooth curve B is replaced by Brownian motion on g .4.1. Multiple Itˆo integrals.
Let h· , ·i g ⊗ nCM denote the inner product on g ⊗ nCM aris-ing from the inner product on g CM . Also, let { k i } ∞ i =1 ⊂ H ∗ be an orthonormalbasis of H , and define P m ∈ Proj( W ) by(4.4) P m ( w ) = m X i =1 h w, k i i H k i , for all w ∈ W, as in equation (3.10), and define(4.5) π m ( w, x ) := π P m ( w, x ) := ( P m ( w ) , x ) ∈ G P m . Of course, dim( G P m ) = m + N , but in a mild abuse of notation, we will use { h i } mi =1 to denote an orthonormal basis of G P m , rather than the more cumbersome { h i } m + Ni =1 = { ( k i , } mi =1 ∪ { (0 , e i ) } Ni =1 , where { e i } Ni =1 is an orthonormal basis of v .Let { B t } t ≥ = { ( β t , β v t ) } t ≥ be a Brownian motion on g = W ⊕ v with variancedetermined by E [ h B s , h i g CM h B t , k i g CM ] = h h, k i g CM min( s, t ) , for all s, t ≥ h = ( A, a ) and k = ( C, c ), such that
A, C ∈ H ∗ and a, c ∈ v .Then π m B = ( P m β, β v ) is a Brownian motion on g P m = P m W ⊕ v ⊂ g CM . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 19
Proposition 4.1.
For ξ ∈ L (∆ n ( t ) , g ⊗ nCM ) a continuous mapping, let J mn ( ξ ) t := Z ∆ n ( t ) h ξ ( s ) , dπ m B s ⊗ · · · ⊗ dπ m B s n i g ⊗ nCM . Then { J mn ( ξ ) t } t ≥ is a continuous L -martingale such that, for all m , E | J mn ( ξ ) t | ≤ k ξ k L (∆ n ( t ) , g ⊗ nCM ) , and there exists a continuous L -martingale { J n ( ξ ) t } t ≥ such that (4.6) lim m →∞ E (cid:20) sup τ ≤ t | J mn ( ξ ) τ − J n ( ξ ) τ | (cid:21) = 0 , for all t < ∞ . In particular, (4.7) J n ( ξ ) t := Z ∆ n ( t ) h ξ ( s ) , dB s ⊗ · · · ⊗ dB s n i g ⊗ nCM , and J n ( ξ ) t is well-defined independent of the choice of orthonormal basis { h i } ∞ i =1 in (4.4). Proof.
Note first that, J mn ( ξ ) t = m X i ,...,i n =1 Z ∆ n ( t ) h ξ ( s ) , h i ⊗ · · · ⊗ h i n i g ⊗ nCM dB i s · · · dB i n s n where { B i } mi =1 are independent real valued Brownian motions. Let ξ i ,...,i n := h ξ, h i ⊗ · · · ⊗ h i n i . Then | ξ i ,...,i n ( s ) | ≤ k ξ ( s ) k g ⊗ nCM and ξ i ,...,i n ∈ L (∆ n ( t )). Thus, J mn ( ξ ) t is defined as a (finite dimensional) vector-valued multiple Wiener-Itˆo integral, see for example [18, 23].Now note that dJ mn ( ξ ) t = Z ∆ n − ( t ) h ξ ( s , . . . , s n − , t ) , dπ m B s ⊗ · · · ⊗ dπ m B s n − ⊗ dπ m B t i g ⊗ nCM = m X i =1 Z ∆ n − ( t ) h ξ ( s , . . . , s n − , t ) , dπ m B s ⊗ · · · ⊗ dπ m B s n − ⊗ h i i g ⊗ nCM dB it . Thus, the quadratic variation h J mn ( ξ ) i t is given by m X i =1 Z t (cid:12)(cid:12)(cid:12)(cid:12) Z ∆ n − ( τ ) h ξ ( s , . . . , s n − , τ ) , dπ m B s ⊗ · · · ⊗ dπ m B s n − ⊗ h i i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) dτ, and E | J mn ( ξ ) t | = E h J mn ( ξ ) i t = m X i =1 Z t E (cid:20) m X i =1 Z τ (cid:12)(cid:12)(cid:12)(cid:12) Z ∆ n − ( τ ) h ξ ( s , . . . , s n − , τ , τ ) , dπ m B s ⊗ · · ·· · · ⊗ dπ m B s n − ⊗ h i ⊗ h i i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:21) dτ . Iterating this procedure n times gives E | J mn ( ξ ) t | = m X i ,...,i n =1 Z ∆ n ( t ) (cid:12)(cid:12)(cid:12) h ξ ( τ , · · · , τ n ) , h i ⊗ · · · ⊗ h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12) dτ · · · dτ n (4.8) = Z ∆ n ( t ) k π ⊗ nm ξ ( s ) k g ⊗ nCM ≤ k ξ k L (∆ n ( t ) , g ⊗ nCM ) , and thus, for each n , J mn ( ξ ) t is bounded uniformly in L independent of m .A similar argument shows that the sequence { J mn ( ξ ) t } ∞ m =1 is Cauchy in L . For m ≤ ℓ , consider(4.9) J ℓn ( ξ ) t − J mn ( ξ ) t = n X j =1 Z ∆ n ( t ) h ξ ( s ) , dπ ℓ B s ⊗ · · ·· · · ⊗ dπ ℓ B s j − ⊗ d ( π ℓ − π m ) B s j ⊗ dπ m B s j +1 ⊗ · · · ⊗ dπ m B s n i . Thus, applying Cauchy-Schwarz and computing as in equation (4.8),(4.10) E (cid:12)(cid:12) J ℓn ( ξ ) t − J mn ( ξ ) t (cid:12)(cid:12) ≤ n n X j =1 ℓ X i ,...,i j − =1 ℓ X i j = m +1 m X i j +1 ,...,i n =1 Z ∆ n ( t ) (cid:12)(cid:12)(cid:12) h ξ ( s ) , h i ⊗ · · · ⊗ h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12) ds → , as ℓ, m → ∞ , since k ξ k L (∆ n ( t ) , g ⊗ nCM ) = Z ∆ n ( t ) k ξ ( s ) k g ⊗ nCM ds = Z ∆ n ( t ) ∞ X i ,...,i n =1 (cid:12)(cid:12)(cid:12) h ξ ( s ) , h i ⊗ · · · ⊗ h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12) ds < ∞ . Since the space of continuous L -martingales is complete in the norm M E | M t | ,there exists a continuous martingale { X t } t ≥ such that(4.11) lim m →∞ E | J mn ( ξ ) t − X t | = 0 . To see that X t is independent of basis, suppose now that { h ′ j } ∞ j =1 ⊂ H ∗ isanother orthonormal basis for H and P ′ m : W → H ∗ and π ′ m : G → G P ′ m are thecorresponding orthogonal projections, that is, P ′ m w := m X i =1 h w, h ′ i i W h ′ i , and π ′ m ( w, x ) = ( P ′ m w, x ). Let J m ′ n ( ξ ) t = Z ∆ n ( t ) h ξ ( s ) , dπ ′ m B s ⊗ · · · ⊗ dπ ′ m B s n i g ⊗ nCM . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 21
Then, using equation (4.9) with π ℓ replaced by π ′ m , applying Cauchy-Schwarz, andagain computing as in (4.8), gives E (cid:12)(cid:12)(cid:12) J mn ( ξ ) t − J m ′ n ( ξ ) t (cid:12)(cid:12)(cid:12) ≤ n n X j =1 Z ∆ n ( t ) ∞ X i ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , π m h i ⊗ · · ·· · · ⊗ π m h i j − ⊗ ( π m − π ′ m ) h i j ⊗ π ′ m h i j +1 ⊗ · · · ⊗ π ′ m h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds. Writing π m − π ′ m = ( π m − I ) + ( I − π ′ m ), and considering terms for each fixed j ,we have Z ∆ n ( t ) ∞ X i ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , π m h i ⊗ · · · ⊗ π m h i j − ⊗ ( π m − I ) h i j ⊗ π ′ m h i j +1 ⊗ · · · ⊗ π ′ m h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds = Z ∆ n ( t ) m X i ,...,i j =1 ∞ X i j = m +1 ∞ X i j +1 ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , h i ⊗ · · · ⊗ h i j − ⊗ h i j ⊗ π ′ m h i j +1 ⊗ · · · ⊗ π ′ m h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ Z ∆ n ( t ) m X i ,...,i j =1 ∞ X i j = m +1 ∞ X i j +1 ,...,i n =1 (cid:12)(cid:12)(cid:12) h ξ ( s ) , h i ⊗ · · · ⊗ h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12) ds → , as m → ∞ . Similarly, Z ∆ n ( t ) ∞ X i ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , π m h i ⊗ · · · ⊗ π m h i j − ⊗ ( π ′ m − I ) h i j ⊗ π ′ m h i j +1 ⊗ · · · ⊗ π ′ m h i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds = Z ∆ n ( t ) ∞ X i ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , π m h ′ i ⊗ · · · ⊗ π m h ′ i j − ⊗ ( π ′ m − I ) h ′ i j ⊗ π ′ m h ′ i j +1 ⊗ · · · ⊗ π ′ m h ′ i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds = Z ∆ n ( t ) ∞ X i ,...,i j − =1 ∞ X i j = m +1 m X i j +1 ,...,i n =1 (cid:12)(cid:12)(cid:12)(cid:12) h ξ ( s ) , π m h ′ i ⊗ · · · ⊗ π m h ′ i j − ⊗ h ′ i j ⊗ h ′ i j +1 ⊗ · · · ⊗ h ′ i n i g ⊗ nCM (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ Z ∆ n ( t ) ∞ X i ,...,i j − =1 ∞ X i j = m +1 m X i j +1 ,...,i n =1 (cid:12)(cid:12)(cid:12) h ξ ( s ) , h ′ i ⊗ · · · ⊗ h ′ i n i g ⊗ nCM (cid:12)(cid:12)(cid:12) ds → , as m → ∞ . Thus, lim m →∞ E (cid:12)(cid:12)(cid:12) J mn ( ξ ) t − J m ′ n ( ξ ) t (cid:12)(cid:12)(cid:12) = 0 , and X t is independent of the choice of orthonormal basis. In particular, replacing J ℓn ( ξ ) t in (4.10) by J n ( ξ ) t as given in equation (4.7), and taking the limit as m → ∞ ,shows that X t = J n ( ξ ) t satisfies (4.11). Combining this with Doob’s maximalinequality proves equation (4.6).A simple linearity argument extends the map J n to functions taking values in( g ∗ CM ) ⊗ n ⊗ v . Corollary 4.2.
Let F ∈ L (∆ n ( t ) , ( g ∗ CM ) ⊗ n ⊗ v ) be a continuous map. That is, F : ∆ n ( t ) × g ⊗ nCM → v is a map continuous in s and linear on g ⊗ nCM such that Z ∆ n ( t ) k F ( s ) k ds = Z ∆ n ( t ) ∞ X j ,...,j n =1 k F ( s )( h j ⊗ · · · ⊗ h j n ) k v ds < ∞ . Then J mn ( F ) t := Z ∆ n ( t ) F ( s )( dπ m B s ⊗ · · · ⊗ dπ m B s n ) is a continuous L -martingale, and there exists a continuous v -valued L -martingale { J n ( F ) t } t ≥ such that lim m →∞ E (cid:20) sup τ ≤ t k J mn ( ξ ) τ − J n ( ξ ) τ k v (cid:21) = 0 , for all t < ∞ . The martingale J n ( ξ ) t is well-defined independent of the choice oforthonormal basis { h i } ∞ i =1 in (4.4), and will be denoted by J n ( F ) t := Z ∆ n ( t ) F ( s )( dB s ⊗ · · · ⊗ dB s n ) . Proof.
Let { e j } Nj =1 be an orthonormal basis of v . Then for any k , . . . , k n ∈ g CM , F ( s )( k ⊗ · · · ⊗ k n ) = N X j =1 h F ( s )( k ⊗ · · · ⊗ k n ) , e j i e j . Since h F ( s )( · ) , e j i is linear on g ⊗ nCM , for each s there exists ξ j ( s ) ∈ g ⊗ nCM such that(4.12) h ξ j ( s ) , k ⊗ · · · ⊗ k n i = h F ( s )( k ⊗ · · · ⊗ k n ) , e j i . If ξ j : ∆ n ( t ) → g ⊗ nCM is defined by equation (4.12), then k ξ j k L (∆ n ( t ) , g ⊗ nCM ) ≤ Z ∆ n ( t ) k F ( s ) k ds < ∞ . Thus, J n ( F ) t = N X j =1 Z ∆ n ( t ) h ξ j ( s ) , dB s ⊗ · · · ⊗ dB s n i e j = N X j =1 J n ( ξ j ) t e j , is well-defined, and, for each j , J n ( ξ j ) is a martingale as defined in Proposition 4.1. EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 23
Brownian motion and finite dimensional approximations.
Again let B t denote Brownian motion on g . By equation (4.1), the solution to the Stratonovichstochastic differential equation δg t = L g t ∗ δB t , with g = e , should be given by g t = r − X n =1 X σ ∈S n c σn Z ∆ n ( t ) [[ · · · [ δB s σ (1) , δB s σ (2) ] , · · · ] , δB s σ ( n ) ] , for coefficients c σn determined by equation (4.1).To see that this process is well-defined, consider the following. Let { M n ( t ) } t ≥ denote the process in g ⊗ n defined by M n ( t ) := Z ∆ n ( t ) δB s ⊗ · · · ⊗ δB s n . By repeatedly applying the definition of the Stratonovich integral, the iteratedStratonovich integral M n ( t ) may be realized as a linear combination of iterated Itˆointegrals: M n ( t ) = n X m = ⌈ n/ ⌉ n − m X α ∈J mn I nt ( α ) , where J mn := ( ( α , . . . , α m ) ∈ { , } m : m X i =1 α i = n ) , and, for α ∈ J mn , I nt ( α ) is the iterated Itˆo integral I nt ( α ) = Z ∆ m ( t ) dX s ⊗ · · · ⊗ dX ms m with dX is = (cid:26) dB s if α i = 1 P ∞ j =1 h j ⊗ h j ds if α i = 2 ;compare with Proposition 1 of [4].As in equation (4.2), letting F σn ( k ⊗ · · · ⊗ k n ) := [[ · · · [ k σ (1) , k σ (2) ] , · · · ] , k σ ( n ) ] , we may write g t = r − X n =1 X σ ∈S n c σn F σn ( M n ( t ))= r − X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn F σn ( I nt ( α )) , presuming the integrals F σn ( I nt ( α )) are defined.For each α , let p α = { i : α i = 1 } and q α = { i : α i = 2 } (so that p α + q α = m when α ∈ J mn ), and let J n := n [ m = ⌈ n/ ⌉ J mn . Then, for each σ ∈ S n and α ∈ J n , F σn ( I nt ( α )) = Z ∆ pα ( t ) f α ( s, t ) ˆ F σ,αn ( dB s ⊗ · · · ⊗ dB s pα ) , where ˆ F σ,αn and f α are defined as follows. ˆ F σ,αn : g ⊗ p α → g is defined by(4.13) ˆ F σ,αn ( k ⊗ · · · ⊗ k p α ):= ∞ X j ,...,j qα =1 F σ ′ n ( k ⊗ · · · ⊗ k p α ⊗ h j ⊗ h j ⊗ · · · ⊗ h j qα ⊗ h j qα ) , for { h j } ∞ j =1 an orthonormal basis of g CM and σ ′ = σ ′ ( α ) ∈ S n given by σ ′ = σ ◦ τ − ,for any τ ∈ S n such that τ ( dX s ⊗ · · · ⊗ dX ms m )= ∞ X j , ··· ,j qα =1 dB s ⊗ · · · ⊗ dB s pα ⊗ h j ⊗ h j ⊗ · · · ⊗ h j qα ⊗ h j qα ds · · · ds q α . The function f α is a polynomial of order q α in s = ( s , . . . , s p α ) and t . Thus, f α may be written as(4.14) f α ( s, t ) = q α X a =0 b aα t a ˜ f α,a ( s ) , for some coefficients b aα ∈ R and polynomials ˜ f α,a of degree q α − a in s . If ˆ F σ,αn isHilbert-Schmidt on g ⊗ p α CM , then Z ∆ pα ( t ) (cid:13)(cid:13)(cid:13) ˜ f α,a ( s ) ˆ F σ,αn (cid:13)(cid:13)(cid:13) ds = (cid:13)(cid:13)(cid:13) ˜ f α,a (cid:13)(cid:13)(cid:13) L (∆ pα ( t )) (cid:13)(cid:13)(cid:13) ˆ F σ,αn (cid:13)(cid:13)(cid:13) < ∞ , and Corollary 4.2 implies that F σn ( I nt ( α )) = q α X a =0 b aα t a J n ( ˜ f α,a ˆ F σ,αn ) t (4.15)is well-defined. In particular, if α m = 1, then f α = f α ( s ) does not depend on t ,and Corollary 4.2 implies that F σn ( I nt ( α )) is a v -valued L -martingale.The next two results show that ˆ F σ,αn is Hilbert-Schmidt as desired. Lemma 4.3.
Let n ∈ { , . . . , r } , σ ∈ S n , and α ∈ J n . For any v ∈ v , h ˆ F σ,αn , v i isa Hilbert-Schmidt operator on g ⊗ p α CM . Proof.
First consider the case n = 2. In this case, p α = 0 or p α = 2. If p α = 0,then ˆ F σ,α = P ∞ i =1 F σ ( h i ⊗ h i ) = 0. If p α = 2, then ˆ F σ,α ( k ⊗ k ) = F σ ( k ⊗ k ) =[ k σ (1) , k σ (2) )] is Hilbert-Schmidt by Corollary 3.12, and thus h ˆ F σ,α , v i is Hilbert-Schmidt. For n = 3, p α = 1 or p α = 3. If p α = 3, then α = (1 , ,
1) andˆ F σ,α ( k ⊗ k ⊗ k ) = F σ ′ ( k ⊗ k ⊗ k ) = [[ k σ (1) , k σ (2) ] , k σ (3) ]is Hilbert-Schmidt, again by Corollary 3.12. If p α = 1, then α = (1 ,
2) or α = (2 , F σ,α ( k ) = ∞ X i =1 F σ ′ ( k ⊗ h i ⊗ h i ) , EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 25 and we need only consider the case that F σ ′ ( k ⊗ h ⊗ h ) = [[ h, k ] , h ] . So let { k i } ∞ i =1 be an orthonormal basis of g CM and { e ℓ } Nℓ =1 be an orthonormal basisof v . As in the proof of Corollary 3.12, expanding terms in an orthonormal basis of v and applying the Cauchy-Schwarz inequality gives kh ˆ F σ,α , v ik = ∞ X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =1 h [[ h j , k i ] , h j ] , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∞ X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =1 N X ℓ =1 h [ e ℓ , h j ] , v ih e ℓ , [ h j , k i ] i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ∞ X i =1 N X ℓ =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =1 h [ e ℓ , h j ] , v ih e ℓ , [ h j , k i ] i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ∞ X i =1 N X ℓ =1 ∞ X j =1 |h [ e ℓ , h j ] , v i| ∞ X j =1 |h e ℓ , [ h j , k i ] i| ≤ N ∞ X j =1 N X ℓ =1 |h [ e ℓ , h j ] , v i| ∞ X i,j =1 N X ℓ =1 |h e ℓ , [ h j , k i ] i| ≤ N k v k < D − > k [ · , · ] k · k [ · , · ] k . Now assume h ˆ F σ,αn − , v i is Hilbert-Schmidt for all σ ∈ S n − and α ∈ J n − , andconsider h ˆ F σ,αn , v i for some σ ∈ S n and α ∈ J mn . Let a = p α and b = q α , and notethat either a ≥ F σ,αn ( k ⊗ · · · ⊗ k a )= ∞ X j ,...,j b =1 F σ ′ n ( k ⊗ · · · ⊗ k a ⊗ h j ⊗ h j ⊗ · · · ⊗ h j b ⊗ h j b )= ∞ X j ,...,j b =1 [ F σ ′′ n − ( k ⊗ · · · ⊗ k d − ⊗ k d +1 ⊗ · · · ⊗ k a ⊗ h j ⊗ · · · ⊗ h j b ) , k d ]= [ ˆ F τ,βn − ( k ⊗ · · · ⊗ k d − ⊗ k d +1 ⊗ · · · ⊗ k a ) , k d ] , (4.16)for some d ∈ { , . . . , a } , σ ′′ , τ ∈ S n − , and β ∈ J m − n − such that p β = p α − q β = q α , or b ≥ F σ,αn ( k ⊗ · · · ⊗ k a )= ∞ X j ,...,j b =1 [ F σ ′′ n − ( k ⊗ · · · ⊗ k a ⊗ h j ⊗ · · · ⊗ h j d − ⊗ h j d ⊗ h j d +1 ⊗ · · · ⊗ h j b ) , h j d ]= ∞ X j d =1 [ ˆ F τ,βn − ( k ⊗ · · · ⊗ k a ⊗ h j d ) , h j d ] , (4.17) for some d ∈ { , . . . , b } , σ ′′ , τ ∈ S n − and β ∈ J mn − such that p β = p α + 1 and q β = q α −
1. In the first case, working as above for n = 3, (cid:13)(cid:13)(cid:13) h ˆ F σ,αn , v i (cid:13)(cid:13)(cid:13) = ∞ X i ,...,i a =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h F σ ′ n ( k i ⊗ · · · ⊗ k i a ⊗ h j ⊗ · · · ⊗ h j b ) , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∞ X i ,...,i a =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h [ F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , k i d ] , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ∞ X i ,...,i a =1 N X ℓ =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ ih [ e ℓ , k i d ] , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N ∞ X i ,...,i a =1 N X ℓ =1 |h [ e ℓ , k i d ] , v i| (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N k v k k [ · , · ] k N X ℓ =1 (cid:13)(cid:13)(cid:13) h ˆ F τ,βn − , e ℓ i (cid:13)(cid:13)(cid:13) , which is finite by the induction hypothesis. Similarly, in the second case (cid:13)(cid:13)(cid:13) h ˆ F σ,αn , v i (cid:13)(cid:13)(cid:13) = ∞ X i ,...,i a =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h [ F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , h j d ] , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ∞ X i ,...,i a =1 N X ℓ =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j b =1 h F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ ih [ e ℓ , h j d ] , v i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ∞ X i ,...,i a =1 N X ℓ =1 ∞ X j d =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j d − ,j d +1 ,...,j b =1 h F σ ′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × N X ℓ =1 ∞ X j d =1 |h [ e ℓ , h j d ] , v i| ≤ N N X ℓ =1 (cid:13)(cid:13)(cid:13) h ˆ F τ,βn − , e ℓ i (cid:13)(cid:13)(cid:13) · k v k k [ · , · ] k . Proposition 4.4.
Let n ∈ { , . . . , r } , σ ∈ S n , and α ∈ J n . Then ˆ F σ,αn : g ⊗ p α CM → v is Hilbert-Schmidt. EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 27
Proof.
This proof is analogous to that of Lemma 4.3. For ˆ F σ,αn as in equation(4.17), we have k ˆ F σ,αn k = ∞ X i ,...,i a =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j ,...,j b =1 [ F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , h j d ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ N ∞ X i ,...,i a =1 N X ℓ =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j ,...,j b =1 h F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ i [ e ℓ , h j d ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ N ∞ X ℓ =1 ∞ X j ℓ =1 k [ e ℓ , h j d ] k × ∞ X i ,...,i a =1 N X ℓ =1 ∞ X j d =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j ,...,j d − ,j d +1 ,...,j b =1 h F σ ′′ n − ( k i ⊗ · · · ⊗ h j b ) , e ℓ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N k [ · , · ] k N X d =1 (cid:13)(cid:13)(cid:13) h ˆ F τ,βn − , e ℓ i (cid:13)(cid:13)(cid:13) , which is finite by Corollary 3.12 and Lemma 4.3. In a similar way, one may showthat, for ˆ F σ,αn as in equation (4.16), k ˆ F σ,αn k ≤ N k [ · , · ] k N X d =1 (cid:13)(cid:13)(cid:13) h ˆ F τ,βn − , e ℓ i (cid:13)(cid:13)(cid:13) . Remark . The proofs of the previous propositions rely strongly on v being finitedimensional. Thus, if we were to extend the results of this paper to v an infinitedimensional Lie algebra, another proof would be required here, or more likely, sometrace class requirements on the Lie bracket of g .Proposition 4.4 allows us to make the following definition. Definition 4.6. A Brownian motion on G is the continuous G -valued processdefined by g t = r X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn Z ∆ pα ( t ) f α ( s, t ) ˆ F σ,αn ( dB s ⊗ · · · ⊗ dB s pα ) , where c σn = ( − e ( σ ) (cid:30) n (cid:20) n − e ( σ ) (cid:21) , ˆ F σ,αn is as defined in (4.13) and f α is a polynomial of degree q α in s = ( s , . . . , s p α )and t as described in (4.14). For t >
0, let ν t = Law( g t ) be the heat kernel measureat time t , a probability measure on G . Example . Suppose that g is nilpotent of step 3. Then g t = X n =1 X σ ∈S n c σn F σn ( M n ( t ))= X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn F σn ( I nt ( α ))= X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn Z ∆ pα ( t ) f α ( s, t ) ˆ F σ,αn ( dB s ⊗ · · · ⊗ dB s pα ) . For n = 1, there is the single term given by M ( t ) = Z t δB s = B t . For n = 2, J = { (1 , , (2) } , and so M ( t ) = I t ((1 , I t ((2))= Z ∆ ( t ) dB s ⊗ dB s + 12 Z t h i ⊗ h i ds = Z ∆ ( t ) dB s ⊗ dB s + 12 t ∞ X i =1 h i ⊗ h i . There are of course just two permutations: σ = (12) with e ( σ ) = 0 and c σ = , and τ = (21) with e ( τ ) = 1 and c τ = − , and, by the antisymmetry of the Lie bracket, X σ ∈S c σ F σ ( M ( t )) = 14 [ dB s , dB s ] −
14 [ dB s , dB s ] = 12 [ dB s , dB s ] . For n = 3, the permutations are (123) with e = 0, (213), (132), (312), (231) with e = 1, and (321) with e = 2. Thus, X σ ∈S c σ F σ ( k ⊗ k ⊗ k ) = 19 [[ k , k ] , k ] −
118 [[ k , k , ] , k ] −
118 [[ k , k ] , k ] −
118 [[ k , k ] , k ] −
118 [[ k , k ] , k ] + 19 [[ k , k , ] , k ]= 16 [[ k , k ] , k ] + 16 [[ k , k , ] , k ] . (4.18) EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 29
Also, J = { (1 , , , (1 , , (2 , } , and so M ( t ) = I t ((1 , , I t ((1 , I t ((2 , Z ∆ ( t ) dB s ⊗ dB s ⊗ dB s + 12 Z ∆ ( t ) ∞ X i =1 dB s ⊗ h i ⊗ h i ds + 12 Z t ∞ X i =1 s h i ⊗ h i ⊗ dB s = Z ∆ ( t ) dB s ⊗ dB s ⊗ dB s + 12 Z t ∞ X i =1 ( t − s ) dB s ⊗ h i ⊗ h i + 12 Z t ∞ X i =1 s h i ⊗ h i ⊗ dB s . Note that f (1 , ( s, t ) = t − s and f (2 , ( s, t ) = s . Plugging this into equation(4.18) gives, for the α = (1 , , ∈ J term, X σ ∈S c σ F σ ( I t ((1 , , X σ ∈S c σ Z ∆ ( t ) F σ ( dB s ⊗ dB s ⊗ dB s )= 16 Z ∆ ( t ) ([[ dB s , dB s ] , dB s ] + [[ dB s , dB s ] , dB s ]) . For α = (1 , ∈ J , X σ ∈S c σ F σ ( I t (1 , Z t ∞ X i =1 ( t − s )[[ dB s , h i ] , h i ] , and ˆ F σ, (1 , ( k ) = ∞ X i =1 F σ ( k ⊗ h i ⊗ h i )with σ ′ = σ . For α = (2 , ∈ J , X σ ∈S c σ F σ ( I t ((2 , Z t ∞ X i =1 s [[ dB s , h i ] , h i ] , and note that, in this case,ˆ F σ, (2 , ( k ) = ∞ X i =1 F σ ′ ( k ⊗ h i ⊗ h i ) = ∞ X i =1 F σ ( h i ⊗ h i ⊗ k ) , and so σ ′ = σ ◦ (231) (or σ ′ = σ ◦ (321)). Combining the above, Brownian motionon G may be written as g t = B t + 12 Z ∆ ( t ) [ dB s , dB s ]+ 112 Z ∆ ( t ) ([[ dB s , dB s ] , dB s ] + [[ dB s , dB s ] , dB s ])+ 124 ∞ X i =1 Z t (( t − s )[[ dB s , h i ] , h i ] + s [[ dB s , h i ] , h i ])= B t + 12 Z t [ B s , dB s ] + 112 Z ∆ ( t ) ([[ B s , dB s ] , dB s ] + [[ dB s , dB s ] , B s ])+ 124 ∞ X i =1 t [[ B t , h i ] , h i ] . Remark . In principle, the Brownian motion on G has generator∆ = ∞ X i =1 ˜ h i , where { h i } ∞ i =1 is an orthonormal basis of g CM = H ⊕ v and ˜ h is the unique leftinvariant vector field on G such that ˜ h ( e ) = h , and ∆ is well-defined independentof the choice of orthonormal basis. Then the heat kernel measure { ν t } t> has thestandard characterization as the unique family of probability measures such that ν t ( f ) := R G f dν t is continuously differentiable in t for all f ∈ C b ( G ) and satisfies ddt ν t ( f ) = 12 ν t (∆ f ) with lim t ↓ ν t ( f ) = f ( e ) . However, this realization of ν t is not necessary for our results. Proposition 4.9 (Finite dimensional approximations) . For P ∈ Proj( W ) , let g Pt be the continuous process on G P defined by g Pt = r X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn Z ∆ pα ( t ) f α ( s, t ) ˆ F σ,αn ( dπB s ⊗ · · · ⊗ dπB s pα ) , for π ( w, x ) = ( P w, x ) . Then g Pt is Brownian motion on G P . In particular, let g ℓt = g P ℓ t , for projections { P ℓ } ∞ ℓ =1 ⊂ Proj( W ) as in equation (4.4). Then, for all p ∈ [1 , ∞ ) and t < ∞ , (4.19) lim ℓ →∞ E (cid:20) sup τ ≤ t (cid:13)(cid:13) g ℓτ − g τ (cid:13)(cid:13) p g (cid:21) = 0 . Proof.
First note that g Pt solves the Stratonovich equation δg Pt = L g Pt ∗ δP B t with g P = e , see [4, 8, 3]. Thus, g Pt is a G P -valued Brownian motion.Now, if β t a Brownian motion on W , then, for all p ∈ [1 , ∞ ),lim ℓ →∞ E (cid:20) sup τ ≤ t k P ℓ β τ − β τ k pW (cid:21) = 0; EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 31 see, for example, Proposition 4.6 of [10]. Thus,lim ℓ →∞ E (cid:20) sup τ ≤ t k π ℓ B τ − B τ k p g (cid:21) = 0 . By equation (4.15) and its preceding discussion, g ℓt = r X n =1 X σ ∈S n n X m = ⌈ n/ ⌉ c σn n − m X α ∈J mn q α X a =0 b aα t a J ℓn ( ˜ f α ˆ F σ,αn ) t , and thus, to verify (4.19), it suffices to show that, for all p ∈ [1 , ∞ ),lim ℓ →∞ E (cid:20) sup τ ≤ t (cid:13)(cid:13)(cid:13) J ℓn ( ˜ f α ˆ F σ,αn ) τ − J n ( ˜ f α ˆ F σ,αn ) τ (cid:13)(cid:13)(cid:13) p v (cid:21) = 0 , for all n ∈ { , . . . , r } , σ ∈ S n and α ∈ J n . By Proposition 4.4, ˆ F σ,αn is Hilbert-Schmidt, and recall that ˜ f α is a deterministic polynomial function in s . Thus J ℓn ( ˜ f α ˆ F σ,αn ) and J n ( ˜ f α ˆ F σ,αn ) are v -valued martingales as defined in Corollary 4.2.So, by Doob’s maximal inequality, it suffices to show thatlim ℓ →∞ E (cid:13)(cid:13)(cid:13) J ℓn ( ˜ f α ˆ F σ,αn ) t − J n ( ˜ f α ˆ F σ,αn ) t (cid:13)(cid:13)(cid:13) p v = 0Corollary 4.2 gives the limit for p = 2. For p >
2, since each J ℓn ( ˜ f α ˆ F σ,αn ) and J n ( ˜ f α ˆ F σ,αn ) has chaos expansion terminating at degree n , a theorem of Nelson (seeLemma 2 of [21] and pp. 216-217 of [20]) implies that, for each j ∈ N , there exists c j < ∞ such that E (cid:13)(cid:13)(cid:13) J ℓn ( ˜ f α ˆ F σ,αn ) t − J n ( ˜ f α ˆ F σ,αn ) t (cid:13)(cid:13)(cid:13) j v ≤ c j (cid:18) E (cid:13)(cid:13)(cid:13) J ℓn ( ˜ f α ˆ F σ,αn ) t − J n ( ˜ f α ˆ F σ,αn ) t (cid:13)(cid:13)(cid:13) v (cid:19) j . Heat kernel measure
We collect here some properties of the heat kernel measure on G . The followingresults are completely analogous to Corollary 4.9 of [10] and Proposition 4.6 in [12].The proofs are included here for the convenience of the reader. Proposition 5.1.
For any t > , the heat kernel measure ν t is invariant under theinversion map g g − for any g ∈ G . Proof.
The heat kernel measures ν P n t = Law( g nt ) on the finite dimensionalgroups G P n are invariant under inversion (see, for example, [13]). Suppose that f : G → R is a bounded continuous function. By passing to a subsequence if necessary,we may assume that the sequence of G P n -valued random variables { g nt } ∞ n =1 inProposition 4.9 converges almost surely to g t . Thus, by dominated convergence, E (cid:2) f (cid:0) g − t (cid:1)(cid:3) = lim n →∞ E (cid:2) f (cid:0) ( g nt ) − (cid:1)(cid:3) = lim n →∞ E [ f ( g nt )] = E [ f ( g t )] . Since ν t is the law of g t , this completes the proof. Proposition 5.2.
For all t > , ν t ( G CM ) = 0 . Proof.
Let µ t denote Wiener measure on W with variance t . Then for a boundedmeasurable function f on G = W ⊕ v such that f ( w, x ) = f ( w ), Z G f ( w ) dν t ( w, x ) = E [ f ( β t )] = Z W f ( w ) dµ t ( w ) . Let π : W × v → W be the projection π ( w, x ) = w . Then π ∗ ν t = µ t , and thus ν t ( G CM ) = ν t (cid:0) π − ( H ) (cid:1) = π ∗ ν t ( H ) = µ t ( H ) = 0 . This proposition gives some justification to our calling G CM the Cameron-Martinsubgroup of G . In the next section, we further justify this by showing that aCameron-Martin type quasi-invariance theorem holds for ν t .5.1. Quasi-invariance and Radon-Nikodym derivative estimates.
The fol-lowing theorem states that the heat kernel measure ν t = Law( g t ) is quasi-invariantunder left and right translation by elements of G CM and gives estimates for theRadon-Nikodym derivatives of the translated measures. Theorem 5.3.
For all h ∈ G CM and t > , ν t ◦ L − h and ν t ◦ R − h are absolutelycontinuous with respect to ν t . Let Z lh := d ( ν t ◦ L − h ) dν t and Z rh := d ( ν t ◦ R − h ) dν t be the Radon-Nikodym derivatives, K be lower bound on the Ricci curvature of G as in Corollary 3.23, and c ( t ) := te t − , for all t ∈ R , with the convention that c (0) = 1 . Then, Z lh , Z rh ∈ L p ( ν t ) for all p ∈ [1 , ∞ ) , andboth satisfy the estimate k Z ∗ h k L p ( ν t ) ≤ exp (cid:18) c ( Kt )( p − t d CM ( e , h ) (cid:19) , where ∗ = l or ∗ = r . Proof.
As in [10], the proof of this theorem is an application of Theorem 7.3 andCorollary 7.4 in [11] on the quasi-invariance of heat kernel measures for inductivelimits of finite dimensional Lie groups. In applying these results, the reader shouldtake G = G CM , A = Proj( W ), s P = π P , ν P = Law( g Pt ), and ν = ν t = Law( g t ).We now verify that the hypotheses of Theorem 7.3 of [11] are satisfied.By Corollary 3.20, the inductive limit group ∪ P ∈ Proj( W ) G P is a dense subgroupof G CM . By Proposition 4.9, for any { P n } ∞ n =1 ⊂ Proj( W ) with P n | H ↑ I H and f : G → R a bounded continuous function,(5.1) Z G f dν = lim n →∞ Z G Pn ( f ◦ i P n ) dν P n , and thus the heat kernel measure is consistent on finite dimensional projections of G CM . Corollary 3.23 says that K > −∞ and Ric P ≥ Kg P , for all P ∈ Proj( W ),and thus the Ricci curvature is uniformly bounded on these projections. Lastly, thelength of a path in the inductive limit group can be approximated by the lengthsof paths in the finite dimensional projections. That is, for any P ∈ Proj( W ) and ϕ ∈ C ([0 , , G CM ) with ϕ (0) = e , there exists an increasing sequence { P n } ∞ n =1 ⊂ Proj( W ) such that P ⊂ P n , P n | H ↑ I H , and ℓ CM ( ϕ ) = lim n →∞ ℓ G Pn ( π n ◦ ϕ ) . EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 33
To see this, let ϕ ( t ) = ( A ( t ) , a ( t )) be a path in G CM , and recall that, by equation(3.8), ℓ G Pn ( π n ◦ ϕ ) = Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) π n ϕ ′ ( s ) + r − X ℓ =1 d ℓ ad ℓπ n ϕ ( s ) π n ϕ ′ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) g CM ds = Z vuut k P n A ′ ( s ) k H + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a ′ ( s ) + r − X ℓ =1 d ℓ ad ℓπ n ϕ ( s ) π n ϕ ′ ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v ds Applying dominated convergence to this equation shows that (5.1) holds for anysuch choice of P n | H ↑ I H such that P ⊂ P n .We also have the usual strong converse to quasi-invariance of ν t under transla-tions by elements in G CM . Proposition 5.4.
For h ∈ G \ G CM and t > , ( ν t ◦ L − h ) and ν t are singular and ( ν t ◦ R − h ) and ν t are singular. Proof.
Again, let µ t denote Wiener measure on W with variance t . Let h =( A, a ) ∈ G \ G CM with A ∈ W \ H and a ∈ v . Given a measurable subset U ⊂ W , ν t ( U × v ) = P ( β t ∈ U ) = µ t ( U ) . If A ∈ W \ H , µ t ( · − A ) and µ t are singular; for example, see Corollary 2.5.3 of[6]. Thus, there are disjoint subsets W and W of W such that µ t ( W ) = 1 = µ t ( W − A ). Note that L − k ( U × v ) = R − k ( U × v ) = ( U − A ) × v . Thus, for G i := W i × v for i = 0 , G is the disjoint union of G and G , and ν t ( G ) = µ t ( W ) = 1 while ν t (cid:0) R − k ( G ) (cid:1) = ν t (cid:0) L − k ( G ) (cid:1) = ν t (( W − A ) × v ) = µ t ( W − A ) = 1 . Proposition 5.5.
For all h ∈ G CM and t > , Z rh ( g ) = Z lh − ( g − ) . Proof.
By Proposition 5.1, ν t is invariant under inversions. Thus Z G f ( g · h ) dν t ( g ) = Z G f (cid:0) g − · h (cid:1) dν t ( g ) = Z G f (cid:16)(cid:0) h − · g (cid:1) − (cid:17) dν t ( g )= Z G f (cid:0) g − (cid:1) Z lh − ( g ) dν t ( g ) = Z G f ( g ) Z lh − (cid:0) g − (cid:1) dν t ( g ) . Logarithmic Sobolev inequality.Definition 5.6.
A function f : G → R is said to be a (smooth) cylinder function if f = F ◦ π P for some P ∈ Proj( W ) and some (smooth) function F : G P → R .Also, f is a cylinder polynomial if f = F ◦ π P for F a polynomial function on G P . Theorem 5.7.
Given a cylinder polynomial f on G , let ∇ f : G → g CM be thegradient of f , the unique element of g CM such that h∇ f ( g ) , h i g CM = ˜ hf ( g ) := f ′ ( g )( L g ∗ h e ) , for all h ∈ g CM . Then for K as in Corollary 3.23, Z G ( f ln f ) dν t − (cid:18)Z G f dν t (cid:19) · ln (cid:18)Z G f dν t (cid:19) ≤ − e − Kt K Z G k∇ f k g CM dν t . Proof.
Following the method of Bakry and Ledoux applied to G P (see Theorem2.9 of [14] for the case needed here) shows that E (cid:2)(cid:0) f ln f (cid:1) (cid:0) g Pt (cid:1)(cid:3) − E (cid:2) f (cid:0) g Pt (cid:1)(cid:3) ln E (cid:2) f (cid:0) g Pt (cid:1)(cid:3) ≤ − e − K p t K P E (cid:13)(cid:13) ( ∇ P f ) (cid:0) g Pt (cid:1)(cid:13)(cid:13) g PCM , for K P as in equation (3.12). Since the function x (1 − e − x ) /x is decreasing and K ≤ K P for all P ∈ Proj( W ), this estimate also holds with K P replaced with K .Now applying Proposition 4.9 to pass to the limit as P ↑ I gives the desired result. Remark . It is desirable to state Theorem 5.7 for a larger class of functionsin L ( ν t ). To do this, one must prove that the gradient operator ∇ : L ( ν t ) → L ( ν t ) ⊗ g CM is closable. Unfortunately, Theorem 5.3 doesn’t give good informationon the dependence of the Radon-Nikodym derivatives Z lh and Z rh on h , and so at thispoint we can’t prove the necessary integration by parts formulae to show closability. References
1. H´el`ene Airault and Paul Malliavin,
Quasi-invariance of Brownian measures on the group ofcircle homeomorphisms and infinite-dimensional Riemannian geometry , J. Funct. Anal. (2006), no. 1, 99–142. MR MR2264248 (2008b:60119)2. Dmitri Alekseevsky, Peter W. Michor, and Wolfgang A.F. Ruppert,
Extensions of Lie algebras ,2000.3. Fabrice Baudoin,
An introduction to the geometry of stochastic flows , Imperial College Press,London, 2004. MR MR2154760 (2006f:60003)4. G´erard Ben Arous,
Flots et s´eries de Taylor stochastiques , Probab. Theory Related Fields (1989), no. 1, 29–77. MR MR981567 (90a:60106)5. Arthur L. Besse, Einstein manifolds , Ergebnisse der Mathematik und ihrer Grenzgebiete(3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987.MR MR867684 (88f:53087)6. Vladimir I. Bogachev,
Gaussian measures , Mathematical Surveys and Monographs, vol. 62,American Mathematical Society, Providence, RI, 1998. MR MR1642391 (2000a:60004)7. Henri Cartan and Samuel Eilenberg,
Homological algebra , Princeton Landmarks in Math-ematics, Princeton University Press, Princeton, NJ, 1999, With an appendix by David A.Buchsbaum, Reprint of the 1956 original. MR MR1731415 (2000h:18022)8. Fabienne Castell,
Asymptotic expansion of stochastic flows , Probab. Theory Related Fields (1993), no. 2, 225–239. MR MR1227033 (94g:60110)9. Lawrence J. Corwin and Frederick P. Greenleaf, Representations of nilpotent Lie groups andtheir applications. Part I , Cambridge Studies in Advanced Mathematics, vol. 18, CambridgeUniversity Press, Cambridge, 1990, Basic theory and examples. MR MR1070979 (92b:22007)10. B. Driver and M. Gordina,
Heat kernel analysis on infinite-dimensional Heisenberg groups ,J. Funct. Anal. (2008), no. 2, 2395–2461.11. ,
Integrated Harnack inequalities on Lie groups , Submitted, arXiv:0711.4392 (2008).12. ,
Square integrable holomorphic functions on infinite-dimensional Heisenberg typegroups , to appear in Probab. Theory Related Fields (2008).13. Bruce K. Driver,
Integration by parts and quasi-invariance for heat kernel measures on loopgroups , J. Funct. Anal. (1997), no. 2, 470–547. MR MR1472366 (99a:60054a)14. Bruce K. Driver and Terry Lohrenz,
Logarithmic Sobolev inequalities for pinned loop groups ,J. Funct. Anal. (1996), no. 2, 381–448. MR MR1409043 (97h:58176)15. J. J. Duistermaat and J. A. C. Kolk,
Lie groups , Universitext, Springer-Verlag, Berlin, 2000.MR MR1738431 (2001j:22008)
EAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 35
16. D. B. Fuks,
Cohomology of infinite-dimensional Lie algebras , Contemporary Soviet Mathe-matics, Consultants Bureau, New York, 1986, Translated from the Russian by A. B. Sosinski˘ı.MR MR874337 (88b:17001)17. Yuzuru Inahama,
Logarithmic Sobolev inequality for H s -metric on pinned loop groups , In-fin. Dimens. Anal. Quantum Probab. Relat. Top. (2004), no. 1, 1–26. MR MR2021645(2005d:58061)18. Kiyosi Itˆo, Multiple Wiener integral , J. Math. Soc. Japan (1951), 157–169. MR MR0044064(13,364a)19. Hui Hsiung Kuo, Gaussian measures in Banach spaces , Lecture Notes in Mathematics, Vol.463, Springer-Verlag, Berlin, 1975. MR MR0461643 (57
The free Markoff field , J. Functional Analysis (1973), 211–227.MR MR0343816 (49 Quantum fields and Markoff fields , Partial differential equations (Proc. Sympos. PureMath., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence,R.I., 1973, pp. 413–420. MR MR0337206 (49
On extensions of Lie algebras by the Heisenberg algebra , Mat. Zametki (2005), no. 5, 745–747. MR MR2252954 (2007e:17016)23. Ichiro Shigekawa, Stochastic analysis , Translations of Mathematical Monographs, vol. 224,American Mathematical Society, Providence, RI, 2004, Translated from the 1998 Japan-ese original by the author, Iwanami Series in Modern Mathematics. MR MR2060917(2005k:60002)24. Robert S. Strichartz,
The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differ-ential equations , J. Funct. Anal. (1987), no. 2, 320–345. MR MR886816 (89b:22011)25. V. S. Varadarajan, Lie groups, Lie algebras, and their representations , Graduate Textsin Mathematics, vol. 102, Springer-Verlag, New York, 1984, Reprint of the 1974 edition.MR 85e:2200126. Bill Yankosky,
On nilpotent extensions of Lie algebras , Houston J. Math. (2001), no. 4,719–724. MR MR1874666 (2002k:17026) Department of Mathematics, University of Virginia, Charlottesville, VA 22936
E-mail address ::