HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS

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1 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS TAI MELCHER Abstract. This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-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 logarithmic Sobolev inequality holds in this setting. Contents 1. Introduction 1 Acknowledgement.. Preliminaries 3.1. Abstract Wiener spaces 3.. Extensions of Lie algebras 3 3. Semi-infinite Lie algebras and groups Examples Hilbert-Schmidt norms Length and distance Ricci curvature Brownian motion Multiple Itô integrals Brownian motion and finite dimensional approximations 5. Heat kernel measure Quasi-invariance and Radon-Nikodym derivative estimates Logarithmic Sobolev inequality 3 References Introduction We define Brownian motion on a class of infinite dimensional Lie groups which we call semi-infinite Lie groups. We then prove a Cameron-Martin type quasiinvariance result for the associated heat kernel measure, as well as a logarithmic Sobolev inequality. A particular example of these semi-infinite Lie algebras was treated in [1], and we build on the methods used there. We briefly describe here the main results and give an outline of the paper; see Sections and 3 for definitions. Let (W, H, µ) be an abstract Wiener space and v be a finite dimensional nilpotent Lie algebra equipped with an inner product. Mathematics Subject Classification. Primary 6J65 8D5; Secondary 58J65 E65. Key words and phrases. Heat kernel measure, infinite dimensional Lie group, quasi-invariance, logarithmic Sobolev inequality. 1

2 TAI MELCHER Let g W v be a nilpotent Lie algebra extension of W by v, and we will call g H v the Cameron-Martin Lie subalgebra of g. Since g is nilpotent, we may define an explicit group operation on g via the Baker-Campbell-Hausdorff-Dynkin formula, and W v equipped with this group operation will be denoted by G. Similarly, G H v with the same group operation is called the Cameron- Martin subgroup of G, and we equip G with the left invariant Riemannian metric which agrees with the inner product (A, a), (B, b) g A, B H + a, b v on g Te G. In Section, we set the notation and give some standard facts needed about abstract Wiener spaces and extensions of Lie algebras. In Section 3, we construct (nilpotent) semi-infinite Lie algebras and give some examples. We make some additional requirements so that the Lie bracket on g is continuous, making g into a Banach Lie algebra. In Section 3., this continuity gives bounded Hilbert-Schmidt norms for the Lie bracket, and, in Section 3.4, lower bounds on the Ricci curvature of G and a uniform lower bound on certain finite dimensional approximations of G. In Section 4, we define Brownian motion on G as the solution to a stochastic differential equation with respect to a Wiener process on g. For a sketch of this construction, let B t denote Brownian motion on g. Then, Brownian motion on G is the solution to the Stratonovich stochastic differential equation δg t g t δb t : L gt δb t, with g e (, ). (Note that here and throughout this paper, δg t and δb t denote Stratonovich differentials.) For t >, let n (t) denote the simplex in R n given by {s (s 1,, s n ) R n : < s 1 < 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(),, σ(n)), that is, e(σ) #{j < n : σ(j) > σ(j + 1)}. Then the Brownian motion on G may be written as r 1 / [ ]) g t (( 1) e(σ) n n 1 [[ [δb e(σ) sσ(1), δb sσ() ], ], δb sσ(n) ], n(t) n1 σ S n where this sum is finite since g is assumed to be nilpotent. In Section 4, we show that these stochastic integrals are well-defined and each may be expressed as a sum of iterated Itô integrals. We also show that g t may be realized as a limit of Brownian motions living on the finite dimensional approximations to G. In particular, we show 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 prove that ν t Law(g t ) is invariant under (right or left) translation by elements of G. Moreover, this theorem gives good bounds on the L p -norms of the Radon- Nikodym derivatives. These results are important for future applications to spaces of holomorphic functions on G, as in [1]. We also show in Theorem 5.7 that a logarithmic Sobolev inequality holds for polynomial cylinder functions on G. For heat kernel analysis, quasi-invariance results, and logarithmic Sobolev inequalities in related infinite dimensional settings, see [1, 19]. Acknowledgement. The author thanks the anonymous referee for the careful reading and the valuable comments and suggestions to improve this paper.

3 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 3. Preliminaries.1. Abstract Wiener spaces. In this section, we summarize several well known properties of Gaussian measures and abstract Wiener spaces that are required for the sequel. For proofs of these results, see Section of [1]. Also see [6, 1] for more on abstract Wiener spaces and some particular examples. Suppose that W is a real separable Banach space and B W is the Borel σ-algebra on W. Definition.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 ˆµ(u) : e iu(x) dµ(x) e 1 q(u,u), for all u W, 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.. Let µ be a Gaussian measure on a real separable Banach space W. For 1 p <, let (.1) C p : w p W dµ(w). For w W, let w H : W sup u W \{} u(w) q(u, u) and define the Cameron-Martin subspace H W by H : {h W : h H < }. Then (1) For all 1 p <, C p <. () H is a dense subspace of W. (3) There exists a unique inner product, H on H such that h H h, h H for all h H, and H is a separable Hilbert space with respect to this inner product. (4) For any h H, h W C h H. (5) If {k j } j1 is an orthonormal basis of H and ϕ is a bounded linear map from W to a real Hilbert space C, then (.) ϕ H C : ϕ(k j ) C ϕ(w) C dµ(w) <. j1 A simple consequence of (.) is that (.3) ϕ H C ϕ W C w W dµ(w) C ϕ W C. W.. Extensions of Lie algebras. Suppose v is a Lie algebra and Der(v) is the set of derivations on v. That is, Der(v) consists of all linear maps ρ : v v satisfying Leibniz 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 1, ρ ] ρ 1 ρ ρ ρ 1, for ρ 1, ρ Der(v). W

4 4 TAI MELCHER Der(v) is a subset of linear maps on v, so if v is a normed vector space, one may equip Der(v) with the usual norm (.4) ρ sup{ ρ(x) v : X v 1}. Now suppose that h and v are Lie algebras, and that there is a linear mapping and a skew-symmetric bilinear mapping satisfying, for all X, Y, Z h, α : h Der(v) ω : h h v, (B1) [α X, α Y ] α [X,Y ]h ad ω(x,y ) and (B) (α X ω(y, Z) ω([x, Y ] h, Z)). cyclic Then, one may verify that, for X 1 + V 1, X + V h v, [X 1 + V 1, X + V ] g : [X 1, X ] h + ω(x 1, X ) + α X1 V α X V 1 + [V 1, V ] v defines a Lie bracket on g : h v, and we say g is an extension of h over v. That is, g is the Lie algebra with ideal v and quotient algebra g/v h. The associated exact sequence is v ι1 g π h, where ι 1 is inclusion and π is projection. In fact, the following theorem (see, for example, []) states that these are the only extensions of h over v. Theorem.3. Isomorphism classes of extensions of h over v (that is, short exact sequences of Lie algebras v g h ) modulo the equivalence described by the commutative diagram of Lie algebra homomorphisms v g h id ϕ id 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 (B), where (α, ω) (α, ω ) if there exists a linear b : h v such that and α X α X + ad b(x), 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 algebra homomorphism α : h gl(v ) and ω H (h, V ), a Chevalley cohomology class with 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 [] also gives a nice (although unpublished) summary. Reference [8] gives some conditions under which the extension of h over v is nilpotent (when h

5 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 5 and v are nilpotent); [4] gives a characterization of extensions of a Lie algebra over a Heisenberg Lie algebra. 3. Semi-infinite Lie algebras and groups Throughout the rest of this paper (W, H, µ) will denote a real abstract Wiener space, and v will denote a nilpotent Lie algebra with dim(v) N <, equipped with an inner product, v and a Lie bracket [, ] v. Since v is finite dimensional, its bracket is necessarily continuous and there exists a constant c < such that [X, Y ] v c X v Y v, for all X, Y v. For simplicity, we will assume that c 1. Also, Der(v) will denote the derivations of v, equipped with the norm defined in (.4). We will consider the vector spaces g : W v and g : H v. Note that g is a Banach space in the norm (w, v) g : w W + v v, and g is a Hilbert space with respect to the inner product (A, a), (B, b) g : A, B H + a, b v. The associated Hilbertian norm on g is given by (A, a) g : A H + a v. Motivated by the discussion in Section., we may consider W as an abelian Lie algebra and construct extensions of W over v. So suppose there is a skew-symmetric continuous bilinear mapping ω : W W v and a continuous linear mapping α : W Der(v) such that α and ω satisfy (B1) and (B), which in this setting become (C1) [α X, α Y ] ad ω(x,y ) and (C) α X ω(y, Z) + α Y ω(z, X) + α Z ω(x, Y ), for all X, Y, Z W. Then we may define a Lie algebra structure on g W v via the Lie bracket [(X 1, V 1 ), (X, V )] g : (, ω(x 1, X ) + α X1 V α X V 1 + [V 1, V ] v ). Theorem.3 indicates that these are the only extensions of W over v. Since v is nilpotent, we may choose ω and α so that g is a nilpotent Lie algebra (see Section 3.1 for some examples). Thus, we make the following definition. Definition 3.1. Let (W, H, µ) be an abstract Wiener space and v a finite dimensional nilpotent Lie algebra. Then g W v endowed with a Lie bracket satisfying (1) [g, g] v, () [, ] : g g g is continuous, and (3) there exists r N such that ad m x, for all m r and x g will be called a semi-infinite Lie algebra. Each extension g will depend on (the equivalence class of) a given ω and α.

6 6 TAI MELCHER Notation 3.. Let ω : sup{ ω(w 1, w ) v : w 1 W w W 1} and α : sup{ α w v v : w W v v 1} be the uniform norms of ω and α, which are finite by their assumed continuity. It will be useful to note that (3.1) [, ] : sup{ [g 1, g ] v : g 1 g g g 1} ω + α + 1 <, and similarly (3.) C : C(ω, α) : sup{ [h, k] v : h g k g 1} [, ] <. Thus, for all h, k g, ad l hk v C l h l g k g, for l 1,, r 1. The Baker-Campbell-Hausdorff-Dynkin formula implies that r 1 log(e A e B ) A + B + for all A, B g, where k1 (3.3) a k n,m : a k n,mad n1 A adm1 B (n,m) I k ( 1) k (k + 1)m!n!( n + 1), adn k A adm k B A, I k : {(n, m) Z k + Z k + : n i + m i > for all 1 i k}, and for each multi-index n Z k +, n! n 1! n k! and n n n k ; see, for example, [15]. Since g is nilpotent of step r, ad n1 A adm1 B adn k A adm k B A if n + m r. for A, B g. Since g is simply connected and nilpotent, the exponential map is a global diffeomorphism (see, for example, Theorems 3.6. of [7] or 1..1 of [9]). In particular, we may view g as both a Lie algebra and Lie group, and one may verify that (3.4) r 1 g h g + h + k1 a k n,mad n1 g (n,m) I k ad m1 h adn k g ad m k h g defines a group structure on g. Note that g 1 g and the identity e (, ). Definition 3.3. When we wish to emphasize the group structure on g, we will denote g by G. Similarly, when we wish to view g as a subgroup of G, it will be denoted by G and will be called the Cameron-Martin subgroup. Remark 3.4. Note that, for the purpose of making Definition 3.1, it is not really necessary to assume the continuity of the bracket or that g be nilpotent. That is, Definition 3.1 is reasonable if v is a (not necessarily nilpotent) finite dimensional Lie algebra and we only require that [g, g] v. However, the group operation given here, and, in fact, all subsequent results included in this paper, rely on the nilpotence, and many results require the continuity of the bracket. Thus, we include these assumptions in the definition above.

7 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 7 Lemma 3.5. The Banach space topologies on g and g make G and G into topological groups. Proof. Since g and g are topological vector spaces, g g 1 g and (g 1, g ) g 1 + g are continuous by definition. The map (g 1, g ) [g 1, g ] is continuous in both the g and g topologies by the estimates in equations (3.1) and (3.). It then follows from (3.4) that (g 1, g ) g 1 g is continuous as well Examples. In this section, we give a few simple examples of semi-infinite Lie algebras. Example 3.6. If v is a finite dimensional inner product space, we may consider v as an abelian Lie algebra, and taking α yields the infinite dimensional (step, stratified) Heisenberg like Lie algebras described in [1]. Example 3.7. Suppose v is an N-dimensional nilpotent Lie algebra. One standard way to construct Lie algebra extensions is as follows. Let β : W v be a continuous linear map, and define α : W Der(v) as the inner derivation α X : ad β(x). In this case, (C1) and (C) 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 (, [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 decomposition W Nul(β) Nul(β), where dim(nul(β) ) dim(v) N. Thus, for X X 1 + X, Y Y 1 + Y W, ω(x 1 + X, Y 1 + Y ) [β(x 1 + X ), β(y 1 + 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 3.8. 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 by a nilpotent Lie algebra. For the sake of a concrete example, consider the following. Let and H W W (R 3 ) {σ : [, 1] R 3 : σ is continuous and σ() } { σ W : σ is absolutely continuous and 1 } σ(s) ds <, so that (W, H) is classical Wiener space. Let v R 3 be an abelian Lie algebra. Let σ 1 σ(s) ds ( σ 1, σ, σ 3 ), and define ω : W W R 3 by ω(σ, τ) ( σ 1 τ τ 1 σ, σ τ 3 τ σ 3, ) and α σ : R 3 R 3 by α σ (x, y, z) (,, σ 1 y σ 3 x). Then α σ α τ and (C1) is trivially satisfied. Using that α κ ω(σ, τ) (,, κ 1 ( σ τ 3 τ σ 3 ) κ 3 ( σ 1 τ τ 1 σ ))

8 8 TAI MELCHER one may verify that (C) is satisfied. g W R 3 is given by Thus, the Lie bracket for this extension [(σ, v), (τ, u)] (, σ 1 τ τ 1 σ, σ τ 3 τ σ 3, σ 1 u σ 3 u 1 τ 1 v + τ 3 v 1 ), [(κ, w), [(σ, v), (τ, u)]] (,,, κ 1 ( σ τ 3 τ σ 3 ) κ 3 ( σ 1 τ τ 1 σ )), and all higher order brackets are. As an aside, note that the bracket in this extension is essentially defined as the bracket of a linear Lie algebra, and the extension itself is analogous to a (standard) construction of T 4 {4 4 strictly upper triangular matrices} as an extension of R 3 by R 3. To see this, for A (a, b, c) R 3, define the isomorphisms a a c f(a) b c à and g(a) b Ā. Let U R 3 and V v R 3, and define ω : U U g(v ) by and α : U V g(v ) by ω (A, A ) f(a)f(a ) f(a )f(a) Ãà à à α AA f(a)g(a ) g(a )f(a) ÃĀ Ā Ã. Thus, T 4 R 6 U V with bracket determined by the pair (g 1 ω, g 1 α ). In particular, for the extension g W R 3 as given in this example, we have that ω g 1 ω β and α g 1 α β where β : W U is given by β(σ) ( σ 1, σ, σ 3 ). Example 3.9. Consider v R n R as an abelian Lie algebra. For ω : W W R n, we may write ω (ω 1,, 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 } n i1 is the standard basis for Rn. Thus, n α w (a 1,..., a n ) a i α i (w). Then α and ω satisfy (C) as long as i1 α 1 ω α n ω n. In the case n 1, this is not very interesting, since α ω implies that ω α β for some β W. For n, 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 1 γ and α γ, so that, for any u, w W and v (v 1, v ) R, ω(w, u) (Ω(w, u), Ω(w, u)) and α w v γ(w)(v 1 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)). Thus, for any (w, v, x), (w, v, x ), (w, v, x ) W v, [(w, v, x), (w, v, x )] (, ω(w, w ), α w v α w v) (, (Ω(w, w ), Ω(w, w )), γ(w)(v 1 v ) γ(w )(v 1 v )),

9 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 9 [(w, v, x ), [(w, v, x), (w, v, x )]] (,, α w ω(w, w )), and g is a step Lie algebra. The group operation is given by (w, v, x) (w, v, x ) (w + w, v + v + 1 (Ω(w, w ), Ω(w, w )), x + x + 1 (γ(w)(v 1 v ) γ(w )(v 1 v )). As an example of a particular appropriate Ω and γ, again let W W (R 3 ) and H be as in Example 3.8. Suppose ϕ is an anti-symmetric bilinear form on R 3, ρ : R 3 R is a linear map, and η is a finite measure on [, 1]. Then we may define and Ω(σ, τ) γ(σ) 1 1 ϕ(σ(s), τ(s)) dη(s) ρ(σ(s)) dη(s). Example 3.1. Here we make a slight modification on the previous example to construct a stratified step 3 Lie algebra. Let v R 6 R 3 R R be an abelian Lie algebra. Let Ω and γ be as in the previous example. Define ω : W W R 3 by ω(w, u) (Ω(w, u), Ω(w, u), Ω(w, u)) and α : W v v by α w ((v 1, v, v 3 ), (x 1, x ), y) (, (γ(w)(v 1 v ), γ(w)(v v 3 )), γ(w)(x 1 x )) (so α w is a particular element of the 6 6 strictly lower triangular matrices). Then α w α u α u α w and so α satisfies (C1), and also α v ω(w, u) (, (γ(v)(ω(w, u) Ω(w, u)), γ(v)(ω(w, u) Ω(w, u))), ), so α and ω satisfy (C) trivially. The Lie bracket is given by [(w, v, x, y), (w, v, x, y )] (, 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 3, [(w, v, x, y), (w, v, x, y )] 3 (γ(w)(v 1 v ) γ(w )(v 1 v ), γ(w)(v v 3) γ(w )(v v 3 )) R, and [(w, v, x, y), (w, v, x, y )] 4 γ(w)(x 1 x ) γ(w )(x 1 x ) R. Thus, [(w, v, x, y ),[(w, v, x, y), (w, v, x, y )]] (,, α w ω(w, w ), α w (α w v α w v)) (,,, α w α w v α w α w v) (,,, γ(w )γ(w)(v 1 v 3) γ(w )γ(w )(v 1 v 3 )),

10 1 TAI MELCHER and all higher order brackets are. So for g (w, v, x, y) and g (w, v, x, y ), the group operation is given by (g g ) 1 w + w (g g ) v + v + 1 ω(w, w ) (g g ) 3 x + x + 1 (α wv α w v) (g g ) 4 y + y + 1 (α wx α w x) (α wv + α w v α wα w (v v )). Clearly, this example may be further modified to make nilpotent Lie algebras of arbitrary step. 3.. Hilbert-Schmidt norms. In this section, we will show that the assumed continuity of ω and α makes the Lie bracket into a Hilbert-Schmidt operator on g. This result will be needed later in guaranteeing that our stochastic integrals are well-defined. Notation Let H 1,..., H n and V be Hilbert spaces, and let {h i j }dim(hi) j1 denote an orthonormal basis for each H i. If ρ : H 1 H n V is a multilinear map, then the Hilbert-Schmidt norm of ρ is defined by ρ : ρ H1 H n V ρ(h 1 j 1,..., h n j n ) V. j 1,...,j n In particular, for H an infinite dimensional Hilbert space with orthonormal basis {h i } i1, ρ : H n V is Hilbert-Schmidt if ρ ρ (H ) n V j 1,...,j n1 ρ(h j1,..., h jn ) V <. One may verify directly that these norms are independent of the chosen bases. Proposition 3.1. For all w W and x v, α w v v N α w W and α x H v C α x v, where N dim(v), C is as in equation (.1), and is as defined in Notation 3.. Also, ω(w, ) H v C ω w W. Furthermore, α NC α < and ω C ω <. Proof. Let {e i } N i1 be an orthonormal basis of v. Then, for any w W, N N α w v v α w e i v α w W e i v N α w W. i1 i1 For fixed x v, α x : W v is a continuous linear map. Thus, equation (.) gives α x H v α w x v dµ(w) W α w W x v dµ(w) C α x v. W

11 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 11 Similarly, for fixed w W and ω(w, ) : W v, ω(w, ) H v ω(w, w ) v dµ(w ) W ω w W w W dµ(w ) C ω w W. W Since w α w is a continuous linear map from W to v v, it follows from equations (.) and (.3) that α W α w v v dµ(w) W N α w W dµ(w) NC α, and since w ω(w, ) is a continuous linear map from W to H v, ω h ω(h, ) H (H v) ω(w, ) H v dµ(w) W C ω w W dµ(w) C ω. W This proposition easily gives the following result. Corollary For all m, [[[, ],...], ] : g m v is Hilbert-Schmidt. Proof. For m, this follows from the previous proposition and the continuity of the Lie bracket on v, since taking {h i } i1 {k i} i1 {e j} N j1, where {k i} i1 and {e j } N j1 are orthonormal bases of H and v, respectively, gives [, ] [, ] g g v [h i1, h i ] v i 1,i 1 N ω(k i1, k i ) v + α ki1 e j v i 1,i 1 i 11 j 1 N N + α ki e j1 v + [e j1, e j ] v i 1 j 11 j 1,j 1 ω + α + N <. Now assume the statement is true for all m,..., l. Consider m l+1. Writing [[h i1, h i ],, h il ] v in terms of the orthonormal basis {e j } N j1 and using multiple

12 1 TAI MELCHER applications of the Cauchy-Schwarz inequality gives [[[, ],...], ] [[[, ],...], ] (g ) l+1 v [[[h i1, h i ],, h il ], h il+1 ] v N i 1,...,i l+1 1 i 1,...,i l+1 1 N N [e j, h il+1 ] e j, [[h i1, h i ],, h il ] j1 i 1,...,i l+1 1 j1 i l+1 1 j1 N [e j, h il+1 ] v e j, [[h i1, h i ],, h il ] N [e j, h il+1 ] v i 1,...,i l 1 j1 N [, ] g v [[[, ],...], ] g l v, v N e j, [[h i1, h i ],, h il ] where in the penultimate inequality we have used that all terms in the sums are positive. The last line is finite by the induction hypothesis Length and distance. In this section, we define the Riemannian distance on G and show that the topology induced by this metric is equivalent to the Hilbert topology induced by g. For g G, let L g : G G and R g : G G denote left and right multiplication by g, respectively. As G is a vector space, to each g G we can associate the tangent space T g G to G at g, which is naturally isomorphic to G. Notation For f : G R a Frechét smooth function and v, x G and h g, let f (x)h : h f(x) d dt 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 any smooth curve in G such that σ() x and σ() v (for example, σ(t) x + tv), then L g v x d dt g σ(t). Notation Let T > and C 1 ([, T ], G ) denote the collection of C 1 -paths g : [, T ] G. The length of g is defined as l (g) : T L g 1 (s) g (s) g ds. The Riemannian distance between x, y G then takes the usual form d (x, y) : inf{l (g) : g C 1 ([, T ], G ) such that g() x and g(t ) y}. Note that the value of T in the definition of d functional is invariant under reparameterization. is irrelevant since the length

13 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 13 Proposition For g, x G and v x T x G, (3.5) r 1 L g v x v + a k n,m k1 (n,m) I k j {1,..., k} m j > m j 1 l adg n1 ad m1 x ad nj g ad l xad v ad mj l 1 x ad nj 1 g ad n k g ad m k x g, where a k n,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 note that d dt ad n1 g ad m1 x(t) adn k g ad m k x(t) g j {1,..., k} m j > m j 1 l adg n1 ad m1 x ad nj g Then using (3.4) and plugging this into L g v x d dt g x(t) d r 1 g + x(t) + dt yields the desired result. k1 ad l xad v ad mj l 1 x ad nj 1 g a k n,mad n1 g (n,m) I k ad m1 x(t) adn k g ad m k Example 3.17 (The step 3 case). When r 3, the group operation is g h g + h [g, h] + ([g, [g, h]] + [h, [h, g]]). 1 ad n k g ad m k x g. x(t) g Thus, L g v x d dt g x(t) d ( dt g + x(t) + 1 ) 1 [g, x(t)] + ([g, [g, x(t)]] + [x(t), [x(t), g]]) 1 v [g, v] + ([g, [g, v]] + [v, [x, g]] + [x, [v, g]]). 1 Lemma There exists a continuous decreasing function ε > such that, for all g G and v g, L g 1 v g g ε( g g ) v g.

14 14 TAI MELCHER (3.6) Proof. Let g G and v g. By equation (3.5), where with r 1 L g 1 v g v + m j 1 a k n,m k1 (n,m) I k m j> l ad n1 g ad m1 1 g ad nj g ad l gad 1 v ad mj l 1 g ad n k g ad m 1 k g g 1 r 1 v + ( 1) n 1 {mk >}a k n,mad m + n g v (I Λ(g))v, k1 (n,m) I k d l : l k1 r 1 Λ(g) : d l ad l g (n, m) I k m + n l l1 ( 1) n 1 {mk >}a k n,m. Since g is nilpotent, the operator Λ(g) is nilpotent. Thus, there exists M N so that M (I Λ(g)) 1 Λ(g) k, and v (I Λ(g)) 1 L g 1 v g. Note that the operator norm satisfies k r 1 Λ(g) d l C l g l g <, where C C(ω, α) is as defined in (3.). Therefore, taking ( M r 1 ) k ε( g g ) : d l C l g l g satisfies the desired estimate. l1 k l1 Proposition For all x G and R >, there exists δ δ(x, R) such that d (x, y) < δ implies that y x g < R. Proof. Fix x G and R >. We will determine δ δ(x, R) so that y x g R implies d (x, y) δ(x, R). Let B(x, R) : {z G : z x g < R}, and consider y G such that y / B(x, R). Then, for any C 1 -path g : [, 1] G such that g() x and g(1) y, there is some first time t such that g exits B(x, R), and l (g) 1 L g 1 (s) g (s) g ds ε( x g + R) t t g (s) g ds 1 L g 1 (s) g (s) g ds ε( x g + R) g(t ) x g ε( x g + R)R : δ(x, R), for ε as given in Lemma Since this estimate is true for any C 1 -path g from x to y / B(x, R), optimizing over all such paths gives the desired result.

15 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 15 Proposition 3.. There exists a continuous increasing function K < such that K() and, for all x, y G, d (x, y) (1 + K( x g y g )) y x g + o ( y x g ), as y x g, where the implied constants in o ( y x g ) also depend on x g y g. Proof. For notational simplicity, let T 1. Let g(s) be a path in C 1 ([, 1], G ). By equation (3.6), taking g g(s) and v g(s) g (s), 1 r 1 (3.7) l (g) g (s) + d l ad l g(s)g (s) ds. l1 g Now suppose that x, y G, and take g(s) x + s(y x) for s 1. Then (3.7) gives d (x, y) l (g) 1 r 1 (y x) + d l ad l x+s(y x)(y x) ds l1 g 1 r 1 (y x) + d l s n ad m1 x ady x n1 ad m l x ad n l y x(y x) ds. g l1 (n, m) I l m + n l Splitting off all terms in the sum of order two or higher and evaluating the integral, r 1 d (x, y) (y x) + d l ad l x(y x) l1 r 1 1 { n >} + d l n + 1 adm1 x ady x n1 ad m l x ad n l y x(y x) l1 r l1 (n, m) I l m + n l (n, m) I l m + n l g d l C l x l g y x g + o ( y x g ), where C C(ω, α) is as defined in (3.). Interchanging the roles of x and y in g(s), and thus in this inequality, completes the proof. Propositions 3.19 and 3. yield the following corollary. Corollary 3.1. The topology on G induced by d is equivalent to the Hilbert topology induced by g Ricci curvature. In this section, we compute the Ricci curvature of certain finite 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 derivatives of ν t. It will also be applied in Section 5. to prove a logarithmic Sobolev inequality 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. That is, i l : l i for all l W. Also, let H : {h H :, h H Range(i ) H }.

16 16 TAI MELCHER That is, for h H, h H if and only if, h H H extends to a continuous linear functional on W, which we will continue to denote by, h H. Because H is a dense subspace of W, i is injective and thus has a dense range. Since h, h H as a map from H to H is a linear isometric isomorphism, it follows that H h, h H W is a linear isomorphism also, and so H is a dense subspace of H. Now suppose that P : H H is a finite rank orthogonal projection such that P H H. Let {k j } m j1 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 m (3.8) P w : w, k j H k j j1 for all w W. For later purposes, we will also define π P (w, x) : (P w, x). Notation 3.. 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 (3.8). Further, let G P : P W v (a subgroup of G ), and we equip G P with the left invariant Riemannian metric induced from the restriction of the inner product on g H v to Lie(G P ) P H v : g P. Let Ric P denote the associated Ricci tensor at the identity in G P. Proposition 3.3. For X (A, a) g P, Ric P X, X g P 1 4 a, [, ] (g P ) (g P ) 1 [, X] (g P ) v, where (g P ) (P H) v. Proof. For g any nilpotent Lie algebra with orthonormal basis Γ, (3.9) Ric X, X 1 ad Y X 1 ad Y X, 4 Y Γ for all X g; see for example Theorem 7.3 and Corollary 7.33 of [5]. So let Γ m : {h i } m+n i1 {(k i, )} m i1 {(, e j)} N j1 be an orthonormal basis of g P P H v, where {k i} m i1 and {e j} N j1 are orthonormal bases of P H and v, respectively. Then, for Y g P, ad Y X ad Y X, h i g h i X, ad Y h i g h i. h i Γ m h i Γ m Thus, h i Γ m ad h i X g h i Γ m Y Γ h j Γ m X, ad hi h j g h i,h j Γ m X, [h i, h j ] g. Plugging this into (3.9) gives Ric P X, X g P 1 X, [h i, h j ] g 4 1 [h i, X] g h i,h j Γ m h i Γ m 1 a, [h i, h j ] v 1 [h i, X] 4 v. h i,h j Γ m h i Γ m

17 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 17 Corollary 3.4. Let K : 1 } { [, sup X] g v : X g 1. Then K > and K is the largest constant such that Ric P X, X g P K X g, for all X g P P, holds uniformly for all P Proj(W ). Proof. The first assertion is simple, since K 1 [, ] >, by Corollary Now, for P Proj(W ) as in Notation 3., Proposition 3.3 implies that Thus, Ric P X, X g P 1 [, X] (g P ) v. Ric P X, X g P X 1 [, X] (g P ) v X g P g P 1 { } (3.1) sup [, X] (g P ) v : X g P 1 : K P. Noting that the infimum of K P over all P Proj(W ) is K completes the proof. Remark 3.5. Of course, one can compute the Ricci curvature for G just as in Proposition 3.3. Choose an orthonormal basis Γ {h i } i1 {(k i, )} i1 {(, e j )} N j1 of g H v, where {k i } i1 is an orthonormal basis of H, and {e j } N j1 is an orthonormal basis of v. Then, for all X (A, a) g, Ric X, X g 1 4 a, [h i, h j ] v 1 [h i, X] v i,j1 i1 1 4 a, [, ] g g 1 [, X] g v K X g. 4. Brownian motion Suppose that B t is a smooth curve in g with B, and consider the differential equation ġ t g t Ḃ t : L gt Ḃt, with g e. The solution g t may be written as follows (see [6]): For t >, let n (t) denote the simplex in R n given by {s (s 1,, s n ) R n : < s 1 < 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(),, σ(n)), that is,

18 18 TAI MELCHER e(σ) #{j < n : σ(j) > σ(j + 1)}. Then (4.1) g t r n1 σ S n / [ (( 1) e(σ) n n 1 e(σ) ]) n(t) [ [Ḃs σ(1), Ḃs σ() ],..., ]Ḃs σ(n) ] ds. For n {1,, r} and σ S n, let Fn σ : g n v be the linear map given by (4.) F σ n (k 1 k n ) : [[ [k σ(1), k σ() ], ], k σ(n) ]. Recall that Fn σ is Hilbert-Schmidt by Corollary Then we may write ( r 1 ) (4.3) g t c σ nfn σ Ḃ s1 Ḃs n ds, n1 σ S n n(t) where the coefficients c σ n are determined by (4.1). Using this as our motivation, we first explore stochastic integral analogues of equation (4.3) where the smooth curve B is replaced by Brownian motion on g Multiple Itô integrals. Let, g n denote the inner product on g n arising from the inner product on g. Also, let {k i } i1 H be an orthonormal basis of H, and define P m Proj(W ) by m (4.4) P m (w) w, k i H k i, for all w W, i1 as in equation (3.8), and define π m (w, x) : π Pm (w, x) (P m (w), x) G Pm. Of course, dim(g Pm ) m + N, but in a mild abuse of notation, in this section we will use {h i } m i1 to denote an orthonormal basis of G P m, rather than the more cumbersome {h i } m+n i1 {(k i, )} m i1 {(, e i)} N i1, where {e i} N i1 is an orthonormal basis of v. Let {B t } t {(β t, βt v )} t be a Brownian motion on g W v with variance determined by E [ B s, h g B t, k g ] h, k g min(s, t), for all s, t and 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 Pm P m W v g. Proposition 4.1. For ξ L ( n (t), g n ) a continuous mapping, let Jn m (ξ) t : πm n ξ(s), db s1 db sn g n n(t) ξ(s), dπ m B s1 dπ m B sn g n. n(t) Then {J m n (ξ) t } t is a continuous L -martingale such that, for all m, E J m n (ξ) t ξ L ( n(t),g n ),

19 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 19 and there exists a continuous L -martingale {J n (ξ) t } t such that [ (4.5) lim m E ] sup Jn m (ξ) τ J n (ξ) τ, τ t for all t <. J n (ξ) t is well-defined independent of the choice of orthonormal basis {h i } i1 in (4.4), and so will be denoted by J n (ξ) t ξ(s), db s1 db sn g n. Proof. Note first that, m Jn m (ξ) t i 1,...,i n1 n(t) n(t) ξ(s), h i1 h in g n dbs i1 1 dbs in n where {B i } m i1 are independent real valued Brownian motions. Let ξ i 1,...,i n : ξ, h i1 h in. Then ξ i1,...,i n (s) ξ(s) g n and ξ i1,...,i n L ( n (t)). Thus, Jn m (ξ) t is defined as a (finite dimensional) vectorvalued multiple Wiener-Itô integral, see for example [, 5]. Now note that dj m n (ξ) t ξ(s 1,..., s n 1, t), dπ m B s1 dπ m B sn 1 dπ m B t g n n 1(t) m ξ(s 1,..., s n 1, t), dπ m B s1 dπ m B sn 1 h i g n dbt. i i1 n 1(t) Thus, the quadratic variation Jn m (ξ) t is given by m t ξ(s 1,..., s n 1, τ), dπ m B s1 dπ m B sn 1 h i g n dτ, and i1 n 1(τ) E Jn m (ξ) t E Jn m (ξ) t m t [ m τ1 E i 11 i 1 n (τ ) ξ(s 1,..., s n, τ, τ 1 ), dπ m B s1 ] dπ m B sn h i h i1 g n dτ dτ 1. Iterating this procedure n times gives m E Jn m (ξ) t ξ(τ 1,, τ n ), h i1 h in g n dτ 1 dτ n i 1,...,i n1 n(t) n(t) πm n ξ(s) ξ g n L ( n(t),g n ), and thus, for each n, J m n (ξ) t is bounded uniformly in L independent of m.

20 TAI MELCHER Now, for P Proj(W ), let Jn P (ξ) t : π n n(t) P ξ(s), db s 1 db sn g n. For P, Q Proj(W ), a similar argument to the above implies that (4.6) E Jn P (ξ) t Jn Q (ξ) t π n P n(t) ξ(s) π n ξ(s) ds. g n In particular, take P P m and Q P l with l m, and note that (4.7) π n m ξ(s) π n l ξ(s) g n n m n n i 1,...,i n1 n j1 π j 1 π j 1 m j1 i 1,...,i n1 n m m j1 i 1,...,i j 11 i jl+1 i j+1,...,i n1 Q (π m π l ) π n j 1 l ξ(s), h i1 h in g n (π m π l ) π n j 1 l ξ(s), h i1 h in g n l ξ(s), h i1 h in g n, as l, m, for all s n (t), since ξ <. Thus, equation (4.6) L ( n(t),g n ) implies that lim E J n m (ξ) t Jn(ξ) l t, l,m by dominated convergence, and {J m n (ξ) t } m1 is Cauchy in L. Since the space of continuous L -martingales is complete in the norm M E M t, there exists a continuous martingale {J n (ξ) t } t such that lim E J n m (ξ) t J n (ξ) t. m Combining this with Doob s maximal inequality proves equation (4.5). To see that J n (ξ) t is independent of the choice of basis, suppose now that {h j } j1 H is another orthonormal basis for H, and let P m : W H and π m : G G P m be the corresponding orthogonal projections. Consider the inequality (4.7) with π l replaced by π m. Writing π m π m (π m I) + (I π m), and considering terms for each fixed j, we have π j 1 m i 1,...,i n1 m (π m I) (π m) n j 1 ξ(s), h i1 h in g n i 1,...,i j 11 i jm+1 i j+1,...,i n1 m i 1,...,i j 11 i jm+1 i j+1,...,i n1 ξ(s), h i 1 h ij π mh ij+1 π mh in g n ξ(s), h i1 h in g n,

21 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 1 as m. Similarly, π j 1 m (I π m) (π m) n j 1 ξ(s), h i1 h in g n i 1,...,i n1 π j 1 m i 1,...,i n1 as m. Thus, (I π m) (π m) n j 1 ξ(s), h i 1 h i n g n lim m π n m ξ(s) (π m) n ξ(s), g n, for each s n (t). Thus, for Jn m (ξ) t : J P m n (ξ) t, using equation (4.6) with P P m and Q P m shows that lim m E J m n (ξ) t J m n (ξ) t, again by dominated convergence. A simple linearity argument extends the map J n to functions taking values in (g ) n v. Corollary 4.. Let F L ( n (t), (g ) n v) be a continuous map. That is, F : n (t) g n v is a map continuous in s and linear on g n such that F (s) ds F (s)(h j1 h jn ) v ds <. Then n(t) Jn m (F ) t : n(t) j 1,...,j n1 n(t) F (s)(dπ m B s1 dπ m B sn ) is a continuous L -martingale, and there exists a continuous v-valued L -martingale {J n (F ) t } t such that [ ] lim E sup Jn m (ξ) τ J n (ξ) τ v, m τ t for all t <. The martingale J n (ξ) t is well-defined independent of the choice of orthonormal basis {h i } i1 in (4.4), and thus will be denoted by J n (F ) t F (s)(db s1 db sn ). n(t) Proof. Let {e j } N j1 be an orthonormal basis of v. Then for any k 1,..., k n g, N F (s)(k 1 k n ) F (s)(k 1 k n ), e j e j. Since F (s)( ), e j is linear on g n, for each s there exists ξ j(s) g n (4.8) ξ j (s), k 1 k n F (s)(k 1 k n ), e j. If ξ j : n (t) g n j1 is defined by equation (4.8), then ξ j L ( n(t),g n ) F (s) ds <. n(t) such that

22 TAI MELCHER Thus, N N J n (F ) t ξ j (s), db s1 db sn e j J n (ξ j ) t e j, j1 n(t) j1 is well-defined, and, for each j, J n (ξ j ) is a martingale as defined in Proposition Brownian motion and finite dimensional approximations. Again let B t denote Brownian motion on g. By equation (4.1), the solution to the Stratonovich stochastic differential equation should be given by (4.9) g t r 1 c σ n n1 σ S n δg t L gt δb t, with g e, n(t) [[ [δb sσ(1), δb sσ() ], ], δb sσ(n) ], for coefficients c σ n determined by equation (4.1). To understand the integrals in (4.9), consider the following heuristic computation. Let {M n (t)} t denote the process in g n defined by M n (t) : δb s1 δb sn. n(t) By repeatedly applying the definition of the Stratonovich integral, the iterated Stratonovich integral M n (t) may be realized as a linear combination of iterated Itô integrals: n 1 M n (t) n m It n (α), where J m n : { m n/ α J m n (α 1,..., α m ) {1, } m : } m α i n, i1 and, for α Jn m, It n (α) is the iterated Itô integral It n (α) dxs 1 1 dxs m m with { dxs i m(t) db s if α i 1 j1 h j h j ds if α i ; compare with Proposition 1 of [4]. This change from multiple Stratonovich integrals to multiple Itô integrals may also be recognized as a specific case of the Hu-Meyer formulas [17, 18], but we will compute more explicitly to verify that our integrals are well-defined. As in equation (4.), letting we may write g t r 1 n1 F σ n (k 1 k n ) : [[ [k σ(1), k σ() ], ], k σ(n) ], σ S n c σ nf σ n (M n (t)) r 1 n n1 σ S n m n/ c σ n n m α J m n F σ n (I n t (α)),

23 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 3 presuming we can make sense of the integrals Fn σ (It n (α)). For each α, let p α #{i : α i 1} and q α #{i : α i } (so that p α + q α m when α Jn m ), and let n J n : Jn m. pα (t) m n/ Then, for each σ S n and α J n, Fn σ (It n (α)) f α (s, t) ˆF n (db s1 db spα ), ˆF n where and f α are as follows. The map ˆF n : g pα g is defined by (4.1) ˆF n (k 1 k pα ) : Fn σ (k 1 k pα h j1 h j1 h jqα h jqα ), j 1,...,j qα 1 for {h j } j1 an orthonormal basis of g and σ σ (α) S n given by σ σ τ 1, for any τ S n such that τ(dxs 1 1 dxs m m ) db s1 db spα h j1 h j1 h jqα h jqα ds 1 ds qα. j 1,,j qα 1 To define f α, first consider the polynomial of order q α, in the variables s i with i such that α i 1 and in the variable t, given by evaluating the integral (4.11) f α((s i : α i 1), t) ds i, qα (t) i:α i where q α (t) {s i 1 < s i < s i+1 : α i } with s and s m+1 t. Then f α is f α with the variables reindexed by the bijection {i : α i 1} {1,..., p α } that maintains the natural ordering of these sets. (For example, for α (1,, 1, ) J6 4, f α(s 1, s 3, t) ds ds 4 (t s 3 )(s 3 s 1 ), {s 1<s <s 3,s 3<s 4<t} so that f α (s 1, s, t) (t s )(s s 1 ).) This explicit realization of f α is not critical to the sequel. It is really only necessary to know that f α is a polynomial of order q α in s (s 1,..., s pα ) and t, and thus may be written as f α (s, t) q α a b a αt a fα,a (s), for some coefficients b a α R and polynomials f α,a of degree q α a in s. Now, if ˆF n is Hilbert-Schmidt on g pα, then f α,a (s) ˆF n ds f L α,a ˆF n <, ( pα (t)) pα (t)

24 4 TAI MELCHER and (4.1) F σ n (I n t (α)) q α a b a αt a J n ( f α,a ˆF n ) t may be understood in the sense of the limit integrals in Corollary 4.. (In particular, if α m 1, then f α f α (s) does not depend on t, and Corollary 4. implies that Fn σ (It n (α)) is a v-valued L -martingale.) The above computations show that, if for all n σ S n and α J n, then we may rewrite (4.9) as r 1 n c σ q α n g t n m n1 σ S n m n/ α J m n a ˆF n is Hilbert-Schmidt for all b a αt a J n ( f α,a ˆF n ) t, where J n is as defined in Corollary 4.. The next two results show that Hilbert-Schmidt as desired, and thus g t in (4.9) is well-defined. Lemma 4.3. Let n {,..., r}, σ S n, and α J n. For any v v, ˆF n, v is a Hilbert-Schmidt operator on g pα. Proof. First consider the case n. In this case, p α or p α. If p α, then ˆF i1 F σ (h i h i ). If p α, then ˆF (k 1 k ) F σ (k 1 k ) [k σ(1), k σ() )] is Hilbert-Schmidt by Corollary 3.13, and thus ˆF, v is Hilbert- Schmidt. For n 3, p α 1 or p α 3. If p α 3, then α (1, 1, 1) and ˆF 3 (k 1 k k 3 ) F3 σ (k 1 k k 3 ) [[k σ(1), k σ() ], k σ(3) ] is Hilbert-Schmidt, again by Corollary If p α 1, then α (1, ) or α (, 1) and ˆF 3 (k) i1 and we need only consider the case that F σ 3 (k h i h i ), F3 σ (k h h) [[h, k], h]. So let {k i } i1 be an orthonormal basis of g and {e l } N l1 be an orthonormal basis of v. As in the proof of Corollary 3.13, expanding terms in an orthonormal basis of v and applying the Cauchy-Schwarz inequality gives N ˆF 3, v [[h j, k i ], h j ], v [e l, h j ], v e l, [h j, k i ] i1 j1 i1 j1 l1 N N [e l, h j ], v e l, [h j, k i ] i1 l1 j1 N N [e l, h j ], v e l, [h j, k i ] i1 l1 j1 l1 j1 N N [e l, h j ], v N v [, ] [, ]. j1 i,j1 l1 N e l, [h j, k i ] ˆF n is

25 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 5 ˆF Now assume n 1, v is Hilbert-Schmidt for all σ S n 1 and α J n 1, and consider ˆF n, v for some σ S n and α Jn m. Let a p α and b q α, and note that either a 1 and ˆF (4.13) n (k 1 k a ) Fn σ (k 1 k a h j1 h j1 h jb h jb ) j 1,...,j b 1 j 1,...,j b 1 [F σ n 1(k 1 k d 1 k d+1 k a h j1 h jb ), k d ] τ,β [ ˆF n 1 (k 1 k d 1 k d+1 k a ), k d ], for some d {1,..., a}, σ, τ S n 1, and β Jn 1 m 1 q β q α, or b 1 and such that p β p α 1 and ˆF n (k 1 k a ) [Fn 1(k σ 1 k a h j1 h jd 1 h jd h jd+1 h jb ), h jd ] j 1,...,j b 1 (4.14) τ,β [ ˆF n 1 (k 1 k a h jd ), h jd ], j d 1 for some d {1,..., b}, σ, τ S n 1 and β J m n 1 such that p β p α + 1 and q β q α 1. In the first case, working as above for n 3, ˆF n, v N N i 1,...,i a1 Fn σ (k i1 k ia h j1 h jb ), v j 1,...,j b 1 [Fn 1(k σ i1 h jb ), k id ], v N Fn 1(k σ i1 h jb ), e l [e l, k id ], v j 1,...,j b 1 N [e l, k id ], v Fn 1(k σ i1 h jb ), e l i 1,...,i a1 j 1,...,j b 1 i 1,...,i a1 l1 i 1,...,i a1 l1 N N v [, ] τ,β ˆF n 1, e l, l1 j 1,...,j b 1

26 6 TAI MELCHER which is finite by the induction hypothesis. Similarly, in the second case ˆF n, v [Fn 1(k σ i1 h jb ), h jd ], v i 1,...,i a1 j 1,...,j b 1 N N Fn 1(k σ i1 h jb ), e l [e l, h jd ], v i 1,...,i a1 l1 j 1,...,j b 1 N N Fn 1(k σ i1 h jb ), e l i 1,...,i a1 l1 j d 1 j 1,...,j d 1,j d+1,...,j b 1 N [e l, h jd ], v l1 l1 j d 1 N τ,β N ˆF n 1, e l v [, ]. Proposition 4.4. Let n {,..., r}, σ S n, and α J n. Then is Hilbert-Schmidt. ˆF n : g pα v Proof. This proof is analogous to that of Lemma 4.3. For as in equation (4.14), we have ˆF n [Fn 1(k σ i1 h jb ), h jd ] i 1,...,i a1 j 1,...,j b 1 N N Fn 1(k σ i1 h jb ), e l [e l, h jd ] i 1,...,i a1 l1 j 1,...,j b 1 N [e l, h jd ] l1 j l 1 N i 1,...,i a1 l1 j d 1 j 1,...,j d 1,j d+1,...,j b 1 N N [, ] τ,β ˆF n 1, e l, d1 F σ ˆF n n 1(k i1 h jb ), e l which is finite by Corollary 3.13 and Lemma 4.3. In a similar way, one may show in the second case that the same estimate holds for ˆF n as in equation (4.13). Remark 4.5. The proofs of the previous propositions rely strongly on v being finite dimensional. Thus, if we wished to extend the results of this paper to v an infinite dimensional Lie algebra, another proof would be required here, or more likely, some trace class requirements on the Lie bracket of g.

27 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 7 Proposition 4.4 allows us to make the following definition. Definition 4.6. A Brownian motion on G is the continuous G-valued process defined by g t r n n1 σ S n m n/ c σ n n m α J m n pα (t) f α (s, t) ˆF n (db s1 db spα ), where / [ ] c σ n ( 1) e(σ) n n 1, e(σ) ˆF n is as defined in equation (4.1), and f α is as defined below equation (4.11). For t >, let ν t Law(g t ) be the heat kernel measure at time t, a probability measure on G. Example 4.7 (The step 3 case). Suppose that g is nilpotent of step 3. Then g t 3 n1 3 σ S n c σ nf σ n (M n (t)) n n1 σ S n m n/ 3 n n1 σ S n m n/ c σ n n m α J m n c σ n n m α J m n F σ n (I n t (α)) pα (t) f α (s, t) ˆF n (db s1 db spα ). For n 1, there is the single term given by M 1 (t) For n, J {(1, 1), ()}, and so t δb s B t. M (t) It ((1, 1)) + 1 I t (()) db s1 db s + 1 t h i h i ds (t) db s1 db s + 1 t h i h i. (t) i1 (Again, we use slightly heuristic computations to determine the correct form for the Brownian motion, but the integrals in the end are well-defined.) There are of course just two permutations: σ (1) with e(σ) and c σ 1 4, and τ (1) with e(τ) 1 and c τ 1 4, and, by the antisymmetry of the Lie bracket, c σ F σ (M (t)) 1 4 [db s 1, db s ] 1 4 [db s, db s1 ] 1 [db s 1, db s ]. σ S

28 8 TAI MELCHER For n 3, the permutations are (13) with e, (13), (13), (31), (31) with e 1, and (31) with e. Thus, c σ 3 F3 σ (k 1 k k 3 ) 1 9 [[k 1, k ], k 3 ] 1 18 [[k, k 1, ], k 3 ] 1 18 [[k 1, k 3 ], k ] σ S 3 (4.15) Also, J 3 {(1, 1, 1), (1, ), (, 1)}, and so 1 18 [[k 3, k 1 ], k ] 1 18 [[k, k 3 ], k 1 ] [[k 3, k, ], k 1 ] 1 6 [[k 1, k ], k 3 ] [[k 3, k, ], k 1 ]. M 3 (t) It 3 ((1, 1, 1)) + 1 I3 t ((1, )) + 1 I3 t ((, 1)) db s1 db s db s3 + 1 db s1 h i h i ds 3 3(t) (t) i1 + 1 t s 3 h i h i db s3 i1 db s1 db s db s (t) + 1 t s 3 h i h i db s3. i1 t (t s 1 )db s1 h i h i Note that f (1,) (s, t) t s 1 and f (,1) (s, t) s 3. Plugging this into equation (4.15) gives, for the α (1, 1, 1) J3 3 term, c σ 3 F3 σ (It 3 ((1, 1, 1))) c σ 3 F3 σ (db s1 db s db s3 ) σ S 3 σ S 3 3(t) 1 ([[db s1, db s ], db s3 ] + [[db s3, db s ], db s1 ]). 6 For α (1, ) J 3, and 3(t) σ S 3 c σ 3 F σ 3 (I t (1, )) 1 6 ˆF σ,(1,) 3 (k) with σ σ. For α (, 1) J 3, t i1 (t s 1 )[[db s1, h i ], h i ], i1 F3 σ (k h i h i ) i1 σ S 3 c σ 3 F σ 3 (I t ((, 1))) 1 6 t s 3 [[db s3, h i ], h i ], i1 and note that, in this case, ˆF σ,(,1) 3 (k) F3 σ (k h i h i ) i1 F3 σ (h i h i k), i1

29 HEAT KERNEL ANALYSIS ON SEMI-INFINITE LIE GROUPS 9 and so σ σ (31) (or σ σ (31)). Combining the above, Brownian motion on G may be written as g t B t + 1 [db s1, db s ] (t) + 1 ([[db s1, db s ], db s3 ] + [[db s3, db s ], db s1 ]) 1 3(t) + 1 t ((t s)[[db s, h i ], h i ] + s[[db s, h i ], h i ]) 4 i1 B t + 1 t [B s, db s ] + 1 ([[B s1, db s1 ], db s ] + [[db s, db s1 ], B s1 ]) 1 (t) + 1 t[[b t, h i ], h i ]. 4 i1 Remark 4.8. In principle, the Brownian motion on G has generator h i, i1 where {h i } i1 is an orthonormal basis of g H v and h is the unique left invariant vector field on G such that h(e) h, and is well-defined independent of the choice of orthonormal basis. Then the heat kernel measure {ν t } t> has the standard characterization as the unique family of probability measures such that ν t (f) : G f dν t is continuously differentiable in t for all f Cb (G) and satisfies d dt ν t(f) 1 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 gt P be the continuous process on G P defined by r n gt P c σ n n m f α (s, t) ˆF n (dπb s1 dπb spα ), n1 σ S n m n/ α J m n pα (t) for π(w, x) (P w, x). Then gt P is Brownian motion on G P. In particular, let gt l g P l t, for projections {P l } l1 Proj(W ) as in equation (4.4). Then, for all p [1, ) and t <, [ ] (4.16) lim E g l τ g τ p. l g sup τ t Proof. First note that gt P solves the Stratonovich equation δgt P L g P t δp B t with g P e, see [4, 8, 3]. Thus, gt P is a G P -valued Brownian motion. Now, if β t a Brownian motion on W, then, for all p [1, ), [ ] lim E l sup P l β τ β τ p W τ t ;

30 3 TAI MELCHER see, for example, Proposition 4.6 of [1]. Thus, [ lim E l sup π l B τ B τ p g τ t By equation (4.1) and its preceding discussion, r n gt l c σ q α n n m n1 σ S n m n/ α J m n a ]. b a αt a J l n( f α ˆF n ) t, and thus, to verify (4.16), it suffices to show that, for all p [1, ), [ lim E sup Jn( l f α ˆF n ) τ J n ( f ] p α ˆF n ) τ, l τ t for all n {,..., r}, σ S n and α J n. By Proposition 4.4, ˆF n is Hilbert- Schmidt, and recall that f α is a deterministic polynomial function in s. Thus Jn( l f α ˆF n ) and J n ( f α ˆF n ) are v-valued martingales as defined in Corollary 4.. So, by Doob s maximal inequality, it suffices to show that J lim E l n ( f α ˆF n ) t J n ( f p α ˆF n ) t. l Corollary 4. gives the limit for p. For p >, since each Jn( l f α ˆF n ) and J n ( f α ˆF n ) has chaos expansion terminating at degree n, a theorem of Nelson (see Lemma of [3] and pp of []) implies that, for each j N, there exists c j < such that E Jn( l f α ˆF ) t J n ( f j α ˆF ) t c j (E Jn( l f α ˆF n ) t J n ( f ) j α ˆF n ) t. n n v v v v 5. Heat kernel measure We collect here some properties of the heat kernel measure on G. The following two propositions are completely analogous to Corollary 4.9 of [1] and Proposition 4.6 in [1]. 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 the inversion map g g 1 for any g G. Proof. The heat kernel measures νt Pn Law(gt n ) on the finite dimensional groups G Pn 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 Pn -valued random variables {gt n } n1 in Proposition 4.9 converges almost surely to g t. Thus, by dominated convergence, E [ f ( gt 1 )] lim E [ f ( (gt n ) 1)] lim E [f n n (gn t )] E [f (g t )]. Since ν t is the law of g t, this completes the proof. Proposition 5.. For all t >, ν t (G ). Proof. Let µ t denote Wiener measure on W with variance t. Then for a bounded measurable function f on G W v such that f(w, x) f(w), f(w) dν t (w, x) E[f(β t )] f(w) dµ t (w). G W