STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 2

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1 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L MARIA GORDINA Abstract. We prove that a class of stochastic differential equations with multiplicative noise has a unique solution in a noncommutative L space associated with a von Neumann algebra. As examples we consider usual L on a measure space, Hilbert-Schmidt operators and a hyperfinite II 1 -factor. A problem of finding an inverse of the solution is then discussed. Finally, we explain how a stochastic differential equation can be used to construct a heat kernel measure on an infinite dimensional group. Table of Contents 1. Introduction 1. Noncommutative L p spaces 3. Preliminary remarks 4 4. Stochastic differential equations 6 5. Kolmogorov s backward equation 1 References 1 1. Introduction The main goal of this paper is to describe how a certain class of stochastic differential equations can be solved on noncommutative L spaces. Our results concern two different fields of mathematics: von Neumann algebras and stochastic differential equations on infinite dimensional spaces. In Section we give an overview of noncommutative L p spaces. These spaces are completions of von Neumann algebras with respect to a norm induced by a trace. The idea of such spaces can be traced back to I.E.Segal in [1] for L and L 1, and then extended and simplified by E.Nelson in [11]. Later U. Haagerup defined more general noncommutative L p spaces (for example, in [9]). In this paper we will use construction which is closer to Segal s original one. These abstract completions can be realized as (unbounded) operators, and one of these realizations can be used to describe examples of noncommutative L p spaces in Section. In particular, the standard L p on a measurable space can be viewed as a particular case of this construction for an Abelian von Neumann algebra. Another example is the space 1991 Mathematics Subject Classification. Primary 46L5, 46B9, 6H15; Secondary E3, 58G3. Key words and phrases. noncommutative L p, stochastic differential equation. Research was partially supported by the National Science Foundation Postdoctoral Fellowship. 1

2 MARIA GORDINA of Hilbert-Schmidt operators realized as a noncommutative L space with respect to the ordinary trace. The fact that a noncommutative L space is a Hilbert space allows us to use stochastic differential equations in a Hilbert space. These equations are with multiplicative noise, though in the case of the ordinary trace we consider an equation with both multiplicative and additive noises. We refer readers to [] for a comprehensive review of what is known about stochastic differential equations in infinite dimensions. The main difference from the case of a general Hilbert space is that we use the operator composition as a multiplication on noncommutative L p spaces. In general a Hilbert space is not an algebra, therefore the stochastic differential equations we consider are different from ones in []. One of applications of stochastic differential equations on noncommutative L spaces is the construction of a heat kernel measure carried out by the author in [4], [5], [6]. This heat kernel measure lives in some infinite dimensional Lie group, and its Lie algebra is a subspace of a certain noncommutative L. Acknowledgement. The subject of this paper combines two different branches of mathematics: noncommutative integration and infinite dimensional stochastic analysis. Both of these topics appeared in L.Gross work. I thank him for introducing me to the field and encouraging my efforts. I also thank B. Driver for his help throughout the process of preparation of this work and an anonymous referee for useful suggestions.. Noncommutative L p spaces Let H be a separable Hilbert space, and B(H) be the space of bounded linear operators on H. Denote by M a von Neumann algebra on the Hilbert space H, that is, M B(H) is a C -algebra closed in the weak-operator topology and contains the identity operator I. The commutant M of M is the set of those elements of B(H) that commute with all elements of M. Now let us define a trace on M +, the set of all positive elements of M. Definition.1. Let M be a von Neumann algebra. (1) A trace τ on M + is a function τ : M + [, ] such that (a) τ(a + B) = τ(a) + τ(b) for any A, B M +. (b) τ(ca) = cτ(a) for any c and A M + with the usual convention =. (c) τ(uau 1 ) = τ(a) for any unitary operator U in M and A M +. This condition is equivalent to the usual trace property of being central τ(ab) = τ(ba) for any A, B M +. () A trace τ is called faithful if A M +, τ(a) = imply A =. (3) A trace τ is called finite if τ(a) < for any A M +. (4) A trace τ is called normal if for any A M + and any increasing net A α converging to A in the strong operator topology τ(a α ) τ(a). (5) A trace τ is called semifinite if τ(a) = sup{τ(b) : B M +, B A, τ(b) < } for any A M +. Throughout this paper we assume that the von Neumann algebra M is equipped with a faithful normal semifinite trace τ. Let A p = (τ((a A) p/ )) 1/p

3 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 3 for 1 p <. Then we can define a noncommutative L p -spaces in the following way. Definition.. For 1 p < we denote by L p (M, τ) the Banach space completion of the two-sided ideal I = {A M : A 1 < }. The algebra M is equipped with the norm =, the operator norm. Elements of the space L p (M, τ) may be identified with certain (unbounded) operators. In what follows D(A) denotes the domain of an operator A on H. Definition.3. (1) A linear operator A on H is called affiliated with M if UA AU for any unitary U M. We will write A η M if A is affiliated with M. The set of all closed densely defined operators affiliated with M will be denoted by M. () A subspace S of H is called τ-measurable if for any ε there exists a projection P M such that P H S, τ(p ) ε. (3) An operator A M on H is called τ-measurable or strongly measurable if D(A) is τ-dense. The set of all τ-measurable operators will be denoted by M. Note that we can extend the trace to any positive self-adjoint operator A affiliated with M by ( n ) τ(a) = sup τ λde λ, n where A = λde λ is the spectral resolution of A. Then the following statement holds. A proof can be found in [7], [11], [14]. Statement.4. For 1 p < L p (M, τ) = {A M : τ( A p ) < } M. Example.5. Abelian case. Let M = L (X, Ω, µ) for a measure space (X, Ω, µ), where µ is a finite measure. Note that M can be identified with the multiplicative algebra A = {M f, f L (X, µ)} B(L (X, µ)), where M f g = fg, g L (X, µ). Define τ(f) = fdµ. Then τ is a faithful finite normal trace on M, X and L p (M, τ) = L p (X, µ). Example.6. Ordinary trace. Let M = B(H) and τ(a) = tra is the ordinary operator trace. Then L 1 (M, τ) is the space of trace-class operators, and L (M, τ) is the space of Hilbert-Schmidt operators. Note that if H is finite-dimensional, then tr is just the matrix trace. Example.7. A hyperfinite II 1 -factor. We use a representation of a II 1 -factor as the weak closure of a subalgebra of the CAR-algebra. In this description we follow, in particular, L. Gross [8] and I. E. Segal [13]. Let H be a complex Hilbert space with an inner product (, ). Denote by Λ n (H) the space of skew symmetric tensors of rank n over H. Put Λ (H) = C and write Λ(H) = n=λ n (H). The bare vacuum Ω = 1 is the element in Λ (H) Λ(H). For any x in H there exists a bounded operator C x such that C x u = (n + 1) 1 x u, u Λ n (H), where x u denotes P a (x u) and P a is the antisymmetric projection. If we denote A x = C x, then C x A y + A y C x = (x, y)i, which are the canonical anti-commutation relations.

4 4 MARIA GORDINA We fix a conjugation J on H, that is, J is antilinear, antiunitary and idempotent. We will call an element x in H real with respect to J if x = Jx. Let us define def B x = C x + A Jx for any x in H. Note that B x B y + B y B x = x, y I, where x, y = (x, Jy). Let C be the smallest weakly closed algebra of operators on Λ(H) containing all the operators B x, x H. We consider the trace on C given by tr(u) = (uω, Ω). By Corollary 3.4 of [13] the space C with this trace is a hyperfinite II 1 -factor, and there is just one such a factor due to Connes result in [1]. By Theorem 5 of [8] the map u uω extends to a unitary map from L (C) onto Λ(H). In particular if we choose a real (with respect to J) orthonormal basis {x 1, x,..., x n,...} in H, then {B xi1 B xi...b xin, i 1 < i <... < i n, n} is an orthonormal basis in L (C). 3. Preliminary remarks Let Q : L (M, τ) L (M, τ) be a bounded linear symmetric nonnegative operator, and W t be an L (M, τ)-valued Wiener process with Q as its covariance operator. We assume that Q is a trace class operator on L (M, τ). Let Q 1/ L (M, τ) = L Q (M, τ) be a Hilbert space equipped with the norm A L Q (M,τ) = Q 1/ A L (M,τ). In what follows let {ξ n } denote an orthonormal basis of L Q (M, τ) as a real space. We assume that L Q (M, τ) is a subspace of M. We can think of W t in terms of the orthonormal basis {ξ n } in the following way W t = Wt i ξ i, i=1 where Wt i are one-dimensional independent real Wiener processes. Several operators on H defined by the orthonormal basis {ξ n } of LQ (M, τ) play a significant role in the proof of existence and uniqueness of solutions of stochastic differential equations in Section 4. The next statement about these operators follows a similar statement from [6], where it was proved for a hyperfinite II 1 -factor. Lemma 3.1. Let us assume that ξnξ n, ξ n ξn and ξn are bounded operators (and the series are convergent in M), that is, (3.1) Then the operators ξnξ n, the basis {ξ n }. ξnξ n <, ξn <, ξ n ξn <. ξ n ξn and ξn are independent of the choice of Proof. Define a bilinear real form on L (M, τ) L (M, τ) by L(f, g) = τ(h Q 1/ fq 1/ g) = Q 1/ g, (Q 1/ f) h L (M,τ), for some h M, f, g L (M, τ). Then f L(f, g) is a bounded linear functional on L (M, τ) and

5 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 5 L(f, g) = Q 1/ f, h(q 1/ g) L (M,τ) = f, Q 1/ (h(q 1/ g) ) L (M,τ). Let Bg = Q 1/ (h(q 1/ g) ). Then B is a trace class operator from L (M, τ) to L (M, τ) since Q 1/ is a Hilbert-Schmidt operator on L (M, τ), and multiplication by h is a bounded operator on L (M, τ). The trace of B over L (M, τ) is T rb = e n, Be n L (M,τ) = L(e n, e n ) = τ(h ξn), and it does not depend on the choice of {ξ n } for any h M. Use the forms M 1 (f, g) = τ(h (Q 1/ f) Q 1/ g) and M (f, g) = τ(h Q 1/ f(q 1/ g) ) to verify that ξnξ n and ξ n ξn are independent of the choice of the basis. Remark 3.. If there exists an orthonormal basis of L (M, τ) such that its elements are uniformly bounded in the operator norm, and the operator Q is diagonal in this basis, then condition 3.1 is satisfied (see Example 3.1). Example 3.1. Abelian case. Let L (M, τ) = L (, 1) be the Abelian L -space with the Lebesgue measure. Then we can choose an orthonormal basis in such a way that the conditions on the series described above can be satisfied. For example, take an orthonormal (real) basis of L (, 1) to be e n = sin(πnx) for n = 1,,... Let Q be diagonal in this basis, that is, ξ i = λ i e i, and define a function g(x) on [, 1] by g(x) = ξnξ n = ξ n ξn = ξn. Then for any x in [, 1] we have ξnξ n = ξ n ξn = ξ n ξ n = λ n e n λ n and therefore ξn λ n I = trqi B(L (, 1)), if Q is a trace-class operator. In particular, if we choose Q 1 =, where is the Dirichlet Laplacian, then λ n = 1/(πn), and the condition 3.1 is satisfied. Moreover, in this case we can compute the series in 3.1 g(x) = ξn = sin(πnx) π n = 1 x(1 x). This follows from Parseval s identity for the function f(s) defined by f(s) = 1, s x and f(s) =, x s 1. Another way is to look at Green s function G(x, y) for the Dirichlet problem on [, 1] and expand G(x, y) as follows G(x, y) = { x(1 y) x y y(1 x) y x, G(x, y) = G(x, ), sin(nπ ) L (,1) sin(nπy) = sin(πnx) sin(nπy) π n.

6 6 MARIA GORDINA Note that ξ nξ n = G(x, x) is the values on the diagonal of the integral kernel G(x, y) for Q = ( ) 1. Example 3.. Ordinary trace. In the case when L (M, τ) = HS is the space of Hilbert-Schmidt operators (see Section ) we can actually see that the conditions on the series described above can be satisfied. Indeed, elements of this L (M, τ) space can be viewed as infinite matrices whose entries form an l -series. Take an orthonormal basis of HS to be {e i,j } i,j=1, the matrix with the only nonzero entry equal to 1 at the ijth place. We can consider Q to be diagonal in this basis. To simplify calculations, let Qe i,j = λ i λ j e i,j, where λ i > and i=1 λ i <. The last conditions makes Q a trace-class operator. In this setting ξ i,j = λ i λ j e i,j and ξi,jξ i,j = ξ i,j ξi,j = λ i λ j e i,i HS, ξi,j = λ i e i,i HS. i,j=1 i,j=1 i,j=1 and therefore these operators are bounded. i,j=1 4. Stochastic differential equations Let B : L (M, τ) L (M, τ) and F : L (M, τ) L (M, τ). Theorem 4.1. Assume that ξnξ n, is a bounded operator, the series is convergent in M) and denote ξnξ n = K 1 <. Suppose that F and B are Lipschitz continuous on L (M, τ). Then for ξ L (M, τ) the stochastic differential equation (4.1) dx t = F (X t )dt + dw t B(X t ), X = ξ has a unique solution, up to equivalence, among the processes satisfying ( ) T P X s L (M,τ) ds < = 1. Proof of Theorem 4.1. Denote by HS Q = HS(L Q (M, τ), L (M, τ)) the space of the Hilbert-Schmidt operators from L Q (M, τ) to L (M, τ) with the (Hilbert-Schmidt) norm i=1 Ψ HS Q = n Ψξ n L (M,τ), where {ξ n } is an orthonormal basis of L Q (M, τ) as a real space. We will use Theorem 7.4 from the book by DaPrato and Zabczyk []. We will abuse notation by treating B and C as maps from L Q (M, τ) to L (M, τ) by multiplication. Here are the conditions that we need to check to apply the results from [] (1) B(X)( ) is a measurable mapping from L (M, τ) to HS Q. () B(X 1 ) B(X ) HSQ C X 1 X L (M,τ) for any X 1, X HS. (3) B(X) HS Q K(1 + X L (M,τ) ) for any X L (M, τ).

7 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 7 (4) F is Lipschitz continuous on L (M, τ) and F (X) L (M,τ) L(1 + X L (M,τ) ) for any X L (M, τ). By the assumption B and F are Lipschitz continuous on L (M, τ) with some positive constants B and F B(X 1 ) B(X ) L (M,τ) B X 1 X L (M,τ), F (X 1 ) F (X ) L (M,τ) F X 1 X L (M,τ) for any X 1 and X in L (M, τ). First of all, B(X)(U) L (M, τ) for any U L Q (M, τ), since U L Q (M, τ) M and B(X) L (M, τ). Now let us verify that B(X) HS Q. The Hilbert- Schmidt norm of B as an operator from L Q (M, τ) to L (M, τ) can be found as follows B(X) HS Q = ξ n B(X), ξ n B(X) L (M,τ) = τ(b(x) ξnξ n B(X)) ( ) ξnξ n τ(b(x)b(x) ) = K 1 B(X) L (M,τ) <. Then the Lipschitz continuity of B implies that B(X 1 ) B(X ) HSQ K 1 B(X 1 ) B(X ) L (M,τ) K 1 B X 1 X L (M,τ). and B(X) L (M,τ) max{b, B() L (M,τ)}( X L (M,τ) + 1). Thus conditions 1, and 3 are satisfied. Condition 4 can verified similarly to the proof of 3. The following Proposition gives an estimate of the moments of the L -norm of X t. An interesting question whether X t is in L p (M, τ) is still open. Proposition 4.. For any p, t > E X t p L (M,τ) < 1 C p,t (e t C p,t 1), where C p,t = C p M p t p 1 + L p t p 1 p 1, C p = (p(p 1)) p ( p p 1 )p. Proof. 1. First of all, let us estimate E dw sb(x s ) p L (M,τ). From part 3 of the proof of Theorem 4.1 we know that B(X) HS Q M( X L (M,τ) + 1) for a positive constant M. In addition we will use Lemma 7. from the book by DaPrato and Zabczyk [], p.18: for any r 1 and for an arbitrary HS Q -valued predictable process Φ(t), (4.) E( sup s [,t] s dw (u)φ(u) r L (M,τ) ) C re( Φ(s) HS Q ds) r, t [, T ],

8 8 MARIA GORDINA where C r = (r(r 1)) r ( r r 1 )r. Thus (4.3) E dw s B(X s ) p L (M,τ) C p E( B(X s ) HS Q ds) p C p M p E( (( X L (M,τ) + 1)ds) p C p M p p t 1 E ( X L (M,τ) + 1) p ds Now we can use inequality (x + y) q q 1 (x q + y q ) for any x, y for the estimate (4.3) E dw s B(X s ) p L (M,τ) C p M p t p 1 p 1 E. Second, estimate E Finally, (1 + X s p L (M,τ) )ds = C p M p p 1 t p 1 (t + E ( p F (X s )ds p L (M,τ) E F (X s ) L (M,τ)ds) t p 1 E L p t p 1 p 1 E F (X s ) p L (M,τ) ds Lp t p 1 E (1 + X L (M,τ) ) p ds (1 + X s p L (M,τ) )ds Lp t p 1 p 1 (t + E E X t p L (M,τ) E B(X s )dw s p HS C p,t(t + E where C p,t = C p M p t p 1 + L p t p 1 p 1. Thus, E X t p L (M,τ) < 1 C p,t (e t C p,t 1) by Gronwall s lemma. X p HS ds). X p HS ds). X s p HS ds), The next theorem explains how to find a left inverse to solutions of a certain class of stochastic differential equations. The particular coefficients of these equations insure that the solution to the second stochastic equation is a left inverse of the solution of the first equation. Note that the solutions are elements of a certain space of unbounded operators which makes the task of finding inverses very difficult. This inverse is double-sided if L (M, τ) is the space of the Hilbert-Schmidt operators. Theorem 4.3. Assume that ξnξ n, ξ n ξn and ξ n ξ n are bounded operators (and the series are convergent in M) and denote ξnξ n = K 1 <, ξ n ξn = K <. Let X t and Z t be the solutions of the stochastic differential equations dx t = T X t dt + dw t X t, dz t = Z t T dt Z t dw t, X = x, Z = z,

9 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 9 where x and z are in L (M, τ) and T = 1/ ξ n. Then Z t X t = zx with probability 1 for any t >. Proof of Theorem 4.3. First of all, note that the first equation has a unique solution in L (M, τ) by Theorem 4.1, and the second one has a unique solution in L (M, τ) by very similar arguments. To prove the statement of Theorem 4.3 we will apply Itô s formula to G(Z t, X t ), where G is defined as follows: G(Z, X) = λ(zx), where Z, X L (M, τ) and λ is a nonzero linear real bounded functional from L (M, τ) L (M, τ) to R. Here we view G as a function on a Hilbert space L (M, τ) L (M, τ). Then Z t X t = xz if and only if λ(z t X t xz) = for any such λ. In order to use Itô s formula we must verify several properties of the processes Z t and X t and the mapping G: (1) B( X s ) is an HS Q -valued process stochastically integrable on [, T ] () G and the derivatives G t, G Y, G Y Y are uniformly continuous on bounded subsets of [, T ] L (M, τ) L (M, τ), where Y = (Z, X). Proof of 1. See 1 in the proof of Theorem 4.1. Proof of. Let us calculate G t, G Y, G Y Y. First, G t =. For any S = (S 1, S ), V = (V 1, V ) L (M, τ) L (M, τ) and G Y (Y )(S) = λ(s 1 X + ZS ), G Y Y (Y )(S T ) = λ(s 1 V + V 1 S ). Thus condition is satisfied. We will use the following notation G Y (Y )(S) = ḠY Y, S L (M,τ), G Y Y (Y )(S V ) = ḠY Y (Y )S, V L (M,τ), where ḠY is an element of L (M, τ) L (M, τ) and ḠY Y is an operator on L (M, τ) L (M, τ) corresponding to the functionals G Y (L (M, τ) L (M, τ)) and G Y Y ((L (M, τ) L (M, τ)) (L (M, τ) L (M, τ))). Now we can apply Itô s formula to G(Z t, X t ) (4.4) G(Z t, X t ) = 1 ḠY (Z s, X s ), ( Z s dw s, dw s X s ) L (M,τ) + ḠY (Z s, X s ), (Z s T, T X s ) L (M,τ)ds + T r L (M,τ)[ḠY Y (Z s, X s )( Z s Q 1/ ( ), Q 1/ ( )X s )( Z s Q 1/ ( ), Q 1/ ( )X s ) )]ds. Let us calculate the integrands in (4.4) separately. The first integrand is ḠY (Z s, X s ), ( Z s dw s, dw s X s ) L (M,τ) = G Y (Z s, X s )( Z s dw s, dw s X s ) The second integrand is = λ( Z s dw s X s + Z s dw s X s ) =. ḠY (Z s, X s ), (Z s T, T X s ) L (M,τ) = λ(z s T X s + Z s T X s )) = λ(z s T X s ).

10 1 MARIA GORDINA The third integrand is 1 T r L (M,τ)[ḠY Y (Z s, X s )( Z s Q 1/ ( ), Q 1/ ( )X s )( Z s Q 1/ ( ), Q 1/ ( )X s ) )] = 1 ) )) G Y Y (Z s, X s ) (( Z s Q 1/ e n, Q 1/ (e n )X s ( Z s Q 1/ e n, Q 1/ (e n )X s = = 1 λ ( Z s ξ nx s Z s ξnx ) s = λ ( Z s ξnx ) s. This shows that the stochastic differential of G is zero, so G(Z t, X t ) = G(Z, X ) = zx for any t >. Remark 4.4. As we described earlier, if we choose M = B(H) and τ(a) = tra to be the ordinary operator trace, then L 1 (M, τ) is the space of trace-class operators, and L (M, τ) is the space of Hilbert-Schmidt operators. Consider dx t = T (X t + I)dt + dw t (X t + I), X =. Note that this equation has the additive noise as well as the multiplicative one. It is possible to solve this equation since L (M, τ) M = B(H) in this case. Then X t is in GL(H), the invertible elements of B(H), with probability 1 for any t > as was shown in [5]. This means that in this case X t has a double-sided inverse. The shift by the identity operator is necessary to ensure that X t lives in GL(H) since I is not a Hilbert-Schmidt operator. Remark 4.5. This lemma has been proved in the case of a hyperfinite II 1 -factor in[6]. Remark 4.6. In [5] and [6] the stochastic differential equations had no drift terms. The reason for that is that in those cases Q was complex linear, and therefore T = 1/ ξ n =. Indeed, Lemma 3.1 shows that ξn is independent of the choice of the basis {ξ n }, and this basis can be chosen so that the sum is. Namely, we can choose such a basis that ξ k = iξ k 1, where i = Kolmogorov s backward equation First of all, the coefficient B depends only on X L (M, τ), therefore the transition probability satisfies P s,t f(x) = Ef(X(t, s; X)) = P t s f(x). Later in this section we give Kolomogorov s backward equation (Equation 5.) for a stochastic differential equation of the form 4.1. At the moment though let us look at Kolomogorov s equation in the particular case of B(X) = X and F (X) = 1/ ξ nx. Then Equation 5. is the heat equation with the Laplacian which is a half of the sum of second exponential derivatives in the directions of an orthonormal basis of LQ (M, τ). This Lapalcian is an infinite dimensional analogue of the Laplacian on a finite-dimensional Lie group. In the case of the Hilbert-Schmidt operators, this is a Laplacian on an infinite-dimensional (Lie) group, but for a hyperfinite II 1 -factor this is not so. Let v : L (M, τ) R be a function and n v(x) = ( ξ n v)(x) = d dt t= v(e tξn X). Here ξ n is the right-invariant vector field corresponding to ξ n. Then the Laplacian

11 STOCHASTIC DIFFERENTIAL EQUATIONS ON NONCOMMUTATIVE L 11 is (5.1) v = 1 nv = 1 ξ n ξn v, Let us calculate derivatives n of v : L (M, τ) R and therefore Thus the Laplacian is ( v)(x) = 1 ( n v)(x) = v X (X) d dt t= (e tξn X) = v X (X)(ξ n X) ( ξ nξn v)(x) = v XX (X)(ξ n X ξ n X) + v X (ξnx). [v XX (X)(ξ n X ξ n X) + v X (ξnx)] = 1 T r[v XX(t, X)(Q 1/ ( )X)(Q 1/ ( )X) ] + 1 (ξnx, v X (t, X)) L (M,τ). Therefore the Lie group Laplacian is the same differential operator that appears in Kolmogorov s backward equation (Equation 5.), and so this equation can be viewed as a heat equation. This justifies the following definition. Definition 5.1. We define a Borel measure µ t by f(x)µ t (dx) = Ef(X t (I)) = P t, f(i) L (M,τ) for any bounded Borel function f on L (M, τ). Then µ t is called the heat kernel measure on L (M, τ). Now let us look at the general case. According to Theorem 9.16 from [] for any ϕ Cb (L (M, τ)) and X L (M, τ) the function v(t, X) = P t ϕ(x) is a unique strict solution from C 1, b (L (M, τ)) for the parabolic type equation (Kolmogorov s backward equation) (5.) t v(t, X) = 1 T r[v XX(t, X)(Q 1/ ( )B(X))(Q 1/ ( )B(X)) ] + v(, X) = ϕ(x), t >, X L (M, τ). (F (X), v X (t, X)) L (M,τ) Here Cb n(l (M, τ)) denotes the space of all functions from L (M, τ) to R that are n- times continuously Frechet differentiable with all derivatives up to order n bounded and C k,n b (L (M, τ)) denotes the space of all functions from [, T ] L (M, τ) to R that are k-times continuously Frechet differentiable with respect to t and n-times continuously Frechet differentiable with respect to X with all partial derivatives continuous in [, T ] L (M, τ) and bounded.

12 1 MARIA GORDINA Let us rewrite Equation (5.) in the following way T r[v XX (t, X)(Q 1/ ( )B(X))(Q 1/ ( )B(X)) ] = v XX (t, X)(Q 1/ e n B(X) Q 1/ e n B(X)) = v XX (t, X)(ξ n B(X) ξ n B(X)), Thus Kol- where v XX (t, X) is viewed as a functional on L (M, τ) L (M, τ). mogorov s backward equation is t v(t, X) = 1 v XX (t, X)(ξ n B(X) ξ n B(X)) + (F (X), v X (t, X)) L (M,τ) v(, X) = ϕ(x), t >, X L (M, τ), which is a heat equation with a first order term. References [1] A. Connes, Classification of injective factors. Cases II 1, II, III λ, λ 1, Ann. of Math. () 14, 1976, 1, [] G.DaPrato and J.Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 199. [3] J. Dixmier, von Neumann algebras, North-Holland Mathematical Library, 7, [4] M. Gordina, Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group, Potential Analysis, 1,, [5] M. Gordina, Heat kernel analysis and Cameron-Martin subgroup for infinite dimensional groups, J. Funct. Anal., 171,, [6] M. Gordina, Taylor map on groups associated with a II 1 -factor, to appear in Infin. Dimens. Anal. Quantum Probab. Relat. Top.,. [7] M. Gordina, Noncommutative integration in examples, notes,. [8] L. Gross, Existence and uniqueness of physical ground states, J. Funct. Anal., 1, 197, [9] U. Haagerup, L p -spaces associated with an arbitrary von Neumann algebra, Algbres d oprateurs et leurs applications en physique mathmatique (Proc. Colloq., Marseille, 1977), pp , Colloq. Internat. CNRS, 74, CNRS, Paris, [1] R. Kadison, J. Ringrose, Fundamentals of the theory of operator algebras, Vol.I-IV. Graduate Studies in Mathematics, 16, American Mathematical Society, Providence, RI, [11] E. Nelson, Notes on non-commutative integration, Journal of Functional Analysis, 1974, 15, [1] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math., 1953, 57, [13] I. E. Segal, Tensor algebras over Hilbert spaces. II, Ann. of Math., 63, 1956, [14] M. Terp, L p spaces associated with von Neumann algebras, manuscript. Department of Mathematics, University of California at San Diego, 95 Gilman Drive, La Jolla, CA , U.S.A. address: mgordina@math.ucsd.edu