On the absolute continuity of Gaussian measures on locally compact groups

Transcription

1 On the absolute continuity of Gaussian measures on locally compact groups A. Bendikov Department of mathematics Cornell University L. Saloff-Coste Department of mathematics Cornell University August 6, 2001 Abstract This note discusses the topological necessary and sufficient conditions for a locally compact connected group to admit a Gaussian measure that is absolutely continuous with respect to Haar measure. 1 Introduction The aim of this note is to discuss, extend and complement a theorem due to H. Heyer and E. Siebert concerning the existence of Gaussian measures that are absolutely continuous with respect to Haar measure. Let G be a locally compact group equipped with a fixed (left-invariant) Haar measure m. Definition 1.1 A weakly continuous convolution semigroup (µ t ) t>0 of probability measures on G is called Gaussian if 1 lim t 0 t µ t(g \ V ) = 0 for every open neighborhood V of the neutral element e G. Definition 1.2 A probability measure µ on G is called Gaussian if there exists a Gaussian convolution semigroup (µ t ) t>0 such that µ = µ 1. This definition of Gaussian measures is due to Courrège [6]. On locally compact Abelian groups, it coincides with other classical definitions based on duality theory. See also [10, 14]. Research partially supported by NSF grant DMS

2 Let G 0 denote the connected component of the identity in G. Let (µ t ) t>0 be a Gaussian convolution semigroup on G. Then, for each t, µ t is Gaussian and supported on G 0. See [10, 6.2.3, 6.2.4]. Thus, without loss of generality, we will assume in the sequel that G is connected. Our aim is to discuss the following basic theorem in the theory of Gaussian measures on locally compact connected groups. Theorem 1.3 For any locally compact connected group G the following two properties are equivalent: 1. G carries a Gaussian measure which is absolutely continuous w.r.t. Haar measure. 2. G is locally connected and has a countable basis for its topology. Moreover, if G is not locally connected, then any Gaussian measure is singular w.r.t. Haar measure. More precisely, there exists a proper dense Borel measurable subgroup M of G such that any Gaussian measure µ on G satisfies µ(m) = 1. Remark 1.4 Implicit in the last statement of this theorem is the well-known fact that in a connected locally compact group any Borel measurable subgroup is either equal to G or has Haar measure zero. In the case of compact groups (more generally, groups of the type R n K with K compact) this theorem was proved by H. Heyer and E. Siebert, see [10, ,6.412]. The last statement for Abelian groups having a countable basis for their topology was proved independently by G. Feldman [7]. Theorem 1.3 leads to the following open question. Question 1.5 Suppose that G is a locally compact (or even compact) connected group that does not admit a countable basis for its topology. Is any Gaussain measure on G singular w.r.t Haar measure? We note the following simple examples for which the answer is positive. Example 1.6 Let G = T ω where T is the circle R/Z and ω is uncountable. Let µ be a Gaussian product measure on G, µ = α ω µ α where each µ α is a Gaussian measure on T α = T. As the normalized Haar measure on G is the product of the normalized Haar measures on the T α s, a simple application of Kakutani s dichotomy theorem implies that µ is singular w.r.t Haar measure. Note that Question 1.5 is open for general Gaussian measure on T ω. Example 1.7 Let G be a compact connected semisimple group (i.e., G = G, see [11]). Then there exists a family of simple connected simply connected compact Lie groups {G α, α ω} and a closed central subgroup H of Σ = α G α such that G = Σ/H. As the center of G α is 2

3 a finite group Z α, H α Z α = Z. Let µ be a Gaussian measure on G. Assume that µ is central, that is, µ(gug 1 ) = µ(u) for any open set U and g G. Let G α = G α /Z α. G α is a compact connected simple Lie group with trivial center and set Σ = α G α = Σ/Z. Obviously, G is a quotient of G, in fact G = G/(Z/H), and µ projects to a central Gaussian measure µ on G. Moreover, because µ is central and G = α G α, it follows that µ is a product measure µ = α µ α where each µ α is a central Gaussian measure on the Lie group G α. See, e.g., [2]. Now, if G does not have a countable basis for its topology, ω must be uncountable. Then Kakutani s dichotomy theorem shows that µ, hence µ, must be singular with respect to Haar measure. That is, any central Gaussian measure on any connected compact semisimple group without countable basis for its topology must be singular w.r.t Haar measure. 2 Existence of absolutely continuous Gaussian measures The goal of this section is to give a proof of the part 2 1 in Theorem 1.3. That is, a proof of the existence of an absolutely continuous Gaussian measure on any connected, locally connected, locally compact group G having a countable basis for its topology. For compact groups this was proved by Siebert [13]. See also [10]. To treat the non-compact case we use a recent result of Berestovskii and Plaut [4, Proposition 2]. The Gleason-Iwasawa- Yamabe theory of locally compact groups (see [8, Theorem A]) asserts that a connected locally compact group splits locally as a product of a local Lie group and a compact group. We learned the following result from Berestovskii and Plaut [4]. See also [12]. Theorem 2.1 Given a locally compact connected group G, there exist a connected simply connected Lie group L, a compact group K, and a surjective homorphism π : L K G which is a local isomorphism. Moreover, if G is locally connected, then K is connected and locally connected and π is a covering map. Remark 2.2 Note that the kernel of π above is a discrete subgroup of L K. Proof of Theorem 1.3, 2 1 Let G be a connected locally connected, locally compact group having a countable basis for its topology. Let π : L K G be the covering given by Theorem 2.1. On the Lie group L, an absolutely continuous Gaussian semigroup (µ L t ) t>0 having a smooth positive density is easily obtained by considering the heat diffusion semigroup H t = e t associated with some fixed left-invariant Laplace-Beltrami operator on L. As H t commutes with left translations, we can write H t f = f µ t and set µ L = µ L 1. This is a Gaussian measure on L which is absolutely continuous and has a smooth positive density w.r.t. Haar measure. See, e.g., [14, 16]. As K is compact connected and locally connected, a result of 3

4 Siebert [13] shows that K carries a Gaussian measure µ K which is absolutely continuous and has a continuous density w.r.t. Haar measure. Thus µ = µ H µ K is a Gaussian measure on L K which is absolutely continuous w.r.t. Haar measure and so is its projection µ = π(µ) on G since π is a covering. See Lemma 3.3 below. Remark 2.3 In fact, it is possible to choose µ K so that µ is absolutely continuous and has a continuous density. See [1, Theorem 5.3]. The references [1, 2, 3] contain further results in this direction. 3 Proof of Theorem 1.3, 1 2 We will need the following lifting result. Theorem 3.1 Let G be a locally compact group. Let H be a closed totally disconnected normal subgroup of G and let π : G G/H be the canonical projection. Any Gaussian measure ν on G/H can be lifted uniquely to a Gaussian measure µ on G such that ν = π(µ). In the case where H is discrete, this is proved in [10, ]. For the general case, we refer to [2, 3]. It is possible that G/H is connected even if G is not connected. In any case, µ and ν are supported by the connected component of G/H and G respectively. Example 3.2 Let G be a connected locally compact group and let π : L K G be as in Theorem 2.1. By Remark 2.2, the kernel of π is discrete. Hence, any Gaussian measure µ on G lifts uniquely to a Gaussian measure µ on G = L K such that π(µ) = µ. We will need the following lemma which refer to this setting. Lemma 3.3 The measure µ is absolutely continuous w.r.t. Haar measure if and only if µ is absolutely continuous w.r.t. Haar measure. The lemma holds whenever π : G G has discrete kernel. The result may fail if the kernel is totally disconnected but not discrete. Proof Denote by m, m the Haar measures on G and G respectively. First we note that it is of course not true that π(m) = m (unless the kernel is finite). However, by Fubini theorem, m(a) = 0 implies m(π 1 (A)) = 0. Moreover, for any set B G, m(b) = 0 implies m(π(b)) = 0 (this last assertion use the discreteness of the kernel). Indeed, it sufficesses to show that m(π(d)) = 0 for any compact set D contained in B. We can cover D by a finite union of translates of any fixed neighborhood of the neutral element in G. By hypothesis, we can pick a neighborhood U such that π U : U π(u) = U is an isomorphism. Moreover, for any Borel subset V of U, m(v ) = m(π(v )). This proves the claim. To finish the proof of the lemma, note that for any Borel subset B of G, µ(b) µ(π(b)) and for any Borel subset A of G, µ(a) = µ(π 1 (A)). 4

5 3.1 Existence of a countable basis We want to show that the existence of an absolutely continuous Gaussian measure µ on a connected locally compact group G implies that G has a countable basis for its topology. When G is connected and compact, this is a consequence of the special form of the Fourier transform of Gaussian measures and the Riemann-Lebesgue Lemma for compact groups, See [10, ]. We give a proof of the general case assuming that we know that the existence of a Gaussian measure that is absolutely continuous w.r.t Haar measure implies that G is locally connected, a fact that will be proved in next section. Accordingly, it suffices to consider the case where G is locally connected. Thus let G be a locally compact connected locally connected group. Let π : L K G be the covering given by Theorem 2.1. In the present case, K is compact, connected and locally connected. Let µ be the (unique) Gaussian measure on L K such that π(µ) = µ. By Lemma 3.3 and the hypothesis that µ is absolutely continuous, µ is absolutely continuous. By Fubini Theorem, it follows that its projection µ K on K is absolutely continuous. Since K is connected and compact and µ K is a Gaussian measure, this implies that K must have a countable basis for its topology. Hence so do L K and its quotient G = π(l K). 3.2 Local connectedness The part of 1 implies 2 relative to local connectedness and the last statement in Theorem 1.3 are contained in the following result. Theorem 3.4 Let G be a locally compact, connected and not locally connected group. Then any Gaussian measure on G is singular w.r.t. Haar measure. More precisely, there exists a dense proper Borel measurable subgroup M of G such that µ(m) = 1 for any Gaussian measure µ on G. This Theorem is due to Heyer [10, ]. For the sake of completness we comment on the proof. Definition 3.5 A connected locally compact group G is said to be finite dimensional if there exists a totally disconnected compact normal subgroup H G such that G/H is a Lie group. The following non-trivial example illustrate this definition. Example 3.6 Let p be a prime. The p-adic solenoid S p can be defined as follows. Let (T k ) 0 be a sequence of circle groups, T k = {z C : z = 1}. For integers l k, let πk,l : T k T l be given by π k,l (z) = z pk l. The sequence (T k, π k,l ) l k is a projective sequence of compact groups and the solenoid S p is the projective limit of this projective sequence. That is, S p is the 5

6 subgroup of 0 T k of those sequences z = (z k ) 0 satisfying z l = z pk l k is compact and connected, S p is a compact connected group. Let for all l k. As each T k S j p S p 0 T k be the compact subgroup of all sequences (z k ) 0 such that z l = 1 for all l j. The subgroup Sp 0 is the projective limit of the finite cylic groups Sp/S 0 p k. It follows that Sp 0 is totally disconnected. Moreover, one can show that S p /Sp 0 = T. Hence S p is a finite dimensional group (in fact, of dimension 1). Theorem 3.7 ([15, 2.2,2.4]) For each locally compact connected finite dimensional group G there exists a Lie group L, a totally disconnected compact Abelian group H and a surjective homorphism and local isomorphism π : L H G. The restriction of π to L is a continuous homomorphism from L onto the arc-connected component G arc of the neutral element e G. The group G arc is a dense Borel measurable subgroup of G. In addition, G is arcwise connected (locally connected) if and only if G is a Lie group. Example 3.8 Let S p be the p-adic solenoid. It is well-known that S p = R p N where p is the group of the p-adic integers and N = {(n, n) : n N} is the group of the natural integers embedded diagonally in R p. The group p is a compact Abelian totally disconnected group. The map π : R p S p is given by π : (φ, a) (z k ) 0 where φ R, a = 0 a lp l p and z k = exp ( ( 2πi φ + p k )) k a k p l. 0 Thus, π(r {0}) = {(z k ) S p : z k = exp(2πiφ/p k ), φ R} is the arc-connected component of the neutral element in S p. It is a dense proper Borel measurable subgroup of S p. Theorem 3.9 ([15, 2.5]) Let G be a connected locally compact group. Then G is locally connected if and only if every finite dimensional factor group is locally connected. 6

7 We are now ready to prove Theorem 3.4. Proof of Theorem 3.4 Assume first that G is finite dimensional. Let π : L H G be as in Theorem 3.7. Let µ be a Gaussian measure on G. Let µ be the unique Gaussian measure on L H such that π(µ) = µ. Since H is totally disconnected, µ is supported in the connected component L {e H } of L H. As L {e H } is σ-compact, M = π(l {e H }) is Borel measurable subgroup of G. By Theorem 3.7, M is the arc-component of G and is a dense proper subgroup of G. Moreover µ(m) = µ(l {e H }) = 1. This proves Theorem 3.4 in the finite dimensional case. In general, if G is not locally connected, then by Theorem 3.9, there exists a compact normal subgroup K of G such that G/K is finite dimensional, not locally connected. Let M 1 G/K be a proper dense Borel measurable subgroup of G/K such that any Gaussian measure ν on G/K satisfy ν(m 1 ) = 1. If ψ : G G/K is the canonical projection then M = ψ 1 (M 1 ) is a subgroup of G having the desired property. Remark 3.10 Given a locally compact connected not locally connected group G, Theorem 3.4 asserts the existence of a proper dense Borel measurable subgroup M such that any Gaussian measure µ on G satisfies µ(m) = 1. In the case of finite dimensional group the proof shows that M can be taken to be the arc-connected component of the neutral element. The following question arises. Question 3.10 Is it true or not that any Gaussian measure µ on a connected locally compact group G satisfy µ(m) = 1 for some arcwise connected Borel measurable subgroup M? In next section we provide a positive answer when G has a countable basis for its topology. 4 Groups having a countable basis Recall that for a locally compact group, to have a countable basis for its topology is equivalent to be metrizable. Recall also that a locally compact connected group having a countable basis is arcwise connected if and only if it is locally connected. See [11, Th 9.68] for the compact case. The extension to the non-compact case follows from Iwasawa local splitting theorem. The aim of this section is to prove the following result. Theorem 4.1 Let G be a connected, locally compact group. Assume that G has a countable basis for its topology. Then 1. The arc component G arc of G is a Borel set and, for any Gaussian measure µ, µ(g arc ) = For any Gaussian measure µ on G, there exists an arcwise connected subgroup M of G which is a countable union of compact sets in G and such that µ(m) = 1. 7

8 Remark 4.2 Note that we are not claiming that M is independent of µ. In general, it is not clear that G arc is a countable union of compact sets. Proof of Theorem 4.1(1) The fact that G arc is a Borel set is not an obvious fact. For metrizable groups, it is proved by K. Hofmann in [12] who also discusses the logically intricate case of non metrizable groups. Once we know that G arc is a Borel set, it is obvious from a probabilistic point of view that µ(g arc ) = 1. This is because µ is the law of X 1 where (X t ) t>0 is a stochastic process having continuous paths almost surely. See [10, Chapter 5]. Of course, if one starts from the Gaussian measure µ, embeded in a convolution semigroup (µ t ) t>0, constructing the process (X t ) t>0 and showing that it has continuous paths requires some work. Note also that this seems to require being in the metric case (i.e., having a countable basis for the topology). The argument given below for the second statement of Theorem 4.1 gives an alternative proof. Proof of Theorem 4.1(2) The proof of the second statement of Theorem 4.1 is several steps. Step 1 Fix a connected locally compact group G. Theorem 2.1 yields a connected Lie group L and a compact group K such that G is the image of a continuous homorphism π : L K G which is also a local isomorphism. Let K 0 be the connected component of the neutral element in K. Let π 0 be the restriction of π to L K 0. Claim 4.3 For any Gaussian measure µ on G there is a unique Gaussian measure µ 0 on L K 0 such that π(µ 0 ) = µ. This claim follows from Theorem 3.1 and the fact that any Gaussian measure is supported by the the component of the neutral element. Step 2 Consider the connected component Z 0 of the center of K 0 and the commutator group K 0. By the Lévy-Malcev decomposition of connected compact groups (see [11, Theorem 9.24]), we have K 0 = (Z0 K 0)/H where H = Z 0 K 0 is totally disconnected. Consider the natural projection π 1 from G 1 = L Z 0 K 0 onto L K 0. Then the kernel of π 1 is isomorphic to H. Thus, for any Gaussian measure µ 0 on L K 0, there exists a unique Gaussian measure µ 1 on G 1 such that π 1 (µ 1 ) = µ 0. Step 3 The compact connected Abelian group Z 0 is the projective limit of torii T α, α Ω. Denote by R α the Lie algebra of T α. By construction, the family L T α K 0 is projective with limit L Z 0 K 0. Similarly, the family L R α K 0 is projective with limit L R 0 K 0 where R 0 is the projective limit of the R α s. R 0 is a topological vector space which is a closed subspace of α Ω R α. Observe that, if the group G has a countable basis, then the set Ω is finite or countable. Let µ 1 be a Gaussian measure on L Z 0 K 0. For each α, µ 1 yields by projection a Gaussian measure µ α 1 on L T α K 0. Obviously, the projective limit of (µ α 1 ) α is µ 1. By Theorem 3.1, 8

9 each µ α 1 yields a Gaussian measure µ α 2 on L R α K 0. This yields a projective system of probability measures. By Kolmogorov s theorem, there exists a unique probability measure µ 2 on L R 0 K 0 whose projection on L R α K 0 is µ α 2 for each α. Although L R 0 K 0 might not be locally compact, µ 2 is a Gaussian measure in the following sense: Its projection to any Lie quotient of L R 0 K 0 is Gaussian. Step 4 Let µ 2 and L R 0 K 0 be as in step 3 and assume that G has a countable basis. Then R 0 is a closed subspace of R. In fact, it can be shown that R 0 is isomorphic to either R n or R (this easily follows from [5]). Claim 4.4 There exists a Hilbert subspace H of R 0 such that µ 2 is supported by L H K 0. Consider the projection of µ 2 onto R 0. This is a probability measure ν with the property that any finite dimensional projection is Gaussian. The claim above will follow if we can show that ν is supported by some Hilbert subspace of R 0. Without loss of generality, we can assume that ν is symmetric. The case R 0 = R n is trivial, so assume that R 0 = R. Viewing ν as a measure on R, consider its covariance matrix A = (a i,j ). Let (b i ) be a sequence such that i b ia i,i < and set (Bx, x) = i b ix 2 i. Then the well-known formula (Bx, x)dν A (x) = 1 b i a i,i = Tr(BA) < shows that ν = ν A is supported by the Hilbert space i H = {x R : (Bx, x) < }. Step 5 This is the last step. Let G be a connected, locally compact group having a coutable basis. Let µ be a Gaussian measure on G. By steps 1,2,3,4, there exists a connected Lie group L, a separable Hilbert space H, a compact connected semisimple group K 0, a probability measure µ 2 on L H K 0 and a continuous homomorphism such that ψ : L H K 0 G µ = ψ(µ 2 ). Since H is σ-compact in the weak topology, M = ψ(l H K 0) is an arc-connected Borel measurable subgroup of G with µ(m) = 1. Remark 4.5 Hofmann [12] proves that G arc is a Borel set under the asumption that Z 0 /H is metrizable. The proof of part 2 of Theorem 4.1 also works exactly under this hypothesis. Indeed, although Z 0 /H can be metrizable and Z 0 not metrizable, they have the same Lie algebra because H is totally disconnected. It follows that if Z 0 /H is metric, R 0 in steps 4 is again either R n or R. 9

10 Theorem 4.6 Let G be a connected, locally compact group. Referring to the Levy-Malcev decomposition of G, assume that Z 0 /(Z 0 G ) = G/G has a countable basis for its topology. Then the two conclusions of Theorem 4.1 still hold true. References [1] Bendikov A. and Saloff-Coste L. Potential theory on infinite products and groups. Potential Analysis, 11, , [2] Bendikov A. and Saloff-Coste L. Central Gaussian semigroups of measures with continuous density, To appear in J. Funct. Anal. [3] Bendikov A. and Saloff-Coste L. Gaussian semigroups of measures on locally compact locally connected groups, Preprint [4] Berestovskii V.W. and Plaut C. Homogeneous spaces of curvature bounded below. Preprint Univ. Tenesse, Knoxville, 1995, to appear in Journal of Geometric Analysis. [5] Born E. Projective Lie Algebra Bases of a Locally Compact Group and Uniform Differentiability. Math. Zeit. 200, 1989, [6] Courrège Ph. Générateur infinitésimal d un semigroupe de convolution sur R n et formule de Lévy-Khinchine. Bull. Sci. Math. 2ème série, 88, 1964, [7] Feldman G. On Gaussian distributions on locally compact Abelian groups. Theory of Probability and its Application, 3, 1978, [8] Glushkov V.N. The structure of locally compact groups and Hilbert s Fifth Problem. AMS Translations 15, 1960, [9] Grenander U. Probability on Algebraic Structures. Amquist & Wiksell, 1963, Stockholm- Göteborg-Upsalla. [10] Heyer H. Probability Measures on Locally Compact Groups. Ergeb. der Math. und ihren Grenzgeb. 94, Springer, Berlin-Heildelberg-New York, [11] Hofmann K. and Morris S. The structure of compact groups. W. de Gruyter, [12] Hofmann K. Arc components in locally compact groups are Borel sets. Preprint, [13] Siebert E. Absolut-Stetigkeit und Träger von Gauss-Verteilungen auf lokalkompakten Gruppen. Math. Ann. 210, 1977, [14] Siebert E. Absolute continuity, singularity and supports of Gaussian semigroups on a Lie group. Monatshefte für Math. 93, 1982,

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